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User Manual - Numerical Algorithms Group

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1. GO8AFF GO8AHF GO1BJF GO1ECF GO8ACF GO7EAF GO7AAF GO1ALF GO1AEF GO8AAF GO2BNF GO8AGF GO8AGF NAG Subroutine Add In Correspondence Available Add In functions from Chapter G01 GO1AAF GO1AEF GO1ALF GO1ALF G01BJF G01BJF SUMMARY_STATS RUNS_TEST RUNS_TEST S5PT_SUMMARY BINOMIAL_TEST MCNEMAR 102 Statistical Add Ins for Excel User Guide GO1EAF BINOMIAL_TEST GO1EAF NORMAL_PROB GO1EBF T_PROB GO1ECF CHI_PROB GO1ECF CHI_SQ_1 GO1ECF MCNEMAR GO1EDF F_ PROB GO1EEF BETA _PROB GO1EFF GAMMA_PROB GO1FAF NORMAL_PPT GO1FBF T_PPT GO1FCF CHI_PPT GO1FDF F_PPT GO1FEF BETA_PPT GO1FFF GAMMA_PPT GO1HBF MULTI_NORMAL_PROB Available Add In functions from Chapter G02 GO2BNF KENDALL_TAU GO2BNF SPEARMAN_RHO G02BXF CORREL_MAT G02BYF PARTIAL_CORREL_MAT G02DAF MULT_LIN_REG GO2FAF STAND_RESIDS G02FCF DURBIN_WATSON G02GAF NORMAL_GLM G02GBF BINOMIAL_GLM G02GCF POISSON_GLM Available Add In functions from Chapter G03 GO3AAF PRIN_COMP GO3BAF ORTHOG_ROTATION GO3CAF FACTOR GO3CCF FACTOR_SCORE GO3DAF DISCRIM_TEST GO3DBF DISCRIM_DIST GO3DCF DISCRIM_ALLOC GO3EAF DIST_MAT GO3ECF CLUSTER GO3EJF GROUPS_FROM_CLUSTER GO3FAF METRIC_SCAL GO3FCF MDS Available Add In functions from Chapter G04 GO4BBF ANOVA_BLOCK G04BCF ANOVA_ROW_COLUMN G04CAF ANOVA_FACTORIAL G04DAF ANOVA_CONTRASTS G04DBF MUL
2. 5 3 3 2 5 3 3 3 Statistical Add Ins for Excel User Guide The spectrum is then calculated for a subset of the values computed by the FTT as given by the frequency division A divisor of 2r is used when computing the sample spectrum from the FFT TIME_SERIES_DIFF Set 1 TIME SERIES DIFF NAG G13AAF carries out non seasonal and seasonal differencing on a time series Parameters Series Required The time series Note At least one of Differencing or Seasonal_diff must be present Differencing Optional The order of non seasonal differencing If present must be gt 0 Seasonal_diff Optional The order of seasonal differencing If present must be gt 0 Seasonality Optional The seasonality for the seasonal differencing Must be present if Seasonal_diff is present If present must be gt 1 Output Output as a NAGExtract format list Differenced Series Remarks See Using the Add Ins and the Introduction to Time Series Methods for further information ACF Set 1 ACF NAG G13ABF computes the sample autocorrelation function of a time series along with a test for autocorrelation Parameters Series Required The time series No_of_autocorrels Optional The number of lags for which the autocorrelations are required Default min 20 number of observations 1 Output Output as a NAGExtract format list Mean Variance Chi squared Statistic Chi squared Significance ACF 34 5 3
3. YES O Tabulation a Nonparametric tests a Distribution tests a Survival analysis a Further regression facilities U Handling larger data sets C O O O O Z o What other enhancements or additions would be of interest to you Please specify 106 Contacting NAG NAG Ltd Wilkinson House Jordan Hill Road OXFORD OX2 8DR United Kingdom Tel 44 0 1865 511245 Fax 44 0 1865 310139 Web www nag co uk NAG Inc 1400 Opus Place Suite 200 DOWNERS GROVE IL 60515 5702 USA Tel 1 630 971 2337 Fax 1 630 971 2706 Web www nag com NAG GmbH Schleissheimer Strasse 5 85748 GARCHING Deutschland Tel 49 0 89 3207395 Fax 49 0 89 3207396 Email info naggmbh de Web www naggmbh de Nihon NAG KK Yaesu Nagaoka Building No 6 1 9 8 Minato Chuo ku TOKYO Japan Tel 81 0 3 5542 6311 Fax 81 0 3 5542 6312 Email sales nag j co jp Web www nag j co jp 107 Statistical Add Ins for Excel User Guide NAG Ltd Response Centre Email infodesk nag co uk Tel 44 0 1865 311744 Fax 44 0 1865 311755 NAG Inc Response Center Email infodesk nag com Tel 1 630 971 2345 Fax 1 630 971 2346 10 Statistical Add Ins for Excel User Guide References Aitchinson J and Dunsmore R 1975 Statistical Prediction Analysis Cambridge Andrews D F and Herzberg A M 1985 Data A Collection of Problems from Many Fields for the Student and Research Worker Springer Verlag Atkinson A C 1981 Two gra
4. 3 2213 GAMMA PROB Set lesa 19 5 2214 GAMMA PRES aii 19 3 2 3 Summary SUGUISTICS atrial KTE A E E detras 20 5 2 3 1 Introduction rta do a e A 20 5 2 3 2 SUMMARY STATS Oei dass 20 5 2 3 3 SIPE SUMMARY Set Dicc 21 5 2 3 4 TWO WAY TABLE Set 1 r aa aa rA s a EE E Eo rie eni is 21 Statistical Add Ins for Excel User Guide 5 3 BOOK esate a Rot ale Aa e he E fede ees Mat ce Ble 22 5 3 1 INTO UC OM a TEAT e dE Le a RI a AS nett Miter tected A NA 22 5 3 2 Regression and Correlation cccccccccccccccccssccsseescesseseeesseesecseeeecnseesecseeeecnseeecneeescesecseeaeceeeeeeeeeenaeeeeeaeeaes 22 5 3 2 1 IO 0 100 Reta pe A DO RE ren I o dd 22 5 3 2 2 CORRE MAT Set lll o a eed 25 5 3 2 3 PARTIAL CORREL MAT Set 1 ceccecceccccecssesseesseesecneceseeaecnaecsaecaeeeaeeeeeseeeseeesrenseenseeeaeenaees 25 5 3 2 4 MULTIAIN REG Set 1 i 6 o dde e dead 26 5 3 2 5 MM MER SM a a e e as O Paces set a 27 5 3 2 6 DURBIN WATSON Se ibas 29 5 3 2 7 STAND RESIDS Set Di ii 29 5 3 2 8 DUMMY VARS Seti ada 30 5 3 3 TIMES ES an e Tecate eat 31 5 3 3 1 CO OA Ee 31 5 3 3 2 TIME SERIES DIFF Set Di iaa 34 5 3 3 3 AGE Set it 34 5 3 3 4 PAGE Sei a oia 35 5 3 3 5 ARIMA APPROX FIT Set Di 35 5 3 3 6 ARIMA FIT Da ated rederetsi e E E RE AEREE 36 5 3 3 7 ARIMA FORECAST Set Di ie 38 5 3 3 8 TRNS FUNC FORECAST Set Desi aida 38 5 3 3 9 SPECTRALAS iaa 40 DA ABOOK LDS iaon eases sent ac sate seve teeec e
5. 5 5 3 1 ESNE ROYALE TO a EPEE EIEE EAEE a tell intone Pad dates added rd tee ddr Bad A dr the ed dd ND dd tar 68 5 5 3 2 DIST MATL Set D E esc A Sones Satie E dade east E A Ao st ee ace te 73 5 5 3 3 CLUSTER Sed ead Lae ear od ental ste IAN ded panda as als ad eS A Ae itd od hc 73 5 5 3 4 GROUPS FROM CLUSTER Set Desi 75 5 5 3 5 DISCRIM ALLOG Set D sccio E E E EA E isnad 75 5 5 3 6 DISCRIM DIST Set Di 76 5 5 3 7 DISCRIM TEESE Set D E E oe ie ees LO 77 5 5 3 8 MDS Set Di ieee ie hee eves ee en as ees 78 5 5 3 9 METRICS CAL ase oa et 79 Statistical Add Ins for Excel User Guide 6 NAG BOOKS SET 2 NONPARAMETRIC METHODS csccssssssssrssesssssersseeseees SL Ola INTRODUCTION a A AE AI ote tay oted eons 81 622 BOOR A runa ii ads 81 6 2 1 Introduction to Nonparametric Methods cccccccccceseeseeceeeeteeseenecseescsesscesesseesecnseesecseesecnseesenseeeeeaeeaes 81 6 2 2 Index of Nonparametric Methods ccccccccsccesesseeseesseeecuseeseusecseeseceeeceeseeseceeecseeeceseeseeaeseeaeeneeeaeeaeeren 83 6 2 2 1 NONPARAMETRIC INDEX Set Z o coocccicnonicnnononccnonconononnnononn nono nono cono non cn EE iE 83 6 2 3 ONESAMPLE A A A a 84 6 2 3 1 BINOMIAL TEST Set 2 escoicnoniionisocii diocesano diiniita int 84 6 2 3 2 AI A favs coben stages hives bscesteces suave cescvibe ces code scesceubeces coavbsusrvaee ces biveceuceatncesceiven 85 6 2 3 3 COX STUART Set 2 ora ea r it chive ce stvabe ces cuave e
6. Acc_req 0 0 Default 0 00001 Max_steps Optional The maximum number of iterations required If present Max_steps gt 1 Default 30 Output Output as a NAGExtract format list Rotated Loadings Rotation Matrix Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information Individual Directed Methods Introduction Individual directed Methods In general the derived variables will be unique only up to an orthogonal rotation Therefore it may be useful to see if suitable rotations of these variables exist that lead to a simple interpretation of the new variables in terms of the original variables While dealing with the same data matrix as variable directed methods the emphasis is the objects or individuals rather than the variables The methods are generally based on distance or dissimilarity matrices such that the elements give a measure of how far apart individuals are Alternatively a similarity matrix can be used which measures how close individuals are The form of the measure of distance or similarity will depend upon the form of the variables For continuous variables it is usually assumed that some form of squared Euclidean distance is suitable Dy x y Another commonly used distance is the absolute distance or city block metric 68 Statistical Add Ins for Excel User Guide Dyy x yl The function DIST_MAT can be used to compute these d
7. The Rank of Model is returned only when the model is not of full rank The Parameter Estimates consist of the four columns with headings giving Parameter Names Parameter Estimate Standard Error t value The variance covariance matrix of the parameters Cov is returned as an upper triangular matrix 59 5 4 3 5 Statistical Add Ins for Excel User Guide Remarks See Using the Add Ins and the Introduction to Generalized Linear Models for further information GAMMA_GLM Set 1 GAMMA_GLM NAG G02GDF fits a generalized linear model with gamma errors Parameters Link Required Indicates which link function is to be used f an exponent link is used an identity link is used a log link is used a square root link is used a reciprocal link is used DONE M Y Required The response variable X Required Either the array containing the independent variables as columns or the function MM_GLM that computes the model matrix Select_x Optional Indicates which independent variables in array X are to be included in the model If Select_x j gt O then the variable contained in the jth column of array X is included in the model if Select_x j 0 it is excluded If Select_x is present then its elements gt 0 Default all variables included X_names Optional Names for the explanatory variables Default X_1 etc Offsets Optional Offsets for the model Default none Weights n Optional
8. User weights If a Weight is set to zero observation is not included in the model in which case the effective number of observations is the number of observations with non zero weights If present Weights 0 0 Default no weights Mean Optional Indicates if a mean term is to be included M a mean term intercept is to be included in the model Z the model will pass through the origin zero point 60 Statistical Add Ins for Excel User Guide Default mean term included Power Optional If an exponent link is selected the power of the exponent If present Power 0 0 S Optional The scale parameter for the model o If omitted the scale parameter is estimated by the residual mean square If present s gt 0 0 Fitted_values Optional Indicates if fitted values etc are to be returned N No fitted values etc are returned Y Fitted values and residuals are returned A Fitted values residuals leverages linear predictor variance function and working weights are returned Default Y Fitted values and residuals are returned Max_iterations Optional The maximum number of iterations for the iterative weighted least squares fitting If present Max_iterations gt 0 Default 10 Tolerance Optional Indicates the accuracy required for the fit of the model If present Tolerance gt 0 0 Default 0 000001 Eps Optional The value of eps is used to decide if the independent variables are of full ra
9. 0 0 then the ith observation is not included in the analysis If present values must be 0 0 Output Output as a NAGExtract format list Eigenvalues Coefficients Scores Notes The Eigenvalues table consists of the eigenvalues the proportion explained by each component and the cumulative proportion In addition except for when the correlation matrix is analysed test value its degrees of freedom and its significance are returned Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information 65 Statistical Add Ins for Excel User Guide 5 5 2 3 FACTOR Set 1 FACTOR NAG GO3CAF computes the maximum likelihood estimates of the parameters of a factor analysis model Either the data matrix or a correlation covariance matrix may be input Parameters No_of factors Required The number of factors that are required Note Give EITHER Correlation_matrix and Number_of_obs OR Data_matrix and Matrix_type if default not used Correlation_matrix Optional The correlation or variance covariance matrix only the upper triangular part is required Number_of_obs Optional The number of observations used in computing the correlation variance covariance matrix Data_matrix Optional The data matrix with the variables in columns Matrix_type Optional Selects the type of matrix on which factor analysis is to be performed C the factor analysis will be computed for the correlation
10. 3 4 5 3 3 5 Statistical Add Ins for Excel User Guide Remarks See Using the Add Ins and the Introduction to Time Series Methods for further information PACF Set 1 PACF NAG G13ACF calculates the partial autocorrelation function from the autocorrelation function Parameters ACF Required The autocorrelation function Output Output as a NAGExtract format list PACF Predictor Error Variance Ratio Autoregressive Parameters Note The number of partial autocorrelations computed is the maximum that can be computed from the supplied ACF this may be less than the number of autocorrelations The Predictor Error Variance Ratios are defined in the Introduction to Time Series Methods The Autoregressive parameters are for the maximal autoregressive model that can be computed from the autocorrelations Remarks See Using the Add Ins and the Introduction to Time Series Methods for further information ARIMA_APPROX_FIT Set 1 ARIMA_APPROX_FIT NAG G13ADF calculates preliminary estimates of the parameters of an autoregressive integrated moving average ARIMA model from the autocorrelation function of the appropriately differenced time series Parameters Model Required Model specifies the orders p d q P D Q s of the seasonal ARIMA model to be fitted p number of autoregressive terms order of non seasonal differencing number of moving average terms number of seasonal autoregressive terms
11. 3 levels factor 2 pressure would have four levels and the design would be a 3 x 4 factorial giving a total of 12 treatment combinations This design has the advantage of being able to detect the interaction between factors that is the effect of the combination of factors The following are examples of standard experimental designs in the descriptions it is assumed that there are t treatments 1 Completely Randomized Design There are no blocks and the treatments are allocated to units at random 2 Randomized Complete Block Design The experimental units are grouped into b blocks of t units and each treatment occurs once in each block The treatments are allocated to units within blocks at random 3 Latin Square Designs The units can be represented as cells of a square of size t classified by rows and columns The rows and columns represent sources of variation in the experimental material The design allocates the treatments to the units so that each treatment occurs once in each row and each column 42 Statistical Add Ins for Excel User Guide 4 Balanced Incomplete Block Designs The experimental units are grouped into b blocks of t units The treatments are allocated so that each treatment is replicated the same number of times and each treatment occurs in the same block with any other treatment the same number of times The treatments are allocated to units within blocks at random 5 Complete Factorial Experiments If there ar
12. Add Ins for Excel User Guide If however the input were of the wrong type for example if the formula CHI_PPT 5 A were entered then the Excel error message AVALUE would be returned When parameter values for some of the complex functions are being entered an error message may temporarily be displayed until the complete information is given For example with the function MULT_LIN_REG if the Y array has been entered the error message Error y and x of different lengths may be displayed on the function wizard while the X array is being selected until the complete X array has been selected Most NAG error messages should be self explanatory however two need further explanation The error message Catastrophic failure indicates that one of two types of error has occurred An error that should have been trapped and reported has for some reason not been detected If you get such an error first check your inputs to the function It is likely that one of the parameters is incorrect If not check the NAG Web site to see if there is a known problem with the function If it is not recorded please contact NAG preferably by email to report the problem A trial version of the Add Ins has been installed and this has now expired Please contact NAG to obtain a fully licensed copy The error message Error Computational failure indicates that there has been a failure in the underlying algorithm Such an occu
13. GO1AAF T_PPT GO1FBF T_PROB GO1EBF TWO_WAY_TABLE G11AAF Available Add In functions from Book 1 ACF G13ABF ARIMA_APPROX_FIT G13ADF ARIMA_FIT G13BEF ARIMA_FORECAST G13AJF CORREL_MAT G02BXF DUMMY_VARS GO4EAF DURBIN_WATSON G02FCF MULT_LIN_REG GO2DAF PACF G13ACF PARTIAL_CORREL_MAT G02BYF SPECTRAL G13CBF STAND_RESIDS GO2FAF TIME_SERIES_DIFF G13AAF TRNS_FUNC_FORECAST G13BJF Available Add In functions from Book 2 ANOVA_BLOCK GO4BBF ANOVA_CONTRASTS GO4DAF ANOVA_FACTORIAL GO4CAF ANOVA_ROW_COLUMN GO4BCF BINOMIAL_GLM GO2GBF GAMMA_GLM G02GDF MULT_COMPN G04DBF NORMAL_GLM GO2GAF POISSON_GLM GO2GCF 101 8 2 2 Statistical Add Ins for Excel User Guide Available Add In functions from Book 3 CLUSTER DISCRIM_ALLOC DISCRIM_DIST DISCRIM_TEST DIST_MAT FACTOR FACTOR_SCORE GROUPS_FROM_CLUSTER MDS METRIC_SCAL ORTHOG_ROTATION PRIN_COMP Set 2 Nonparametric Methods GO3ECF GO3DCF GO3DBF GO3DAF Available Add In functions from Book 4 BINOMIAL_TEST CHI_SQ_2 CHI_SQ_K CHI_SQ_1 COCHRAN_Q CONTINGENCY_C COX_STUART FRIEDMAN KENDALL_CC KENDALL_TAU KOLMOGOROV_2 KRUSKAL_WALLIS MANN_WHITNEY MCNEMAR MEDIAN_TEST NONPARAMETRIC_INDEX ONE_SAMPLE_CI PROPORTION_CI RUNS_TEST SIGN_PAIRS SPEARMAN_RHO TWO_SAMPLE_CI WILCOXON WILCOXON_PAIRS GO1BJF GO1EAF G11AAF G11AAF GO1ECF GO8ALF G11AAF GO8AAF GO8AEF GO8DAF GO2BNF GO8CDF
14. Generalized Linear Models for further information 57 Statistical Add Ins for Excel User Guide 5 4 3 4 POISSON_GLM Set 1 POISSON_GLM NAG G02GCF fits a generalized linear model with Poisson errors Parameters Link Required Indicates which link function is to be used an exponent link is used and the power must be provided an identity link is used a log link is used a square root link is used a reciprocal link is used AQT m Y Required The response variable Xx Required Either the array containing the independent variables as columns or the function MM_GLM that computes the model matrix Select_x Optional Indicates which independent variables in array X are to be included in the model If Select_x j gt O then the variable contained in the jth column of array X is included in the model if Select_x j 0 it is excluded If Select_x is present then its elements gt 0 Default all variables included X_names Optional Names for the explanatory variables Default X_1 etc Offsets Optional Offsets for model Default none Weights Optional User weights If a weight is set to zero the observation is not included in the model in which case the effective number of observations is the number of observations with non zero weights If present Weights 0 0 Default no weights Mean Optional Indicates if a mean term is to be included M a mean term intercept is to be
15. Ins for Excel User Guide DIST_MAT Set 1 DIST_MAT NAG GO3EAF computes a distance dissimilarity matrix Parameters Distance Required Indicates which type of distances are computed A absolute distances E Euclidean distances S Euclidean squared distances Scaling Required Indicates the standardization of the variables to be used standard deviation range standardizations given in array S unscaled COAL X Required The array of observation for the variable in columns S Optional If Scale G S must be present and contain the scaling factors for variables Input_Dist_Mat Optional Contains the strictly lower triangle of the distance matrix to be updated stored packed by rows i e in the same format as produced by DIST_MAT If omitted no updating takes place Output Output as a NAGExtract format list Distance Matrix Note The matrix is returned as a lower triangular matrix Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information CLUSTER Set 1 CLUSTER NAG GO3ECF performs hierarchical cluster analysis Parameters Method Required Indicates which clustering method is used 1 single link 2 complete link 3 group average 4 centroid 5 median 6 minimum variance Note EITHER the Distance_matrix OR X must be supplied 73 Statistical Add Ins for Excel User Guide Distance_matrix Optional The strictly lower t
16. Link functions For Normal Poisson and gamma distributions identity log exponent square root and reciprocal For Binomial distribution logistic probit and complementary log log The linear model can be formed from the set of input columns or specified by a formula using the NAG MM_GLM function as described below The generalized linear model is fitted by minimizing the deviance which is Deviance 2xLog likelihood a data dependent constant In the case of the Normal error this is simply the residual sum of squares The deviance is minimized using an iterative reweighted least squares algorithm For further details see McCullagh and Nelder 1989 or the NAG Fortran 77 Library manual In the case of gamma errors an adjusted deviance is used which accommodates zero values of the response variable In addition to the parameters of the linear model the Normal distribution has a scale parameter the variance This is estimated by the residual sum of squares divided by the residual degrees of freedom Testing model significance Suppose we have two nested models for example 51 Statistical Add Ins for Excel User Guide n B4 Box and N By Box Bax The significance of the extra terms in this case x can be tested by looking at the difference in the deviance between the two models For the binomial and Poisson distributions the difference is compared with a A distribution with degrees of freedom equal to t
17. See Using the Add Ins and the Introduction to Nonparametric Methods for further information ONE_SAMPLE_CI Set 2 ONE_SAMPLE _Cl NAG GO7EAF computes a rank based confidence interval for the location of a single sample of ordinal data Parameters Data Required The observations from the sample Confidence Optional The required confidence for the interval in the range 0 to 1 Default 0 95 e a 95 confidence interval Method Optional Indicates which algorithm is used to compute the interval E exact algorithm A iterative algorithm Default exact algorithm Output Output as a NAGExtract format list Point estimate Lower limit Upper limit Estimated confidence Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information 87 6 2 3 7 6 2 4 6 2 4 1 Statistical Add Ins for Excel User Guide PROPORTION_CI Set 2 PROPORTION_CI NAG GO7AAF computes a confidence interval for the value of a proportion estimated from a single sample Parameters K Required The observed number of successes N Required The number of observations Confidence Optional The required confidence for the interval in the range 0 to 1 Default 0 95 i e a 95 confidence interval Output Output as a NAGExtract format list Lower limit Upper limit Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further infor
18. b nest two terms e g a b a a b remove term from formula e g a b a b a b power for covariates only gt gt ox The precedence of the operators is At least one of Groups or Covariates must be given along with the corresponding names Groups Optional The array whose columns contain the variables that are to be considered as grouping or classification variables Group_Names Optional The names of the variables in Groups Covariates Optional The array whose columns contain the variables that are to be considered as covariates i e not as classification variables Covariate_Names Optional The names of the variables in Covariates 62 5 5 5 5 1 5 5 2 5 5 2 1 Statistical Add Ins for Excel User Guide Interaction Optional The maximum number of terms to be included in an interaction gt 1 Default all terms included Output To avoid the restriction on the size of returned arrays the function MM_GLM passes the results to the calling function by means of global variables All that is returned directly is an indicator string or any error messages The function computes all required dummy variables with appropriate names The dummy variable corresponding to the first level of a group variable is not included in order to provide a full rank model Remarks See Using the Add Ins and the Introduction to Generalized Linear Models for further information Book 3 Introdu
19. be smaller than Y values U upper tailed test i e X values tend to be larger than Y values Default two tailed test Output Output as a NAGExtract format list Test statistic Z value i e normalised test statistic Significance level Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information MANN_WHITNEY Set 2 MANN_WHITNEY NAG GO8AHF computes the Mann Whitney test for two independent samples of ordinal data Parameters X Required The observations for sample one Y Required The observations for sample two Tail Optional Indicates the critical region and hence the form of H4 T two tailed test i e H1 mx my L lower tailed test i e H1 mx lt my U upper tailed test i e H4 mx gt my where mx and my are the locations of the populations sampled in X and Y 91 6 2 5 4 6 2 5 5 Statistical Add Ins for Excel User Guide Default two tailed test Output Output as a NAGExtract format list Rank sum Normalised test statistic Exact Significance level The significance level is exact for total number of observations lt 40 and uses a Normal approximation for larger numbers of observations Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information MEDIAN_TEST Set 2 MEDIAN_TEST NAG GO8ACF computes the median test for two independent samples of ordinal data Paramete
20. contains two dimensional arrays packed arrays or triangular arrays See Organising the Output Section 3 5 for a perspective of this utility Initialisation deals with getting information into NAGExtract Re locating the Results describes the mechanism of doing a past link using NAGExtract and Setting the Options describes how to set the options for the different types of results obtained from NAG Add Ins Please note that a complete set of appropriate examples are provided for each product in Excel workbook These are installed in the Examples subdirectory of the Add Ins directory and on the CD or other distribution media You might find it helpful to use them to practice the use of NAGExtract Initialisation Before relocating the output NAGExtract needs to be loaded with the data output from the NAG Add Ins Display the NAGExtract dialog box by selecting NAGExtract from the NAG menu The Location of raw NAG results edit box is empty and has the focus i e this box is ready for you to put information into it Do this by typing in the range of cells containing the output from the NAG Add In or alternatively select the range of cells using the mouse Finally press the Search button You should see a number of items appear in the Available data window NAGExtract is now initialized ready for use and the Apply and End buttons are enabled Relocating the Results From the Availa
21. file Choose the Properties option and then the general tab Check the Read only box We get error messages when we load the Add Ins On some systems error messages appear when the Add Ins are loaded Try NOT explicitly loading the NAG Add In utilities despite the message In theory it should not have been necessary to explicitly load these since they are referenced when each of the Add Ins are formed In practice however we found that several machine operating system and Excel version combinations seemed to require explicit loading At least one site has experienced a problem when explicitly loading the utilities Add In which they cured by simply relying on the referencing mechanism Are there any limitations on the size of data sets we can use Most of the NAG Statistical Add Ins for Excel are implemented as array functions This makes it possible for results to be automatically updated on a worksheet when any of the data items change However there is a Microsoft limit in excess of 5000 on the number of elements that may be exported from an array function It follows therefore that for those routines where the amount of data reflects the amount of output there will be a practical limit on the volume of data that can be handled correctly Otherwise as far as we are aware there is no direct restriction on the size of the data sets that may be entered from Excel The Add Ins will not load This is usually indicative
22. i e H1 mx gt my where my and my are the locations of the populations sampled in X and Y Default two tailed test Output Output as a NAGExtract format list Test statistic Significance level Number of non tied pairs Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information WILCOXON_PAIRS Set 2 WILCOXON_PAIRS NAG GO8AGF computes the Wilcoxon test for two matched samples of ordinal data where the differences are also ordinal Parameters Xx Required The observations for sample one Y Required The observations for sample two Tail Optional Indicates the critical region and hence the form of H4 T two tailed test i e H1 Mx My L lower tailed test i e H1 mx lt my U upper tailed test i e Hy mx gt my where mx and my are the locations of the populations sampled in X and Y 89 6 2 5 6 2 5 1 Statistical Add Ins for Excel User Guide Default two tailed test Zeros Optional Indicates whether or not to include cases when the observation equals m in the calculation of the rank sum Y include zeros N do not include zeros Default zeros are not included Output Output as a NAGExtract format list Rank sum Z value i e normalised test statistic Significance level Number of non zero values The significance level is exact for number of observations lt 80 and uses a Normal approximation for larger numb
23. in the array that are not set to parameter values should be set to zero Output Output as a NAGExtract format list Forecasts Standard Errors of Forecasts Remarks See Using the Add Ins and the Introduction to Time Series Methods for further information SPECTRAL Set 1 SPECTRAL NAG G13CBF calculates the smoothed sample spectrum of a univariate time series using spectral smoothing by the trapezium frequency Daniell window Parameters Series Required The time series Width Optional Determines the frequency width of the smoothing window as 2n Width A value of n where n denotes the length of the time series implies no smoothing If present 1 lt Width lt n Default n no smoothing Shape Optional The shape parameter of the trapezium frequency window used when the width of the smoothing window has been set A value of 0 0 gives a triangular window and a value of 1 0 a rectangular window If present 0 0 lt Shape lt 1 0 Default 1 a rectangular window only if Width is set Correction Optional Whether the data are to be initially mean or trend corrected N no correction M mean correction T trend correction Default M mean correction Tapering Optional The proportion of the data totalled over both ends to be initially tapered by the split cosine bell taper A value of 0 0 implies no tapering Default 0 2 Logged Optional Indicates whether unlogged or logged spec
24. in the third location contains the minimum of the input data The is a special delimiter which is detected by a NAG utility NAGExtract which is used in more complicated applications and which will be described later Had the Add In failed an error message would have been produced in the second cell of the selected area Our example with SSPT_SUMMARY presupposed that we knew that we had to select 11 cells to store the result If we did not know that then we observe that the first cell tells us this so if we were to call S5PT_SUMMARY as though it were a normal scalar function then this key information would be returned Specifically select a single cell on the worksheet but make sure that there are vacant cells beneath it Use the function wizard to select cells A9 A14 as before as the argument to S5PT_SUMMARY and press the Enter key or Finish button The selected cell now contains the lt List of 11 gt information Click on this cell to select it and drag down a further additional 10 cells so that 11 cells are now selected Make sure you do not accidentally use the fill handle to drag the cell simply click on the centre of the cell and drag downwards Now click on the formula bar where S5PT_SUMMARY A9 A14 is still displayed and press CONTROL SHIFT ENTER The selected cells now contain the required output To demonstrate the usefulness of array functions change a value in one of the cells A9 A14 Notice that the outpu
25. k largest positive eigenvalues give the best k dimensions in which to represent the objects The function METRIC_SCAL performs this type of scaling If there are negative eigenvalues then the distance matrix cannot be represented in Euclidean space METRIC_SCAL gives the option of computing all the eigenvalues so that this assumption can be checked Instead of the above approach of requiring the distances from the points to match the distances from the objects as closely as possible sometimes only a rank order equivalence is required That is the order of the distances between the points should as far as possible be the same as the distances between the objects they represent This would be appropriate when the dissimilarities are based on subjective rankings For example if the objects were foods and a number of judges ranked the foods for different qualities such as taste and texture the resulting distances would not necessarily obey the metric inequality but the rank order would be significant Alternatively by relaxing the requirement from matching distances to rank order equivalence only the number of dimensions required to represent the distance matrix may be decreased The requirement of rank order equivalence leads to non metric or ordinal multidimensional scaling The criterion used to measure the closeness of the fitted distance matrix to the observed distance matrix is known as STRESS Non metric multidimensional scaling seeks to find t
26. observations Both n and k must be greater than 1 Output Output as a NAGExtract format list Kendall s 7 If there are more than two variables the correlations are returned as a k by k matrix Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information SPEARMAN_RHO Set 2 SPEARMAN_RHO NAG GO2BNF computes the Spearman s rank correlation coefficient r Parameters Data Required The n rows by k columns array of the data where k is the number of variables and n is the number of observations Both n and k must be greater than 1 Output Output as a NAGExtract format list Spearman s r If there are more then two variables the correlations are returned as a k by k matrix Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information 96 7 1 7 2 Statistical Add Ins for Excel User Guide Troubleshooting Guide Introduction This guide is intended to address some of the common problems that users have experienced in the past This revision has tried to remove some of the difficulties and hopefully this manual will also help If nevertheless your problem is not listed please do not hesitate to contact us at one of the addresses given below and we will try to solve your problem It will also tell us where the manual or software needs improvement so we welcome your call Please note that complete sets of examples are provided i
27. of a faulty installation Re install the Add Ins from the supplied media Note that it is not sufficient to copy materials from one machine to another See Installation Have you any examples of use Please see the workbooks BookOE xls Book1E xls Book2E xls Book3E xls or Book4E xls as appropriate These are supplied with the relevant Add Ins product Can obtain more copies of the manual These are available from NAG Ltd please see Addresses Alternatively the manual is available in Acrobat format on the CD on the NAG Web sites and of course via the Windows Help system Will be notified of any updates or additional materials Please register with us to ensure that we can keep you up to date with future developments of this product line If you have purchased the Add Ins via a third party then we are especially keen that you should either complete the registration form in the Appendix see registration or register via the WWW See www nag co uk or www nag com In addition to the benefits of 98 Statistical Add Ins for Excel User Guide being kept up to date with NAG developments this will also expedite our response to any questions you may subsequently have about the product and its usage The Nag menu doesn t appear when load the NAG Add Ins Make sure that all of the NAG Books including NAGExtract are not loaded Now load the NAG Add In Utilities From the User Defined category in the fu
28. orders p d q P D Q and s The function returns the parameter estimates standard errors f values and significance The results can be input into ARIMA_FORECAST to forecast values for the series Transfer Function Models The ARIMA model can be used to describe a single series however if the output series can be related to several input series then a multi input or transfer function model can be used The form of a transfer function model is Yt Z1 Zott t Zme t Ni where n follows an ARIMA model and the components of the model Z are related to the input variables X by either a simple linear model Zit DMoXit or an ARMA like model Zit l Zit1 02 Zit 2 Op Zit p oXit b 1Xitb 1 WpXit b q where b is known as the delay A transfer function model can be fitted using the options of ARIMA_FIT and forecasts can be made using TRNS_FUNC_FORECAST To forecast for the output series of a transfer function model the forecasts of the input series have to be provided Also if the input series have been forecast using ARIMA models these models may be supplied to TRNS_FUNC_FORECAST this will not affect the forecast but will adjust the forecast standard error Spectral Analysis An alternative way of looking at a time series is to consider it as being made up of a number of sine cosine waves of different frequency The power spectrum indicates how much of the variance of the series can be attributed to th
29. runs above values gt cut point Default sample median Output Output as a NAGExtract format list Cut point Number of runs Number of runs below the cut point Number of runs above the cut point Test statistic Significance level The significance uses a Normal approximation Remarks See Using the Add Ins and the Introduction to Nonparametric Methods statistics for further information WILCOXON Set 2 WILCOXON NAG GO8AGF computes the Wilcoxon test for the median for ordinal data from a single sample Parameters Data Required The observations from the sample Median Required The value of the median under the null hypothesis mo Tail Optional Indicates the critical region and hence the form of H4 T two tailed test i e H1 m M L lower tailed test i e Hy m lt mg U upper tailed test i e Hy m gt my Default two tailed test 86 6 2 3 6 Statistical Add Ins for Excel User Guide Zeros Optional Indicates whether or not to include cases when the observation equals mg in the calculation of the rank sum Y include zeros N do not include zeros Default zeros are not included Output Output as a NAGExtract format list Rank sum Z value i e normalised test statistic Significance level Number of non zero values The significance level is exact for number of observations lt 80 and uses a Normal approximation for larger numbers of observations Remarks
30. tested using contrasts For example if there are four treatments in which the first is a control and the other three are different amounts of a chemical then the contrasts that are the difference between no chemical and chemical and the linear effect of chemical could be defined The sums of squares for these contrasts can be computed from which suitable F tests can be performed ANOVA_CONTRASTS computes the sum of squares for a user defined contrast between means In situations where blocks and treatments are not orthogonal both the treatment means ignoring blocks and the treatment means adjusted for blocks have to be provided In the case of factorial experiments the treatment sum of squares and degrees of freedom may be partitioned into main effects for the factors and interactions between factors The main effect of a factor is the effect of the factor averaged over all other factors The interaction between two factors is the additional effect of the combination of the two factors over and above the additive effects of the two factors averaged over all other factors For a factorial experiment in blocks with two factors A and B in which the tth unit of the th block received level of factor A and level k of factor B the model is Vij u amp 07 Bk aplik ej where and are the main effects of factors A and B respectively and aj is the interaction between level of A and level k of B Higher order interactions between three
31. use functions from the other NAG will of course be pleased to accept an order for the other set The items described as belonging to Book 0 Book 1 Book 2 and Book 3 together form the NAG Statistical Add Ins for Excel Modelling and Multivariate Methods product Items belonging to Book 4 constitute the NAG Statistical Add Ins for Excel Nonparametric Methods product NAG is always anxious to receive feed back from its users so please contact us if you have any suggested improvements or if there is any extra functionality you would like Please register with us either via the Web www nag co uk or www nag com or by returning the form printed in the appendix of this manual This form may be printed from the help file if you do not want to take pages from your printed manual Once you are registered we will be able to keep you informed of any updates to this product and other enhancements from NAG Statistical Add Ins for Excel User Guide Installation You should have a CD containing the Add Ins material or a file downloaded from the web If you have a CD then under normal circumstances the CD will autorun If the autorun facility is disabled double click on the setup exe file on the CD from Windows Explorer Follow the prompts on the screen to complete the installation process If you have downloaded a file from the web then double click on the setup exe file from Windows Explorer and follow the prompts on the screen to
32. used to construct the required dummy variables with all levels present CORREL_MAT Set 1 CORREL_MAT NAG G02BXF calculates the product moment correlation matrix and the variance covariance matrix Parameters X Required The array containing the observations on the variables in columns One of the following may be set Frequencies Optional The number of observations associated with each observation The effective number of observations is taken as the sum of the frequencies Weights Optional The weights associated with each observation The number of observations is taken as the number of observations with non zero weights In either case the value zero means that an observation is excluded from the analysis Output Output as a NAGExtract format list Variance Covariance Matrix Correlation Matrix Mean Standard Deviation Remarks See Using the Add Ins and the Introduction to Regression and Correlation for further information PARTIAL_CORREL_MAT Set 1 PARTIAL_CORREL_MAT NAG G02BYF computes a partial correlation variance covariance matrix from a correlation or variance covariance matrix computed by CORREL_MAT Parameters Correlations Required Correlation matrix or the variance covariance matrix 25 5 3 2 4 Statistical Add Ins for Excel User Guide Select_vars Required Indicates which variables belong to set X and Y Y Y or dependent variable X X or independent variable O var
33. x42 fits the quadratic model in x a b fits the terms a b a b that is the main effects of a and b and the a b interaction that is a cross classification model a b fits a a b that is the nested model a b c fits a c plus b c that is a b c a c b c The notation is only available for use with covariates There is also an additional parameter Interaction which sets the maximum number of terms allowed in an interaction So for example a b c with Interaction 2 gives a b c a b a c b c with the three term interaction excluded Note that the model matrix produced by MM_MLR assumes that the mean is also fitted If the mean is to be omitted then DUMMY_VARS should be used to construct the required dummy variables with all levels present NORMAL_GLM Set 1 NORMAL_GLM NAG G02GAF fits a generalized linear model with normal errors Parameters Link Required Indicates which link function is to be used an exponent link is used an identity link is used a log link is used a square root link is used a reciprocal link is used A m Y Required The response variable Xx Required Either the array containing the independent variables as columns or the function MM_GLM that computes the model matrix 53 Statistical Add Ins for Excel User Guide Select_x Optional Indicates which independent variables in array X are to be included in the model If Select_x j gt O then the va
34. 11 5 2 2 12 Statistical Add Ins for Excel User Guide Numerator_df Required The degrees of freedom of the numerator variance must be gt 0 0 Denominator_df Required The degrees of freedom of the denominator variance gt 0 0 Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information BETA_PROB Set 1 BETA_PROB NAG GO1EEF returns as a single value the lower or upper tail probability of the beta distribution Parameters Xx Required The value of the beta variate 0 0 lt X lt 1 0 A Required The first parameter of the required beta distribution 0 0 lt A lt 10 6 B Required The second parameter of the required beta distribution 0 0 lt B lt 10 6 Tail Required Indicates whether an upper or lower tail probability is required U The upper tail probability is returned L The lower tail probability is returned Tolerance Optional The relative accuracy required by the user in the results Default a value of 10 times the machine precision is used Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information BETA PPT Set 1 BETA_PPT NAG GO1FEF returns as a single value the deviate associated with the given lower tail probability of the beta distribution Parameters Probability Required The probability from the required beta distribution A Required The first parameter of the required beta di
35. Indicates for which type of data the functions returned should be suitable A all functions O ordinal N nominal Default all functions Output Output as a NAGExtract format list List of functions Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information One Sample BINOMIAL_TEST Set 2 BINOMIAL_TEST NAG G01BJF GO1EAF computes the binomial test for the value of a proportion estimated from one sample Parameters K Required The observed number of successes N Required The number of observations PO Required The proportion to be tested i e the value p of p under Ho Tail Optional Indicates the critical region and hence the form of H4 T two tailed test i e H1 p Po L lower tailed test i e H1 p lt po U upper tailed test i e H1 p gt po Default two tailed test Output Output as a NAGExtract format list Z value i e normalised test statistic Significance level 84 6 2 3 2 6 2 3 3 Statistical Add Ins for Excel User Guide The significance level is exact for N lt 20 and uses a Normal approximation for larger numbers of observations Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information CHI_SQ_1 Set 2 CHI_SQ_1 NAG G01ECF computes the chi squared test for one sample Parameters Observed Required The observed frequencies for each class Exp
36. L can be used as suitable initial estimates STRESS_type Optional Indicates whether STRESS or Squared STRESS is to be used as the criterion T STRESS is used S squared STRESS is used Max_iter Optional The maximum number of iterations to be used in the fitting process If present Max_iter gt 0 Default 50 Output_distns Optional This optional parameter may be used to select the distances returned Output_distns 0 no distances returned Output_distns 1 fitted distances returned Output_distns gt 1 four sets of distances are returned Fitted distances Fitted distances ordered by input distances Monotonic distances ordered by input distances Monotonic distances Default 0 no distances returned Output Output as a NAGExtract format list Coordinates STRESS Fitted distances Ordered fitted distances 78 5 5 3 9 Statistical Add Ins for Excel User Guide Ordered monotonic distances Monotonic distances Note The fitted distances and monotonic distances are returned in lower triangular form corresponding to the input distance matrix while the Ordered fitted distances and Ordered monotonic distances are returned as one single column Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information METRIC_SCAL Set 1 METRIC_SCAL NAG GO3FAF performs a principal coordinate analysis also known as classical metric scaling Parameters Distance_matrix Req
37. Let there be three clusters A B and C with distances A B Cc A 1 0 2 0 B 3 0 Cc At the next step cluster A is merged with cluster B The distances between the new cluster and cluster C will be 2 the minimum distance from A or B to C if single link is chosen or 3 the maximum distance from A or B to C if complete link is chosen Group average takes into account the size of the clusters A and B So if cluster A had six objects and cluster B four then the new distance to C would be 2 6 10 3 4 10 2 4 The other three methods also take into account the distance between A and B Median clustering takes the average of the distances minus a quarter of the AB distance e g 2 3 2 2 25 Centroid clustering adjusts the group average distance by the AB distance giving 2 6 10 3 4 10 6 10 4 10 1 2 16 Finally minimum variance also takes into account the number of objects in C If there were 5 objects in C minimum variance clustering would give the distance as 5 6 2 5 4 3 5 1 5 6 4 2 93 Details of these methods can be found in Everitt 1974 or Krzanowski 1990 Once the tree has been formed the clusters can be created by either specifying the number of clusters required or specifying the distance at which clusters are formed The function CLUSTER computes a hierarchical cluster analysis tree using one of the six methods described above You can enter either the distance matrix or enter the raw data
38. O1HBF returns as a single value the upper tail lower tail or central probability associated with a multivariate Normal distribution of up to ten dimensions Parameters Mean Required The mean vector of the multivariate Normal distribution Covariance Required The square array containing the variance covariance matrix of the multivariate Normal distribution Only the lower triangle is referenced Lower_bounds Optional If Tail C or U the lower bounds Upper_bounds Optional If Tail C or L the upper bounds Tail Optional Indicates which probability is to be returned U The upper tail probability L The lower tail probability C The central probability is returned Default central probability Tolerance Optional The relative accuracy required in the result Default 0 0001 Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information 15 5 2 2 5 5 2 2 6 5 2 2 7 Statistical Add Ins for Excel User Guide T_PROB Set 1 T_PROB NAG GO1EBF returns as a single value the lower tail upper tail or two tail probability for the Student s t distribution Parameters T Required The value of the Student s f variate Df Required The degrees of freedom of the Student s t distribution must be gt 1 Tail Required Indicates which tail the return probability should represent U The upper tail probability is returned L The l
39. One way is to provide an array with the variables stored in columns This information can then be supplemented by optional names for the variables and 0 1 indicators that allow variables to be dropped from the model Dummy variables can be calculated using DUMMY_VARS but a more general way of specifying the model is to use the MM_MLR function described below This uses an algebraic specification of the model that is found in most statistical packages and automatically generates the required dummy variables The regression fitting returns an analysis of variance table parameter estimates with their standard errors f values and correlations along with fitted values residuals and leverages The leverages provide an indication of how much influence the observation has on the fit and they can also be used to compute confidence intervals for the fitted line Further examination of the residuals can be assisted by the use of standardized residuals computed by STAND_RESIDS which also computes the Cook s D statistics that measure the influence of an observation see Cook and Weiberg 1982 If the observations have been observed serially DURBIN_WATSON computes the Durbin Watson test for serial correlation Specifying the Model Matrix The NAG function MM_MLR can be used to specify a model matrix rather than the user constructing a set of columns The inputs to the model matrix function MM_MLR are 1 Group variables with their names 2 Covariates with
40. Statistical Add Ins for Excel User Guide Statistical Add Ins for Excel User Guide The Numerical Algorithms Group Limited 2000 All rights reserved No part of this manual may be reproduced transcribed stored in a retrieval system translated into any language or computer language or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise without the prior written permission of the copyright owner The copyright owner gives no warranties and makes no representations about the contents of this manual and specifically disclaims any implied warranties or merchantability or fitness for any purpose The copyright owner reserves the right to revise this manual and to make changes from time to time in its contents without notifying any person of such revisions or changes Second Edition ISBN 1 85206 191 X NP3521 NAG is a registered trademark of The Numerical Algorithms Group Limited The Numerical Algorithms Group Inc The Numerical Algorithms Group Deutschland GmbH Nihon Numerical Algorithms Group KK All other trademarks are acknowledged Statistical Add Ins for Excel User Guide Contents LOADINGAN ADD it rt 3 SI A sofecgteatbeenct vecodsaes cae Sack SobecbiapagieaSs eu eshse bbe bade SeSeT Ua ed RU CATG wu oa teed dine bade Sue GANGS ATT wa cei Ho RS 3 PARAMETER INPUT e A A A A eae Poke BBS 4 ARRAY BUNGTIONS iio NEO E O E CLG EA TA e da Tee 4 ORGANISING THE OUTPUT A A ne
41. T Click the dialog box for the first parameter and now with the mouse select the cell containing the value 0 5 The cell address e g A4 will appear in the dialog box as the first parameter Similarly click the cell containing 1 to specify the second parameter Press Enter or click the Finish button Again the value 0 454936 should appear in the output cell If we now change the value 0 5 to say 0 6 in the data cell then with normal settings of Excel that allow automatic updating of cell values the result from the last computation will automatically be updated to 0 708326 to reflect a revised call of CHI_PPT with parameters 0 6 and 1 This is a very powerful feature of Excel functions that we sought to preserve with more complicated routines within the NAG Statistical Add In suite 3 3 3 4 Statistical Add Ins for Excel User Guide If you only have the NAG Statistical Add Ins for Excel Nonparametric Methods product Set 2 then select a cell on a worksheet and enter the formula PROPORTION_CI 5 10 Text lt List of 5 gt appears in the cell Section 3 4 will explain the significance of this suffice to say that this text indicates that the Add In is functioning correctly As in the example described above PROPORTION_CI may be called from the function wizard and may take as its arguments data entered into cell locations in the workbook For example cell A1 may contain the value 5 and cell B1 may contain the value 10 Th
42. T_COMPN GO4EAF DUMMY_VARS 103 Statistical Add Ins for Excel User Guide Available Add In functions from Chapter G07 GO7AAF PROPORTION Cl GO7EAF ONE_SAMPLE_Cl GO7EBF TWO SAMPLE CI Available Add In functions from Chapter G08 GO8AAF COX_STUART GO8AAF SIGN_PAIRS GO8ACF MEDIAN_TEST GO8AEF FRIEDMAN GO8AFF KRUSKAL_WALLIS GO8AGF WILCOXON GO8AGF WILCOXON_PAIRS GO8AHF MANN_WHITNEY GO8ALF COCHRAN_Q GO8CDF KOLMOGOROV_2 GO8DAF KENDALL_CC Available Add In functions from Chapter G11 G11AAF TWO_WAY_TABLE G11AAF CHI_SQ 2 G11AAF CHI_SQ_K G11AAF CONTINGENCY_C Available Add In functions from Chapter G13 G13AAF TIME_SERIES_ DIFF G13ABF ACF G13ACF PACF G13ADF ARIMA_APPROX_FIT G13AJF ARIMA_FORECAST G13BEF ARIMA FIT G13BJF TRNS_FUNC_FORECAST G13CBF SPECTRAL 104 Statistical Add Ins for Excel User Guide Registration Please register your purchase with us in order to be kept up to date with new revisions or related additional materials Registration will also enable us to deal more quickly with any subsequent questions you may have about the Add Ins or their usage Simply register via the WWW at www nag co uk or www nag com or send a completed copy of this form to one of the NAG addresses Name Job Title Department Organization Address Tel Number Fax Number Email Add ins used e g Set 1 and or Set 2 Thank you That is sufficient to re
43. UBLESHOOTING GUIDE ici 7 Tle INTRODUCTION A A A E E N AAA AE 97 TA QUESTIONS A a ote 97 Bohs GLOSSAR Vir laa a ASE Ea E Ica 100 8 2 ALGORITHMIC ASSOCIATION eodeni e E OE E E E E E E E E 101 8 2 1 Add In NAG Subroutine Correspondence cccsccecssceceeseesesseesesseeeecuseesecseeeecuseeeenseeseesesiesesneeeeenseeaees 101 8 2 2 NAG Subroutine Add In Correspondence cccsccccseceveeseesesseenseescueeseesecseeeecuseeecnseesenseseenesesenaeeseenees 102 E A A NR 105 Statistical Add Ins for Excel User Guide Introduction This manual describes the NAG Statistical Add Ins for Excel their installation and their usage The underlying algorithms are based upon routines from the NAG Fortran Library and detailed descriptions of these algorithms are contained in the NAG Fortran Library manuals These manuals may be purchased from NAG Alternatively useful information may be obtained from the NAG web sites The Add Ins have been prepared under Excel version 7 on a PC running Microsoft Windows 95 and tested under Excel 97 and Office 2000 running Windows NT Windows 95 and Windows 98 Both the NAG Statistical Add Ins for Excel Modelling and Multivariate Methods and the NAG Statistical Add Ins for Excel Nonparametric Methods are documented here The Add Ins are referred to as Set 1 and Set 2 respectively in the individual function documents If you have purchased just one of these products then you will not be able to
44. ame effect by typing in the curly braces explicitly the CONTROL SHIFT ENTER mechanism is mandatory If you have the NAG Statistical Add Ins for Excel Modelling and Multivariate Methods product then it is possible to use SSPT_SUMMARY in a similar fashion This returns a set of useful numbers the minimum the lower hinge the median the upper hinge and the maximum relating to an array of data Firstly construct some initial data for example the values 1 2 3 4 5 6 in contiguous cells of the worksheet A9 A14 Select a column of 11 contiguous cells to contain the output from SSPT_SUMMARY we will see why 11 cells and not 5 are required shortly Now use the function wizard to enter the cells A9 A14 as the only argument to S5PT_SUMMARY Press Enter or click on the Finish button As with the matrix multiplication example above we see that only the first cell of the selected cells contains any output we have to enter the formula as an array formula So we click on the formula bar and press CONTROL SHIFT ENTER Now we see all 11 cells filled with useful information and note that just as before the array formula appears in the formula bar surrounded by curly brackets At the head of the column we see lt List of 11 gt This is very useful information which we will utilise later It tells us that to store all the output from the Add In we will require at least 11 cells The second entry is Minimum This tells us that the value underneath
45. an asterisk for those comparisons that are significantly different Remarks See Using the Add Ins and the Introduction to the Analysis of Designed Experiments for further information 49 Statistical Add Ins for Excel User Guide 5 4 2 6 ANOVA_FACTORIAL Set 1 ANOVA_FACTORIAL NAG G04CAF computes an analysis of variance table and treatment means for a complete factorial design Parameters No_of_ blocks Required The number of blocks replicates in the design If there are no blocks set No_of_blocks 0 Factor_names Required The names of the factors in the design Factor_levels Required The number of levels for each factor Data Required The observations ordered by factors within blocks Highest_interaction Optional The maximum number of factors in an interaction term If no interaction terms are to be computed Set Highest_interaction to 0 or 1 If present Highest_interaction gt 0 Default all possible interaction terms computed Df_adjustment Optional This is the adjustment to the degrees of freedom for the residual and total The degrees of freedom for the total is set to number of observations Df_adjustment and the residual degrees of freedom adjusted accordingly If present Df_adjustment gt 0 Output ANOVA table Tables of Treatment Means Tables of Treatment Effects Standard Errors of differences in means Grand Mean Block Means Residuals Note The tables of means and effe
46. and use one of the Euclidean distances The function GROUPS_FROM_CLUSTER can then be used to compute the required clusters Discriminant analysis Discriminant analysis is concerned with the allocation of objects to groups on the basis of observations on those objects using an allocation rule This rule is computed from observations coming from a training set in which group membership is known The allocation rule is based on the distance between the object and an estimate of the location of the groups The usual measure of the distance of an observation from group mean is given by Mahalanobis distance in which the squared distances are standardized by the variance covariance matrix of the variables This can either be the variance covariance matrix for the group or a pooled variance covariance matrix if the group variance covariance matrices can be assumed to be equal This 70 Statistical Add Ins for Excel User Guide assumption can be tested see Morrison 1967 In addition to the distances a set of prior probabilities of group may be used The prior probabilities reflect the user s view as to the likelinood of the objects coming from the different groups It is generally assumed that the variables follow a multivariate Normal distribution By combining the prior probability of an observation s membership of a group with the probability of the observation given a group a posterior probability of belonging to a group can be computed An
47. ar Matrices Option This option allows you to control whether triangular matrices are output as lists or as triangles The default is to output them as triangles Output Headings Option This option controls whether or not the name of the entity as specified in the output list is also placed in the worksheet If so it appears in bold type centred across the columns containing the entity The default action is to output the headings OK Button This button sets the options for the duration of the NAGExtract call or until you reset them 10 Statistical Add Ins for Excel User Guide Help Button The Help button displays information on NAGExtract 11 12 Statistical Add Ins for Excel User Guide 5 1 5 2 5 2 1 5 2 2 5 2 2 1 Statistical Add Ins for Excel User Guide NAG Books Set 1 Modelling and Multivariate Methods Introduction There are 6 Add Ins provided with the NAG Statistical Add Ins for Excel Modelling and Multivariate Methods product NAG Add In Utilities A set of utilities used by the other NAG Books Book 0 Support functions for elementary statistical calculation Book 1 Statistical Modelling 1 Regression and Time Series Book 2 Statistical Modelling 2 Analysis of Experiments and Generalized Linear Models Book 3 Multivariate Methods NAGExtract A utility to reposition output more conveniently on the worksheet Book 0 Introduction Introduction to the Basic St
48. are output when appropriate 47 5 4 2 4 Statistical Add Ins for Excel User Guide ANOVA table Treatment Means Covariances and Standard Errors Replication Efficiency Factors Grand Mean Replicate Means Row Means Column Means Residuals Note In the Covariances and Standard Errors array the upper triangular part contains the variance covariance matrix of the treatment effects and the strictly lower part contains the standard errors of the difference between pairs of treatment effects Remarks If no treatments are specified then the rows of the ANOVA table corresponding to Blocks and Total are computed along with the Block Means and the Grand Mean See Using the Add Ins and the Introduction to the Analysis of Designed Experiments for further information ANOVA_CONTRASTS Set 1 ANOVA_CONTRASTS NAG GO4DAF computes sum of squares for a user defined contrast between treatment means Parameters Treatment_means Required The treatment means as computed by ANOVA_BLOCK or ANOVA_ROW_COLUMN Replication Required The replication for each treatment mean rms Required The residual mean square o rdf Required The residual degrees of freedom Contrasts Required The columns of Contrasts contain the contrasts Tolerance Optional The tolerance used to check is the contrasts are orthogonal and if they are orthogonal to the mean If provided Tolerance gt 0 0 Default 0 0000001 Adj_means O
49. ariables can be shown to be the eigenvectors of the sample variance covariance matrix The amount of variation explained by the eigenvector is proportional to its corresponding eigenvalue The eigenvector corresponding to the largest eigenvalue is known as the first principal component As an alternative to the variance covariance matrix the correlation matrix can be used This has the effect of standardizing the variables to the same scale Ideally only a small 63 Statistical Add Ins for Excel User Guide number of dimensions principal components are needed to represent most of the variation in the data A test on the smaller eigenvalues can be used to investigate the number of dimensions needed The coefficients provided by the eigenvalues indicate how the components relate to the original variables The values of the principal component variables for the individuals are known as the principal component scores These can be standardized so that the variance or the sum of squares of these scores for each principal component is equal to 1 0 or the corresponding eigenvalue The function PRIN_COMP computes the principal component analysis and returns the eigenvalues the proportion of the variation explained by each component and tests for the number of components required The coefficients of the eigenvectors and the scores are also returned Factor analysis is similar to principal component analysis but uses a different underlying model Like
50. arks See Using the Add Ins and the Introduction to Basic Statistics for further information CHI_PPT Set 1 CHI_PPT NAG GO1FCF returns as a single value the deviate associated with the given lower tail probability of the x distribution Parameters Probability Required The probability of the required y distribution Df Required The degrees of freedom of the y distribution must be gt 0 0 Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information F_PROB Set 1 F_PROB NAG GO1EDF returns as a single value the lower or upper tail of the F or variance ratio distribution Parameters Xx Required The value of the F variate must be gt 0 0 Numerator_df Required The degrees of freedom of the numerator variance must be gt 0 0 Denominator_df Required The degrees of freedom of the denominator variance must be gt 0 0 Tail Required Indicates whether an upper or lower tail probability is required U The upper tail probability is returned L The lower tail probability is returned Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information F_PPT Set 1 F_PPT NAG GO1FDF returns as a single value the deviate associated with the given lower tail probability of the F or variance ratio distribution Parameters Probability Required The probability from the required F distribution 17 5 2 2
51. as eas Baile oe ceca COL a Ole BEE A A EAE ole Saat SE 41 5 4 1 ANNA AR REN 41 DADs Analysis Of EAN ents A a a ata 41 5 4 2 1 AV 04 0 F100 1 I er er en E E E A A E E EE E 41 5 4 2 2 ANOVA BLOCK S6t i scissors cht EEE dsd 46 5 4 2 3 ANOVA ROW COLUMN Set Dis a 47 5 4 2 4 ANOVA CONTRASTS Set Mii aida 48 5 4 2 5 MULT ACOMPN Set Di n 49 5 4 2 6 ANOVA FACTORIAL Set Di A 50 5 4 3 Generalized Linear Models rairau r tacos ida 51 5 4 3 1 o seenen EE E E eee 51 5 4 3 2 NORMAL GEM Set D i s ccsecset cei eines there a litio 53 5 4 3 3 BINOMIAL GLEM Seta 55 5 4 3 4 POISSON GEM Set Dial ls 58 5 4 3 5 GAMMA GM Set Dia db 60 5 4 3 6 MM GEM Sef JJ taa 62 Ja A 63 5 5 1 Introduction osrin unnan aa aa thie a a ae o a T ARRE 63 5 5 2 Variable Directed Methods ccccccccccccccscccccccccsssssssscccccessesssssssesccecsesessssscsscessessssssescecsesssssssscesseesenssaesess 63 535 2 E INTO UCA dd a bas Sls rod LL dr ed dd oad oad dN iad ed eta ra 63 535 22 PRIN COMP Set E E O Me ds Se sat selon ee ci 65 5 5 2 3 FACTOR Setllsitamt led ae rara aldo date codes aid adi cobol 66 5 5 2 4 FACTOR SCORES Ue Oates Ma stots baler Le ENE LOE tots 67 5 5 2 5 ORTHOG_ ROTATION Set 1 ccccccecsseesceesceeseeeeceecesecesecseecaeecaecseecaeeeaeseeeseeeseeeeeeseeeseeeeseenaees 67 5 5 3 Individual Directed Methods ccccccccccccccccsssusccesccscussevsscssesccsssessssssesscceceesessassesecessesssssssseseeessssaesecs 68
52. ate routine document from the NAG Fortran Library Printed versions of the manual are available from NAG Alternatively you may wish to look at the documentation available on the NAG Web sites www nag co uk and www nag com 4 2 4 2 1 4 2 2 4 2 3 Statistical Add Ins for Excel User Guide Add Ins in Both Sets NAG Add In Utilities It is not envisaged that you will need to call the NAG utilities directly under normal circumstances Each of the NAG Add Ins needs to perform various common actions such as taking information from the worksheet and putting the values into appropriate Visual Basic arrays These shared utilities are grouped together in the NAG Add In Utilities Add In for greater efficiency Theoretically this Add In should not require explicit loading since it is referenced by each of the NAG Add Ins that require it We have found that it may be necessary to explicitly load the Add In on some systems We recommend therefore that you experiment to see whether this is indeed the case for your own system NAGExtract Introduction NAGExtract is a utility which may be invoked from the Nag menu on the menu bar Its purpose is to ease the paste link burden as you seek to reposition elements of the output to disparate positions Whilst individual elements and possibly even simple vectors can more readily be repositioned using the Microsoft copy and paste link facilities the utility is especially useful if the output
53. atistics Book 0 provides a range of basic statistics and probability functions These are generally intended to support the functions in the other books While much of the functionality is already available in Excel there are advantages with using a common source for your statistical analysis The functions can be split into three categories 1 Distribution functions 2 Summary statistics 3 Contingency tables Please note that a complete set of examples for Book 0 are provided in the workbook BookOE xls This is installed in the Examples subdirectory of the Add Ins directory You might find it helpful to work through these examples to familiarize yourself with the use of the Book 0 Add Ins Distribution Functions Introduction Distribution functions Functions that calculate the probability for a value and functions that calculate the value from the probability percentage points deviates are available for the following distributions Distribution Probability Deviate Normal NORMAL_PROB NORMAL_PPT t T_PROB T_PPT x CHI_PROB CHI_PPT F F_PROB F_PPT Gamma GAMMA_PROB GAMMA_PPT Beta BETA_PROB BETA_PPT 13 5 2 2 2 Statistical Add Ins for Excel User Guide Notes The probability density function of the gamma distribution is given by exp a 1 log x b x b bITa where ais the shape parameter and b is the scale parameter with the mean given by ab The beta distribution is also known as the beta distribution of t
54. bach ita reia 5 GETTING ONTINE HELP dsd 6 ERRORIMESSAGES tonta tits 6 CHANGES FROM RELEASE LD da a al is Ta ras 7 FURTHER TECHNICAL INFORMATION cssssssscccececssssnscecececesssnseaecececsessaaeeesececsesnaeeecececsensaaeaeeececeesenseaeeesecs 8 4 ADD INS IN BOTH SETS osa nad ANS NAG ADDaIN WTIEITIES 5 2iscceccsiacustesesdecegscagsscctsige sesh cuaseneoweceates soadchsswetedepasncecuasmeandecudncusncncogetaghsacusiesuntectsa E 9 AD INAGEXTRA GT wos disens sa cies ons asus E EEE E AT E PEE oe ghstdush tbigestt iba 9 4 2 1 TIT OAUCTION ore occ eas es Pe Rg Me OS MR eek Loi tA tie EN Sea te ae eh WN A oe NN ae aE 9 4 2 2 TNE GUIS ELON es cs es Pecan A E eek aaa cach AEE ENEY 9 42 3 Relocatin the Results it a AA psa 9 ADA 101701 0198 1101 AAA NR ES NN 10 5 NAG BOOKS SET 1 MODELLING AND MULTIVARIATE METHODS 13 Sil gt INTRODUCTION a tenes Eee 8 13 Hz BOOK O Gi dd Oa NTE Sse BRE 13 5 2 1 VALOO TTEA OU EE E I IE E eo OR dead Oe DO a e 13 5 2 2 Distribution ENCINA i OEE 13 5 2 2 1 ICO UCA A da 13 5 2 2 2 NORMAL PROB Set Descuido 14 5 2 2 3 NORMAL PPT SM aorta 15 5 2 2 4 MULTI NORMALE PROB Set Di da 15 AAA T PROB AS a E A E sau eteeree the acanen es canoes desea aed eevee 16 5 2 2 6 A oe 16 S222 Ts CHI PROB Set T ann E sl oe T E E A eaot eae E E 16 5 2 2 8 CHI PREE Sd o 17 5 2 2 9 F PROB St Disc leads 17 5 22210 EPR SE teen 17 5 2211 BETA PROB a E E 18 52212 BETA PRT Sa ais 18
55. ble Data box select the data item that needs to be re positioned Then click on the Destination Cell input box The chances are that the cell location contained therein is not the one you require so delete or select the current contents for overwriting Next specify the new location of the output by typing in or selecting with a mouse a single cell location With default settings this specifies the top left hand corner of the output if the output is a scalar and headings are not required see below then the selected cell is simply the location of the output 4 2 4 Statistical Add Ins for Excel User Guide For scalars and ordinary arrays press the Apply button This paste links the material to the specified location and leaves the NAG dialog box in place ready to re position other elements of the output To copy and close the dialog box press the End button To simply close the dialog box without copying press the Cancel button The Options button is used to tell NAGExtract how to unpack output that represents two dimensional arrays but which is output in some packed form Select the appropriate option The options also allow the user to specify transposition of vectors or to specify whether triangular matrices are to output as a list of elements or as a triangular array See Setting the Options for more detail The Help button brings up help on the NAG Add Ins An alternative approach is to us
56. bution Parameters Probability Required The probability from the required gamma distribution Shape_param Required The shape parameter a of the gamma distribution a gt 0 0 Scale_param Required The scale parameter b of the gamma distribution b gt 0 0 Tolerance Optional The relative accuracy required in the result Default 10 times machine precision 19 5 2 3 5 2 3 1 5 2 3 2 Statistical Add Ins for Excel User Guide Remarks The parameterization of the gamma distribution used by this function is such that if the shape parameter is a and the scale parameter is b the mean is ab and the probability density function is exp a 1 log x b x b bT a See Using the Add Ins and the Introduction to Basic Statistics for further information Summary Statistics Introduction Summary Statistics Two functions are available for the calculation of basic summary statistics SUMMARY_STATS provides the mean standard deviation and coefficients of skewness and kurtosis as well as the maximum and minimum values for a set of ungrouped data The function SSPT_SUMMARY provides the median along with the maximum minimum and the upper and lower hinges The hinges are virtually the same as the quartiles The five point summary is the basis of a box and whisker plot but can also be used directly to assess transformations for symmetry Contingency Tables A two way contingency table arises when a sampl
57. button on the NAGExtract dialog box The Available Data area will now contain 5 entries Minimum Lower Hinge Median Upper Hinge and Maximum This tells us that the items we can handle are five scalar elements so in this instance the values we can relocate consist of 5 real numbers Specify a new destination cell by deleting any default entry in the Destination Cell box in the NAGExtract dialog box and selecting a fresh cell in the workbook By default Minimum is selected as the data item to relocate If desired select a different entity with the mouse in the Available Data list Click on Finish in the dialog box and you should see the required output value paste linked to the new location The new version of NAGExtract provided with both these sets of Add Ins now by default provides a caption to the data items paste linked This facility may be turned off using the Options button of NAGExtract described later If we had chosen a different function then some of the output might have consisted of array information We would have used NAGExtract in exactly the same way noting that when we specified the Destination Cell it was necessary to specify just one location By convention this cell is deemed to be the top left hand corner of any output array Of course if all that NAGExtract did was simply to simulate the copy and Paste Link facility of Excel then it would be unnecessary to provide the macro It
58. ce covariance matrices are assumed to be unequal Default the within group variance covariance matrices are assumed to be unequal Approach Optional Indicates which approach is to be used E the estimative approach is used P the predictive approach is used Default the predictive approach is used Priors Optional The prior probabilities for allocation to the groups Default equal prior probabilities Output Output as a NAGExtract format list Allocated group Posterior probabilities Atypicality index Note Both the posterior probabilities and the atypicality index return one column for each group Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information DISCRIM_DIST Set 1 DISCRIM_DIST NAG GO3DBF computes Mahalanobis squared distances for group or pooled variance covariance matrices It is intended for use after DISCRIM_TEST Parameters Info Required The information provided by DISCRIM_TEST Either the complete output should be input or DISCRIM_TEST should be used as the argument Equal Required Indicates whether or not the within group variance covariance matrices are assumed to be equal and the pooled variance covariance matrix used E the within group variance covariance matrices are assumed to be equal U the within group variance covariance matrices are assumed to be unequal 76 5 5 3 7 Statistical Add Ins for Excel User Guide X Opti
59. cel User Guide quartimax rotation is performed Values of g between zero and one would give a compromise between the two methods PRIN_COMP Set 1 PRIN_COMP NAG GO3AAF performs a principal component analysis on a data matrix Parameters Data Required The array of observed variables in columns Matrix_type Required Indicates for which type of matrix the principal component analysis is to be carried out C for the correlation matrix S for a matrix standardized by values in User_std U for the sums of squares and cross products matrix V for the variance covariance matrix Std_scores Required Indicates if the principal component scores are to be standardized S they are standardized to have sum of squares equal to one U they are unstandardized and have sum of squares equal to the corresponding eigenvalue Z they are standardized so that they have unit variance E they are standardized so that they have variance equal to the corresponding eigenvalue Number_of_scores Optional The number of scores to be returned User_std Optional The standardizations to be used for each variable if Matrix_type S The mean corrected data values are divided by these values If User_std is set equal to the standard deviations it would be equivalent to analysing a correlation matrix Matrix_type C If present values must be gt 0 Weights Optional Contains the weights corresponding to the observations If Weights i
60. coefficients Two common methods of computing factor score coefficients are the regression method and Bartlett s method Bartlett s method gives unbiased estimates of the factor scores whereas estimates from the regression method are biased but have smaller variance than those from Bartlett s method see Lawley and Maxwell 1971 The function FACTOR computes the loadings and communalities Either the raw data or the correlation variance covariance matrix can be input The function FACTOR_SCORE can then be used to compute the factor score coefficients To compute the factor scores from the coefficients the data has to be standardized and then post multiplied by the factor score coefficients With both principal component analysis and factor analysis the pattern of the coefficients or loadings is used as a basis of interpreting the components or factors The interpretation is easier if the coefficients are either very small or large in absolute value This can often be achieved by using an orthogonal rotation Two methods are commonly used varimax and quartimax In both cases the rotation is chosen to maximize a criterion that involves the fourth power of the coefficients for further details see for example Lawley and Maxwell 1971 or Krzanowski 1990 The function ORTHOG_ROTATION computes the orthogonal rotation If the parameter G is set to unity then varimax rotation is performed and if G is set to zero 64 5 5 2 2 Statistical Add Ins for Ex
61. complete the installation process The Add Ins will not work if the installed materials are simply copied from one machine to another If the Add Ins are required on additional machines then check that the licence with NAG covers the additional machine and re install from the CD or file provided The DLL is installed into a directory that must be on the user s PATH and the installation procedure asks for permission to edit the autoexec bat file to ensure this We recommend that you allow this To verify that the installation process has completed correctly re boot the system Open Excel with a new workbook From the Tools Add Ins menu load the NAG Add Ins by ensuring that the appropriate NAG check boxes contain a tick If not use the left button of the mouse to click on the appropriate check box If the NAG Add Ins load without any warnings then the system information has been correctly installed As a simple run time test if you have the NAG Statistical Add Ins for Excel Modelling and Multivariate Methods product then select a cell on a worksheet and enter the formula CHI_PPT 0 5 1 If the cell evaluates to 0 454936 then all is well and the DLL is indeed being correctly accessed If you have the companion product for nonparametric methods then PROPORTION Cl is a suitable test function Select a cell on a worksheet and enter the formula PROPORTION_Cl 2 4 The cell will evaluate to lt List of 5 gt if the product has been s
62. ction Introduction to Multivariate Methods A multivariate data set consists of several variables recorded on a number of objects or individuals For example the amounts of different minerals present in a number of soil samples or different social indicators for regions of a city Multivariate methods can be classified as those that seek to examine the relationships between the variables e g principal components known as variable directed methods and those that seek to examine the relationships between the objects e g cluster analysis known as individual directed methods Please note that a complete set of examples for this Add In are provided in the workbook Book3E xls This is installed in the Examples subdirectory of the Add Ins directory You might find it helpful to work through the examples to familiarize yourself with the use of the Book 3 Add Ins Variable Directed Methods Introduction Variable directed methods Most variable directed methods seek to find derived variables that represent linear relationships between the variables with the aim of reducing the dimensionality of the problem by considering only a small number of these derived variables The two most common methods are principal component analysis and factor analysis Principal component analysis finds new variables that are linear combinations of the observed variables so that they have maximum variation and are orthogonal uncorrelated The required v
63. cts consist of one array for each term and the standard errors array contains all the corresponding standard errors for the difference in pairs of treatment means Remarks See Using the Add Ins and the Introduction to the Analysis of Designed Experiments for further information 50 5 4 3 5 4 3 1 Statistical Add Ins for Excel User Guide Generalized Linear Models Introduction Introduction to Generalized Linear Modelling using the NAG Excel Add Ins Background Generalized Linear Models are an important extension to linear regression models that are fitted by the function MULT_LIN_ REG Examples of generalized linear models are 1 The rational polynomial model with Normal errors y 1 a bx cx E Logistic regression where y has a binomial distribution with mean u where log u N 4 n ZX Bx 3 Log linear models where y can be considered as coming from a Poisson distribution with mean u where log u 7 X BX Dobson 1990 provides an introduction to generalized linear models while further details can be found in McCullagh and Nelder 1989 A generalized linear model consists of three components 1 An error distribution for the response variable y 2 A link function relating the mean 4 of the distribution to a linear predictor 7 n 9 7 3 A linear predictor X 3x For the NAG Excel Add Ins the following range of models is available Error distributions Normal binomial Poisson gamma
64. d G a logistic link is used P a probit link is used C a complementary log log link is used 55 Statistical Add Ins for Excel User Guide Y Required The response variable Denominator Required The binomial denominator X Required Either the array containing the independent variables as columns or the function MM_GLM that computes the model matrix Select_x Optional Indicates which independent variables in array X are to be included in the model If Select_x j gt O then the variable contained in the jth column of array X is included in the model if Select_x j 0 it is excluded If Select_x is present then its elements gt 0 Default all X variables are included X_names Optional Names for the explanatory variables Default X_1 etc Offsets Optional Offsets for model Default none Weights Optional User weights If a weights is set to zero the observation is not included in the model in which case the effective number of observations is the number of observations with non zero weights If present Weights 0 0 Default no weights Mean Optional Indicates if a mean term is to be included M a mean term intercept is to be included in the model Z the model will pass through the origin zero point Default mean term included Fitted_values Optional Indicates if fitted values etc are to be returned N No fitted values etc are returned Y Fitted values and residual
65. d 1 0 0 0 Blue 0 1 0 0 As the sum of the dummy variables is one for each observation the set of dummy variables will not be of full rank if a mean term is fitted equivalent to 1 for each observation So usually dummy variable 1 is not included so that the other dummy variables represent comparisons with this first class in this case red Alternatively the comparison can be with the last or the average of the previous classes Helmert matrix When each level represents some underlying quantitative variable an orthogonal polynomial representation can be used In addition to the effect of a single classification variable there can be the effect of the combination of two classification variables For example if there were also a classification by sex then there would be an effect due to colour and an effect due to sex plus an effect due to 23 Statistical Add Ins for Excel User Guide the combination of a sex and colour For instance if females were in general larger than males and reds were in general larger than blues greens and browns it may be however that green males were even smaller than expected or that blue males were bigger than red females Such effects are known as interactions and indicate that the combination of colour and sex should be examined rather than just examining one variable at a time The function MULT_LIN_REG fits a multiple linear regression model There are two ways of specifying the independent variables
66. e call PROPORTION_CI A1 B1 is then equivalent to the previous usage See Array Functions for more information Parameter Input The parameters of the NAG functions may require input of the following types Single real number Single integer Single string A list or one dimensional array of cells Two dimensional array of cells e rows and columns Single numbers or characters can be entered directly into the function wizard or by reference to cells In the case of the string input only the first character is considered and it is not case sensitive so Y yes Ya will be treated identically When a list of names or a single variable is required this can be input either as a column selection of cells or as a row selection of cells Alternatively it can be entered into the function wizard as a list between curly brackets for example 1 0 1 0 Where two dimensional arrays are expected even if in a particular case there is only one column they must be entered by reference to the cells Variables are always assumed to be stored column wise Array Functions Many statistical calculations return more than a single value as their result Sometimes they might return a number of scalar values as in the case of S5PT_SUMMARY whilst at other times they might return a number of arrays of different shapes and sizes together with scalar output In order to preserve the automatic updating property of functions NAG elected to imple
67. e different frequencies A plot of the spectrum against the frequency is known as a periodogram Spectral analysis is useful in identifying cyclic components in a series The computed sample spectrum can be smoothed to aid interpretation or to provide an estimate of the underlying spectral density The function SPECTRAL smoothes the sample spectrum it calculates by using a trapezium frequency window known as a Daniell Window The shape of the window can vary from triangular to rectangular and the amount of smoothing is controlled by the width of the window While the unsmoothed sample spectrum ordinates are independent for the smoothed spectrum they are only independent if they are the amount given by the bandwidth statistic apart The distribution of the spectrum is approximately that of a scaled y2 variate with degrees of freedom as returned by SPECTRAL This approximate distribution gives a multiplicative or additive if the log spectrum is selected confidence interval for the spectrum The function SPECTRAL computes the sample spectrum by means of a fast Fourier transform FFT For efficiency the length of the series passed to the FFT should be at least twice the length of the observed series and a product of small prime numbers To achieve this with a series of arbitrary length the series is extended with zeros until it is of the required length In order to provide a smooth transition to the zero values the data can be tapered at both ends 33
68. e has been classified by two qualitative variables for example sex male female and occupation type managerial professional skilled unskilled The numbers in each of the resulting 2 by 4 categories form the cells of the two way contingency table these observed numbers are denotes by Oj The standard analysis is the chi squared test This computes what are the expected numbers denoted by Ej in each cell if there was no association between the two qualitative variables that define the table The resulting statistic is xX Sle E E where the sum is over all cells An alternative statistic is the likelihood ratio test statistic given by Luz In the case when both qualitative variables have just two levels the chi squared statistic for the resulting 2 x 2 table can be improved by Yates correction factor and for 2 x 2 tables with small numbers in each cell Fisher s exact test can be computed The function TWO_WAY_TABLE computes both the chi squared and likelihood ratio statistics for a two way table If the total sample size in a 2 x 2 table is less than or equal to 40 Fisher s exact test is performed If the expected value of a cell is less than 0 5 a warning is given since the distribution of the resulting chi squared statistic may not be well approximated by the chi squared distribution For further information see for example Everitt 1977 SUMMARY_STATS Set 1 SUMMARY_STATS NAG GO1AAF calculates the mean standard deviati
69. e t treatment combinations derived from the levels of all factors then either there are no blocks or the blocks are of size t units Other designs include partially balanced incomplete block designs split plot designs factorial designs with confounding and fractional factorial designs For further information on these designs see Cochran and Cox 1957 Davies 1978 or John and Quenouille 1977 Many of the common experimental design can be analysed using the NAG Excel Add Ins Three functions are available 1 ANOVA_BLOCK handles completely Randomized designs as well as any block design with equal sized blocks For example Randomized block BIBs PBIBs can all be handled 2 ANOVA_ROW_COLUMN handles row and column designs For example Latin squares and lattice squares 3 ANOVA_FACTORIAL handles complete factorial designs that may be unconfounded in blocks Other designs can be analysed by combinations of calls to the functions For example a factorial experiment in a Latin square design can be analysed by first using ANOVA_ROW_COLUMN then passing the results to ANOVA_FACTORIAL for further analysis Itis also possible to analyse split plot designs factorial experiments in which some effects have been confounded with blocks or some fractional factorial experiments The Analysis of Variance The analysis of a designed experiment usually consists of two stages The first is the computation of the estimate of variance of the underly
70. e the Excel function INDEX This allows you to pick out a single element of an array So for example INDEX PROPORTION_CI 5 10 3 would calculate the third element of the NAGExtract format list produced by the function PROPORTION_CI Of course INDEX may also be used on the NAGExtract format list itself so for example INDEX A1 A5 3 would copy the third element of the NAGExtract format list stored in A1 A5 Setting the Options The Options Dialog Box The Options dialog box contains a set of radio buttons and check boxes which control the output of arrays and vectors Once your selection has been made click on OK to close the window Changes to the default options are only temporary The next time you call NAGExtract or the options menu of NAGExtract the options will return to the default settings Two dimensional Array Handling Options The default handling of two dimensional arrays is to pack them in row column style or row order the Fortran and Visual Basic convention However if you so wish the two dimensional arrays can either be transposed with the Pack 2 D arrays in Column Order radio button or decomposed into a one dimensional list with the Pack 2 D arrays as a List option Transpose Vectors One dimensional Arrays Option This option allows you to change the way NAGExtract outputs one dimensional arrays from columns to rows If the default is un checked the vectors are output in a column Output Triangul
71. earle S R 1971 Linear Models Wiley Siegel S 1956 Nonparametric Statistics for the Behavioral Sciences McGraw Hill Snedecor G W and Cochran W G 1967 Statistical Methods lowa State University Press Wetherill G B 1972 Elementary Statistical Methods Chapman and Hall Winer B J 1970 Statistical Principles in Experimental Design McGraw Hill 108
72. ected Required The expected values for each class These can be either probabilities or expected frequencies The values will be scaled to sum to the total number observations Estim_param Optional The number of estimated parameters used to calculate the fitted values Default 0 Output Output as a NAGExtract format list Chi squared value Chi squared significance level Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information COX_STUART Set 2 COX_STUART NAG GO8AAF computes the Cox Stuart test for trend with ordinal from one sample Parameters Series Required The observed series Tail Optional Indicates the critical region and hence the form of H7 T two tailed test i e test for increasing or decreasing trend L lower tailed test i e test for decreasing trend U upper tailed test i e test for increasing trend Default two tailed test Output Output as a NAGExtract format list Rank sum Significance level 85 6 2 3 4 6 2 3 5 Statistical Add Ins for Excel User Guide Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information RUNS_TEST Set 2 RUNS_TEST NAG GO1ALF GO1EAF computes the runs test for a single series of ordinal data Parameters Series Required The observed series Cut_Point Optional The cut point that defines the runs below values lt cut point or
73. ed to DISCRIM_ALLOC that will then allocate observations to groups on the basis of this information The choice of pooled or group variances is available as is the choice between the estimative and the predictive approach Posterior predictive probabilities as well as atypicality indices are computed The function DISCRIM_DIST can be used to compute Mahalanobis distances after the use of DISCRIM_TEST Either the between group means or between observations and group means are produced An alternative approach to discrimination is the logistic discrimination This does not depend on the assumption that the data has a Normal distribution In the case of two groups logistic regression can be performed with the response variable indicating the group allocation and the variables in the discriminant analysis being the explanatory variables Allocation can then be made on the basis of the fitted response value This approach can be shown to be valid for a wide range of distributional assumptions The function BINOMIAL_GLM is available in Book 2 Scaling methods Scaling methods seek to represent the observed dissimilarities or distances between objects as distances between points in Euclidean space For example if the distances between objects A B and C were 3 4 and 5 respectively the distances could be represented exactly by three points in two dimensional space Only their relative positions would be important the whole configuration of points could be ro
74. er sales are higher than winter sales or there are more enquires on a Monday morning than on a Friday afternoon The random component is the fluctuation around the trend and cycles Unlike regression models the noise in time series models is not assumed to be independent but is auto correlated Furthermore the noise is usually assumed to have zero mean and to be stationary Being stationary means that the statistical relationship between the observation at time t and time t is the same as the statistical relationship between observations at time t t and t z The relationship can be described in terms of either their joint distribution or in terms of correlation The theoretical correlation between an observation at time t and an observation at time t that is at lag is denoted by p Statistical models for time series relate the correlated noise denoted by w to an independent random component or white noise denoted by e and hence explain the autocorrelation n The most common models for stationary time series are autoregressive models and moving average models An autoregressive model for a series w of order p is Wt O W 1 OWz2 OpWipt E which is denoted by AR p A moving average model of order q is W2 0 Er1 00 9 eS Og t g which is denoted by MA q The difference between the two forms of model is that the autocorrelation for an autoregressive model gradually dies away while the autocorrelation f
75. ers of observations Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information Two Samples CHI_SQ2 Set 2 CHI_SQ_2 NAG G11AAF computes the chi squared test for two independent samples of nominal data Parameters Data Required The n rows by 2 columns array of the observations where n is the number of classes n must be greater or equal to 2 Output Output as a NAGExtract format list Significance Chisquared Degrees of freedom Expected values Chisquared contributions The significance returned is for the chi squared statistic except for a 2 x 2 table when the total number of observations is less than or equal to 40 in which case the probability from Fisher s exact test is returned The chi squared contributions are the values of observed expected expected for each cell which make up the chi squared statistic 90 6 2 5 2 6 2 5 3 Statistical Add Ins for Excel User Guide Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information KOLMOGOROV_2 Set 2 KOLMOGOROV_2 NAG GO8CDF computes the sample Kolmogorov Smirnov test Parameters Xx Required The values of the first sample Y Required The values of the second sample Tail Optional Indicates the critical region and hence the form of H4 T two tailed test i e distributions are not equal L lower tailed test e X values tend to
76. ficients from the result of fitting a factor analysis model by maximum likelihood as performed by FACTOR Parameters Method Required Indicates which method is to be used to compute the factor score loadings R the regression method is used B Bartlett s method is used Loadings Required The matrix of unrotated factor loadings Psi Required The values of the unique component of the variation y as returned by FACTOR Eigenvalues Required The eigenvalues as returned by FACTOR Rotation Optional If rotations have been used the orthogonal rotation matrix as returned by ORTHOG_ROTATION Output Output as a NAGExtract format list Factor Score Coefficients Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information ORTHOG_ROTATION Set 1 ORTHOG_ROTATION NAG GO3BAF computes orthogonal rotations for a matrix of loadings Parameters Loadings Required The array of loadings one column per factor 67 5 5 3 5 5 3 1 Statistical Add Ins for Excel User Guide G Optional G 1 0 gives varimax rotations and G 0 0 giving quartimax rotations If present g 2 0 0 Default G 1 0 i e varimax rotations Stand_rows Optional Indicates if the matrix of loadings is to be row standardized before rotation S loadings are row standardized U loadings are unstandardized Default S standardized Acc_req Optional The accuracy required If present
77. fitted ARIMA model Parameters No_of forecasts Required The required number of forecasts Series Required The original undifferenced time series Model Required The model specification as given to ARIMA_FIT Model must specify the orders p d q P D Q s of the seasonal ARIMA model to be used in the forecast number of autoregressive terms order of non seasonal differencing number of moving average terms number of seasonal autoregressive terms order of seasonal differencing number of seasonal moving average terms the seasonality vn 00UTU2 LS Parameters Required The parameters for the ARIMA model specified in Model That is the p values for the autoregressive parameters the q values of the moving average parameters the P values of the seasonal autoregressive parameters and the Q values of the seasonal moving average parameters in that order Constant Optional The estimated constant of the model fitted to the series Default no constant in model i e fixed at zero Output Output as a NAGExtract format list Forecasts Standard Errors of Forecasts Remarks See Using the Add Ins and the Introduction to Time Series Methods for further information TRNS_FUNC_FORECAST Set 1 TRNS_FUNC_FORECAST NAG G13BJF produces forecasts of a time series which depends on one or more other series via a previously estimated multi input model Parameters Y_series Required The output
78. functions the distribution functions return a single value rather than an array of values so the NAGExtract utility is not required NORMAL_PROB Set 1 NORMAL_PROB NAG G01EAF returns as a single value a one or two tail probability for the standard Normal distribution Parameters X Required The value of the standard Normal variate Tail Required Indicates which tail the return probability should represent U The upper tail probability is returned L The lower tail probability is returned S The two tail significance level is returned C The two tail confidence interval probability is returned Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information 14 5 2 2 3 5 2 2 4 Statistical Add Ins for Excel User Guide NORMAL_PPT Set 1 NORMAL_PPT NAG GO1FAF returns as a single value the deviate associated with the given probability of the standard Normal distribution Parameters Probability Required The probability of the standard Normal distribution as defined by tail Tail Required Indicates which tail the supplied probability represents U The upper tail probability L The lower tail probability S The two tail significance level probability C The two tail confidence interval probability Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information MULTI_NORMAL_PROB Set 1 MULTI_NORMAL_PROB NAG G
79. generalized linear model functions also return the linear predictor and the working weights used in the iterative re weighted least squares The variance standardization is also returned for the binomial Poisson and gamma distributions Specifying the Model Matrix A simple term selection mechanism is given by the optional Select_x parameter of the generalized linear model functions This allows columns from the X array to be excluded from the model A more powerful model specification is provided by the NAG MM_GLM function The NAG function MM_GLM can be used to specify a model matrix rather than the user constructing a set of columns The inputs to the model matrix function MM_GLM are 1 Group variables with their names 2 Covariates with their names 3 A formula Group variables are classification variables e g eye colour sex These variables can be either text e g red green blue or numerical 100 150 300 MM_GLM re codes these values first into an integer representation 1 2 then into the dummy variables required for the model fitting Covariates are ordinary variables e g weight temperature A formula uses the variable 52 5 4 3 2 Statistical Add Ins for Excel User Guide names and the symbols oe Some examples of formulae are Group variables a b and c Covariates x and z a x fits the group variable a and covariate x that is regression model with different intercepts for the different groups x
80. gister If you would be kind enough to complete any or all of the questions below it will help us to continue to enhance and improve the quality products and services that we provide 1 How easy did you find it to order the software Please tick as appropriate Very Easy U Easy UU Quite Difficult O Very Difficult 1 If Quite Difficult or Very Difficult please specify why 2 How easy did you find the installation of NAG Statistical Add Ins for Excel Please tick as appropriate Very Easy U Easy LU Quite Difficult O Very Difficult O If Quite Difficult or Very Difficult please specify why 3 How useful do you find the Windows Help for the NAG Statistical Add Ins Please tick as appropriate 105 a Statistical Add Ins for Excel User Guide Very Useful U Quite Useful 1 Of Some Use U Not Useful If Of Some Use or Not Useful please specify why 4 5 Do you like a printed manual Please tick as appropriate YES J NO 1 How do you use the NAG Statistical Add ins Please specify 6 Did the NAG Statistical Add Ins replace an existing package Please tick as appropriate YES q NO q If YES please specify what Do the NAG Statistical Add Ins meet your needs Please tick as appropriate YES O NO O If NO please specify why 9 Would you be interested in NAG Statistical Add Ins that covered the following areas Please tick as appropriate
81. group must have at least as many members as there are variables in the analysis X Required The array containing variables to be used in the analysis in columns Select_x Optional If Select_x gt O then the variable in the ith column of the array X is included in the analysis otherwise it is excluded Default all columns included Weights Optional If required the weights for each the observations If Weights i 0 0 then the ith observation is not included in the analysis If present Weights gt 0 0 Output Output as a NAGExtract format list 77 5 5 3 8 Statistical Add Ins for Excel User Guide Test statistic Degrees of freedom Significance Discrim Info Number in group Group means Note Since DISCRIM_TEST is intended for use before DISCRIM_ALLOC and DISCRIM_DIST the output contains information required by these functions This is held in Discrim Info Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information MDS Set 1 MDS NAG GO3FCF performs non metric ordinal multidimensional scaling Parameters Distance_matrix Required The array containing strictly lower triangle of the distance matrix i e no diagonal elements Note that negative distances are assumed to represent missing values Initial_x Required The array of the initial estimates of the coordinates for point one column per dimension required Often the results from METRIC_SCA
82. he Dendrogram information produced by CLUSTER Output Output as a NAGExtract format list Number of Clusters Cluster Distance Group Index Note The group index will be for objects in dendrogram order unless Index is provided in which case it will be in the order of the original data Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information DISCRIM_ALLOC Set 1 DISCRIM_ALLOC NAG G03DCF allocates observations to groups according to selected rules It is intended for use after DISCRIM_TEST Parameters Info Required The information provided by DISCRIM_TEST Either the complete output should be input or DISCRIM_TEST should be used as the argument Xx Required The array of observations to be allocated using the information in Info X along with Select_x if present must be compatible with the input to DISCRIM_TEST 75 5 5 3 6 Statistical Add Ins for Excel User Guide Select_x Optional If Select_x gt O then the variable in the ith column of the array X is included in the analysis otherwise it is excluded This should be compatible with the input to DISCRIM_TEST Default all columns are included Equal Optional Indicates whether or not the within group variance covariance matrices are assumed to be equal and the pooled variance covariance matrix used E the within group variance covariance matrices are assumed to be equal U the within group varian
83. he difference in the degrees of freedom associated with the deviances For Normal errors an F statistic is computed by dividing the difference in deviance residual sum of squares by the difference in degrees of freedom and then dividing by the estimate of the scale parameter This is calculated as the residual sum of squares divided by residual degrees of freedom from either the larger model or a suitable maximal model An individual parameter can be tested against the hypothesis that it is zero using the statistic Parameter Estimate Standard Error This may be compared to a Normal distribution in the case of binomial and Poisson errors For Normal errors the t distribution can be used with degrees of freedom taken as the degrees of freedom associated with the deviance or residual sum of squares Model checking In addition to testing the model parameters the residuals from the model should be examined For Normal binomial and Poisson errors the deviance residuals are computed In the case of the Normal errors the deviance residual is simply y u For gamma errors Anscombe residuals that are defined for y O are used In addition to residuals leverages are computed These show the influence of the observation on the fitted value A large value indicates that the fitted model may be unduly influenced by that observation Leverages can also be used to standardize residuals See STAND_RESIDS for more information For advanced use the
84. he first kind The degrees of freedom for the t F and w distributions may be non integer The accuracy of these probability functions varies with their intended use For GAMMA_PPT BETA_PROB and BETA_PPT there is an option to set the required accuracy For NORMAL_PROB NORMAL_PPT and GAMMA_PROB the accuracy should be about 16 decimal places For the remainder the accuracy is about 5 decimal places for normal usage Further information can be found in the NAG Fortran 77 Library Documentation In addition to the above univariate distributions there is also a function that returns the probability for the multivariate normal distribution This requires the input of the variance covariance matrix for the distribution as well as its mean vector Several of the NAG distribution functions have an option for the selection of the tail of the distribution The values available will depend on the function but the possibilities are Lower tail Upper tail Two tail for significance testing Two tail for confidence intervals RAMa The two tail options are only available for the symmetric distributions Normal and t distribution For the significance testing option the sum of the two equal tail probabilities is used that is 0 05 would be used for a two tailed test at the 5 level In the case of the confidence interval option the central symmetric probability is used that is 0 95 would be used for a 95 confidence interval Unlike the rest of the NAG
85. he set of points that minimize the STRESS STRESS is measured by how well the set of points preserve the order of the distances in the original distance matrix The computation of STRESS involves the monotonic regression of the fitted distances on the observed distances For example Observed Distance 1 05 1 25 1 75 2 6 Fitted Distance 0 93 2 62 1 21 3 55 Monotonic Distance 0 93 1 85 1 85 3 55 The STRESS is then computed from the sum of squared differences between the fitted distance and the monotonic distance divided by the sum of squared fitted distances As an alternative to STRESS squared STRESS SSTRESS can be used in which squared distances are preferred instead of the distances themselves The function MDS performs non metric scaling with a choice of STRESS or squared STRESS In addition to the points MDS has the option of returning the fitted distances and the monotonic distances These can be either as a distance matrix corresponding to the input distance matrix or as arrays ordered by the observed distances The latter can be used in plots to assess the appropriateness of the method MDS requires the input of an initial configuration of the points This can usually be provided by METRIC_SCAL Unlike METRIC_SCAL MDS allows missing values in the distance matrix to be specified as negative distances the algorithm may still work provided there are not more than two thirds of the values missing 72 5 5 3 2 5 5 3 3 Statistical Add
86. iable is omitted Output Output as a NAGExtract format list Partial Correlation Matrix Partial Covariance Matrix Note Both are returned as upper triangular matrices The partial covariance matrix contains the true partial variance covariance matrix if a variance covariance matrix is supplied and a variance covariance matrix for standardized variables if a correlation matrix is supplied Remarks See Using the Add Ins and the Introduction to Regression and Correlation for further information MULT_LIN_REG Set 1 MULT_LIN_REG NAG GO02DAF fits a general multiple linear regression model Parameters Y Required Observations on the dependent variable Xx Required Either the array containing the independent variables as columns or the function MM_MLR that computes the model matrix Note that X_names and Select_x should not be used if MM_MLR is used Select_x Optional Indicates which independent variables in array X are to be included in the model if Select_x j gt 0 then the variable contained in the jth column of array X is included in the regression model if Select_x j 0 it is excluded If present Select_x values gt 0 Default all X variables are included X_names Optional Names for the independent variables in array X Default X_1 X_2 etc Mean Optional Indicates whether a mean term is to be included M A mean term intercept will be included in the model Zz The model will pas
87. in either all Os or all 1s there are more than 5 rows and 25 observations Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information FRIEDMAN Set 2 FRIEDMAN NAG GO8AEF computes the Friedman test for ordinal data from k matched samples Parameters Data Required The n rows by k columns array of the observations where n is the number of blocks and k is the number of treatments Both n and k must be greater than 1 Output Output as a NAGExtract format list Test statistic Significance level The significance level is based on a chi squared approximation that is reasonable provided k 2 andn gt 19ork 3andn gt 9ork 4orn gt 4o0rk gt 4 Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information KRUSKAL_WALLIS Set 2 KRUSKAL_WALLIS NAG GO8AFF computes the Kruskal Wallis test for k independent samples or groups of ordinal data Parameters Data Required A column containing the observations for all samples 94 6 2 7 6 2 7 1 6 2 7 2 Statistical Add Ins for Excel User Guide Groups Required A column containing the indicators as to which group sample each observation belongs The indicators can be text or numeric The number of groups will be calculated from the number of distinct values in Groups Output Output as a NAGExtract format list Test statistic Significance level Remarks See Us
88. included in the model Z the model will pass through the origin zero point Default M mean term included Power Optional If an exponent link is selected then the power of the exponent If present Power 0 0 58 Statistical Add Ins for Excel User Guide Fitted_values Optional Indicates if fitted values etc are to be returned N No fitted values etc are returned Y Fitted values and residuals are returned A All of fitted values residuals leverages linear predictor variance function and working weights are returned Default Y Fitted values and residuals are returned Max_iterations Optional The maximum number of iterations for the iterative weighted least squares fitting If present Max_iterations gt 0 Default 10 Tolerance Optional Indicates the accuracy required for the fit of the model If present Tolerance gt 0 0 Default 0 000001 Eps Optional The value of eps is used to decide if the independent variables are of full rank and if not the rank of the independent variables If present eps gt 0 0 Default 0 00000000001 Output Output as a NAGExtract format list Deviance df Rank of Model Parameter Estimates Cov If requested see Fitted_values the following are returned Fitted values Residuals Leverages Linear predictor Variances Working weights Notes The deviance and its degrees of freedom df are returned providing the degrees of freedom are not zero
89. ing random variation in the experiment along with tests for the overall effect of treatments This results in an analysis of variance ANOVA table The second stage is a more detailed examination of the effect of different treatments either by comparing the difference in treatment means with an appropriate standard error or by the use of orthogonal contrasts The analysis assumes a linear model such as Y lt O T Ej where y is the observed value for unit j of block i u is the overall mean is the effect of the ith block 7 is the effect of the th treatment which has been applied to the unit and g is the random error term associated with this unit The expected value of e is zero and its variance o In the analysis of variance the total variation measured by the sum of squares of observations about the overall mean is partitioned into the sum of squares due to blocks SSz the sum of squares due to treatments SSy and a residual or error sum of squares SSe This partition corresponds to the parameters 6 T and o In parallel with the partition of the sum of squares there is a partition of the degrees of freedom associated with the sums of squares From these the mean squares are calculated as the ratio of the sum of squares divided by the degrees of freedom Finally a F test for an overall effect of the treatments can be calculated as the ratio of the treatment mean square to the residual mean square These results are u
90. ing the Add Ins and the Introduction to Nonparametric Methods for further information Correlation and Association CONTINGENCY C Set 2 CONTINGENCY_C NAG G11AAF computes the contingency coefficient for nominal data Parameters Data Required The n rows by k columns array of the observations where k is the number of samples being compared and n is the number of classes n must be greater than 1 and k must be greater or equal to 2 Output Output as a NAGExtract format list Contingency Coefficient Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information KENDALL_CC Set 2 KENDALL_CC NAG GO8DAF computes Kendall s coefficient of concordance Parameters Data Required The n rows by k columns array of the observations where k is the number of objects being compared and n is the number of comparisons Both n and k must be greater than 1 Output Output as a NAGExtract format list Kendall s coefficient of concordance Approximate significance level Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information 95 6 2 7 3 6 2 7 4 Statistical Add Ins for Excel User Guide KENDALL_TAU Set 2 KENDALL_TAU NAG GO2BNF computes the Kendall s rank correlation coefficient lt Parameters Data Required The n rows by k columns array of the data where k is the number of variables and n is the number of
91. ing to the model specified in T_Model Method Optional Indicates the criterion used in estimating the parameters of the model The value 1 gives least squares 2 gives exact likelihood and 3 gives marginal likelihood Default with no input series exact likelihood with input series marginal likelihood Num_iterations Optional The maximum required number of iterations If Num_iterations 0 no change is made to any of the model parameters except that the constant and any transfer function model parameters relating to simple input series are estimated Tolerance Optional Tolerance should be set to the required convergence criterion Default 0 0000001 Output Output as a NAGExtract format list Deviance Degrees of Freedom Variance AIC Parameter Estimates Parameter Correlations Residuals Note The Parameter Estimates array consists of the parameters names the estimates the t values for the estimates and their significance The parameters are in the order ARMA parameters autoregressive moving average seasonal autoregressive and seasonal moving average then transfer function parameters omega parameters followed by delta parameters and finally the constant Remarks See Using the Add Ins and the Introduction to Time Series Methods for further information 37 5 3 3 7 5 3 3 8 Statistical Add Ins for Excel User Guide ARIMA_FORECAST Set 1 ARIMA_FORECAST NAG G13AJF forecasts from a
92. ion MM_MLR passes the results to the calling function by means of global variables All that is returned directly is an indicator string or any error messages The function computes all required dummy variables with appropriate names The dummy variable corresponding to the first level of a group variable is not included in order to provide a full rank model Remarks See Using the Add Ins and the Introduction to Regression and Correlation for further information 28 5 3 2 6 5 3 2 7 Statistical Add Ins for Excel User Guide DURBIN_WATSON Set 1 DURBIN_WATSON NAG G02FCF calculates the Durbin Watson test for serial correlation in linear regression Parameters No_of_params Required The number of independent variables in the regression model including the mean Residuals Required The residuals from the regression Note that the mean of the residuals must be close to zero approximately 10 8 Output Output as a NAGExtract format list Durbin Watson Statistic Lower Bound for Significance Upper Bound for Significance Note The Durbin Watson test does not have an exact level of significance only an upper or lower bound The statistics returned are for the test for positive auto correlation for negative auto correlation the value 4 the Durbin Watson statistic should be calculated and compared with tables Remarks See Using the Add Ins and the Introduction to Regression and Correlati
93. is most useful when the output from a NAG routine includes matrix or vector output The underlying routines present these in different forms which need to be interpreted correctly by NAGExtract and expanded to fill the correct cells of the worksheet In the simplest case a list which is output as part of a column vector may represent a column vector or a row vector The Options button on the NAGExtract dialog box allows you to specify which is required Similarly array information may be stored by row or by column or may represent a triangular matrix Again the representation is specified using the Options button Getting Online Help Online help may be obtained at any time from the Nag menu by clicking the NAGHelp menu item Additionally help on a specific routine may be obtained by clicking the Help button on the function wizard once a particular routine has been selected Error Messages If the input to a NAG Excel function is invalid an error message will be returned rather than any numerical output If the error is one of value or shape rather than type for example if the formula CHI_PPT 2 0 1 was entered the following message would be returned Error Probability gt 1 Similarly if in a regression model fitted by the function MULT_LIN_REG the Y and X arrays were of different lengths the following error message would be returned Error y and x of different lengths 3 8 Statistical
94. istances Often there will be a need to scale the variables to produce satisfactory distances For discrete variables there are various measures of similarity or distance that can easily be computed For example for binary data a measure of dissimilarity could be O if the individuals take the same value 1 otherwise As DIST_MAT has an update option distances computed in this way can be combined with the Euclidean distances given above Given a measure of distance between individuals there are three basic tasks that can be performed 1 Group the individuals that is collect the individuals into groups so that those within a group are closer to each other than they are to members of another group 2 Classify individuals that is if some individuals are known to come from certain groups allocate individuals whose group membership is unknown to the nearest group 3 Map the individuals that is produce a multidimensional diagram in which the distances on the diagram represent the distances between the individuals In the above 1 leads to cluster analysis 2 leads to discriminant analysis and 3 leads to scaling methods Hierarchical cluster analysis Approaches for cluster analysis can be classified into two types hierarchical and non hierarchical Hierarchical cluster analysis produces a series of overlapping groups or clusters ranging from separate individuals to one single cluster This forms a tree The clusters at a le
95. l Runs test RUNS_TEST Wilcoxon one sample test WILCOXON Matched pairs samples McNemar test MCNEMAR Sign test SIGN_PAIRS Wilcoxon matched pairs test WILCOXON_PAIRS Two samples Chi squared test for 2 samples CHI_SQ_2 Kolmogorov Smirnov two sample test KOLMOGOROV_2 Mann Whitney test MANN_WHITNEY Median test MEDIAN_TEST Two sample confidence interval TWO_SAMPLE_CI K samples Cochran Q test COCHRAN_Q Chi squared test for k samples CHI_SQ_K Friedman test FRIEDMAN Kruskal Wallis test KRUSKAL_WALLIS Correlation and association Contingency coefficient CONTINGENCY_C Kendall s Coefficient of Concordance KENDALL_CC Kendall s tau rank correlation KENDALL_TAU Spearmans s rho rank correlation SPEARMAN_RHO An online version of the table is available using the function NONPARAMETRIC_INDEX Index of Nonparametric Methods NONPARAMETRIC_INDEX Set 2 NONPARAMETRIC_INDEX returns a list of the nonparametric functions available in the NAG Add Ins Parameters Class Optional Indicates which class of functions is to be returned 4z0 gt all functions one sample tests and confidence intervals matched pairs tests two sample tests and confidence intervals 83 6 2 3 6 2 3 1 Statistical Add Ins for Excel User Guide K K sample tests C rank correlation coefficients Default all functions Data_type Optional
96. l tests have been derived However there are many circumstances in which the assumption of Normally distributed data may not be valid and for small samples it is very difficult to validate the assumption of Normally distributed data with any degree of confidence The second advantage is that the data may not be actual measurements but are either in the form of classes or in terms of relative comparisons i e ranks Nonparametric tests generally only require the number in a particular category or the ranks for a set of data In order to consider this second point further it is useful to examine a fourfold categorisation of data nominal categorical ordinal interval and ratio 1 The nominal scale is used only to categorise data for each category a name perhaps numeric is assigned so that two different categories will be identified by distinct names for example red green and blue 2 The ordinal scale as well as categorising the observations orders the categories Each is assigned a distinct identifying symbol in such a way that the order of the symbols corresponds to the order of the categories for example small medium and large The most common system for ordinal variables is to assign numerical identifiers to the categories though if they have previously been assigned alphabetic characters these may be transformed to a numerical system by any convenient method which preserves the ordering of the categories 3 The interval sca
97. le not only categorises and orders the observations but also quantifies the comparison between categories this necessitates a common unit of measurement and an arbitrary zero point for example temperature in degrees Celsius 4 The ratio scale is similar to the interval scale except that it has an absolute as opposed to arbitrary zero point for example weight in grams Nonparametric statistics are conveniently divided into those that operate on nominal data and those that require ordinal data Clearly any test can be used on data of a higher level but there may be some loss of information 81 Statistical Add Ins for Excel User Guide Nonparametric statistics can also be categorised by their functions as follows One sample Matched pairs samples Two samples K samples Correlation ORON For the first four categories the most common functions available are for statistical tests In particular for ordinal data the nonparametric tests can be seen as alternatives to the more usual t and F tests for Normally distributed data The rank based correlation methods provide an alternative to the product moment correlation coefficient For nominal data there are two situations One is when there are only two categories binary data in which case the data can be considered to come from a binomial distribution In addition to statistics that explicitly use the binomial distribution specialist tests also exist for this type of data for e
98. limit Upper limit Estimated confidence Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information K Samples CHI_SQ_K Set 2 CHI_SQ_K NAG G11AAF computes the chi squared test for K independent samples of nominal data Parameters Data Required The n rows by k columns array of the observations where k is the number of samples being compared and n is the number of classes n must be greater than 1 and k must be greater than 2 Output Output as a NAGExtract format list Significance Chisquared Degrees of freedom Expected values Chisquared contributions The chi squared contributions are the values of observed expected expected for each cell which make up the chi squared statistic Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information 93 6 2 6 2 6 2 6 3 6 2 6 4 Statistical Add Ins for Excel User Guide COCHRAN_Q Set 2 COCHRAN_Q NAG GO8ALF computes the Cochran Q test for nominal binary data from k samples Parameters Data Required The n rows by k columns array of 0 or 1 values where n is the number of blocks and k is the number of treatments Both n and k must be greater than 1 Output Output as a NAGExtract format list Test statistic Significance level The significance level is based on a chi squared approximation and should be reasonable if after omitting rows which conta
99. ll plan of the experiment the final allocation of treatments to units is performed using a suitable random allocation This avoids the possibility of a systematic bias in the allocation and gives a basis for the statistical analysis of the experiment Replication Each treatment should be observed more than once So in Example 2 more than one sample from each material should be tested Replication allows for an estimate of the variability of the treatment effect to be measured Blocking In many situations the experimental material will not be homogeneous and there may be some form of systematic variation in the experimental material In order to reduce the effect of systematic variation the material can be grouped into blocks so that units within a block are similar but there is variation between blocks For example in an animal experiment litters may be considered as blocks in an industrial experiment blocks may be material from one production batch Factorial designs If more than one type of treatment is under consideration for example the effect of changes in temperature and changes in pressure a factorial design consists of looking at all combinations of temperature and pressure The different types of treatment are known as factors and the different values of the factors that are considered in the experiment are known as levels So if three temperatures and four different pressures were being considered then factor 1 temperature would have
100. mation Matched Pairs Samples MCNEMAR Set 2 MCNEMAR NAG G01BJF GO1ECF computes the McNemar test for two matched samples of nominal binary data Parameters TABLE Required A 2 by 2 array of cells containing the counts of observations on the two matched samples If the observations take the values 0 or 1 then Cell 1 1 number of observations for which sample one is 0 and sample two is 0 Cell 1 2 number of observations for which sample one is 0 and sample two is 1 Cell 2 1 number of observations for which sample one is 1 and sample two is 0 Cell 2 2 number of observations for which sample one is 1 and sample two is 1 Output Output as a NAGExtract format list Z value i e normalised test statistic Significance level The significance level is exact for N lt 20 and uses a Normal approximation for larger numbers of observations Remarks See Using the Add Ins and the Introduction to Nonparametric Methods for further information 88 6 2 4 2 6 2 4 3 Statistical Add Ins for Excel User Guide SIGN_PAIRS Set 2 SIGN_PAIRS NAG GO8AAF computes the sign test for two matched samples of ordinal data Parameters X Required The observations for sample one Y Required The observations for sample two Tail Optional Indicates the critical region and hence the form of H7 T two tailed test i e H1 mx my L lower tailed test i e H1 Myx lt my U upper tailed test
101. matrix V the factor analysis will be computed for the covariance matrix Default C Correlation matrix when data matrix supplied Weights Optional If required the weights for each the observations If Weights i 0 0 then the ith observation is not included in the analysis If present Weights 0 0 Max_steps Optional The maximum number of function evaluations used in computing the estimates Default 100 times number of variables Acc_req Optional The number of decimal places of accuracy required for the estimates of the unique component of the variation y If present O lt Acc_req lt 15 Default 7 Lower_phi Optional The number of decimal places for the lower bound for the values of y i e the lower bound is 10 If present 0 lt Lower_phi lt 15 Default 13 Output Output as a NAGExtract format list Eigenvalues Communalities Psi 66 5 5 2 4 5 5 2 5 Statistical Add Ins for Excel User Guide Loadings Function value Test statistic Degrees of freedom Significance Residuals as lower triangular Note The Test statistic Degrees of freedom and Significance are the value of the chi squared statistic for testing the goodness of fit of the model along with its degrees of freedom and significance Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information FACTOR_SCORE Set 1 FACTOR_SCORE NAG GO3CCF computes factor score coef
102. ment the Add Ins as array functions Microsoft provide a range of these in their own Add Ins and we have followed their conventions To illustrate the use of Excel array functions type in a 2 x 2 array in a block of 2 x 2 adjacent cells within a worksheet For example let us say D1 E2 contains the values Now select another block of 2 x 2 cells and type in another set of values to represent a 2 x 2 array We are going to multiply these two arrays together using the standard Microsoft array function MMULT Since we know that the product of these matrices is itself a 2 x 2 matrix we select a block of 4 cells to contain the result Now in the function wizard select MMULT from the Math amp Trig category Select one of the arrays say D1 E2 with the mouse to satisfy the first parameter 3 5 Statistical Add Ins for Excel User Guide array1 and select the other array to satisfy the second parameter array2 Press the Finish button or simply press return You will only have a single scalar value returned in the 1 1 position at this stage Now click on the formula bar and press Enter whilst holding down both the Control and Shift keys at the same time CONTROL SHIFT ENTER You should now see the result of the matrix multiplication appearing in the four cells selected for this purpose Look also at the formula bar The formula is enclosed in curly braces to denote that it is an array formula You cannot achieve the s
103. n the workbooks BookOE xls Book1E xls Book2E xls Book3E xls for the NAG Statistical Add Ins for Excel Modelling and Multivariate Methods product in and Book4E xls for the companion product providing nonparametric statistical capability These are installed in the Examples subdirectory of the Add Ins directory and are also on the CD You might find it helpful to work through these to familiarize yourself with the use of the Add Ins or to clarify any items not described in sufficient detail Questions What versions of Excel do the Add Ins support The Add Ins were formed under Excel 7 0 using Windows 95 They have been tested and shown to work under Excel 97 and using Windows NT and Windows 98 We have tested them with Office 2000 They will not work under Excel 5 0 operating under Windows 95 or Windows NT If you are interested in a version for this environment please register an interest with NAG Should there be sufficient demand then we would consider doing a specific implementation The first release placed the Add Ins in a NAG Category where are they now We have not discovered a clean way of producing additional categories within Excel We created a category by making a dummy Excel 4 macro This opens up the possibility of creating a new category and we exploited this to create the NAG Books category The scheme worked most of the time but on some systems it seemed to fail Because of this we have reverted to placi
104. nction wizard choose ChangeKey Call it with KeyName set to About and NewValue set to 0 Repeat the process with KeyName set to Help and NewValue set to 0 The Nag menu doesn t disappear when I unload the NAG Add Ins Make sure that all of the NAG Books including NAGExtract are not loaded Now load the NAG Add In Utilities From the User Defined category in the function wizard choose ChangeKey Call it with KeyName set to About and NewValue set to 0 Repeat the process with KeyName set to Help and NewValue set to 0 The Comments on the examples worksheets seem truncated using Excel 97 Select the cell containing the comment From the Insert menu select Comment Resizing handles now appear on the comment box and these may be used to extend the box appropriately We get Trial message boxes appearing when we load the NAG Add Ins This message appears when the software you have loaded is designed to work for a limited period only It is to remind you to contact NAG to obtain a licensed copy before the time period expires After this date the Add Ins will not work We get a Catastrophic failure message when we use the NAG Add Ins This may be because your trial license has expired Please contact NAG to obtain a licensed copy If you have a licensed copy then please contact NAG with the details of your problem 99 Statistical Add Ins for Excel User Guide Ap
105. nd the correlation coefficient sometimes known as the Pearson product moment correlation coefficient measures the linear association between the two variables that is how close the plotted points would be to lying on a straight line The correlation coefficient lies between 1 and 1 with 0 indicating no linear association between the two variables and 1 indicating a perfect linear relationship so that the plotted points would lie on a straight line with positive slope correlation of 1 or negative slope correlation of 1 The function CORREL_MAT computes all the pairs of correlations for a set of variables known as a correlation matrix Correlation can also be seen as a standardized form of the covariance between two variables CORREL_MAT also returns the variance covariance matrix 22 Statistical Add Ins for Excel User Guide Sometimes the correlation between two variables should not be interpreted as a direct relationship between the two variables but as being due in part at least to their being correlated with a third variable For example using historical data there may be a significant correlation between electricity consumption and road accidents This would mainly be due to both increasing over time To remove the effect of the third or other variables a partial correlation coefficient can be computed This measures the linear association between two variables having eliminated the linear effect of other variables The func
106. ng the Add Ins in the Statistical category which already exists We get Microsoft Help when we want NAG Help NAG help may be obtained from the Nag menu by accessing NAGHelp We only obtain a lt list of n gt heading not a set of numbers If you are using a NAG function that you are expecting to return several values then the call to this function must be entered as an Excel ARRAY FUNCTION In practice this means that the destination cells must be pre selected and that once the formula has been formed in the formula bar CONTROL SHIFT ENTER must be pressed to enter the formula You will note that pressing this combination of keys causes the formula to have curly brackets around it in the formula bar You will NOT get the same effect by explicitly typing these brackets 97 Statistical Add Ins for Excel User Guide Will NAG be producing any more Add Ins We are keen to learn of user requirements in this area The Add Ins have been well received and we now have two distinct sets of statistical Add Ins Modelling and Multivariate Methods as one set and Nonparametric Methods as the second set Both are documented together though they are sold as separate products Other Add Ins are contemplated Please let us know what you want We get error messages whenever we open a second workbook Set the properties of the NAG _Library xla to read only To do this find NAG_Library xla using Windows Explorer and right click on the
107. nk and if not the rank of the independent variables If present eps gt 0 0 Default 0 00000000001 Output Output as a NAGExtract format list Deviance df Rank of Model Parameter Estimates Cov If requested see Fitted_values the following are returned Fitted values Residuals Leverages Linear predictor Variances Working weights Notes The adjusted deviance and its degrees of freedom df are returned providing the degrees of freedom are not zero The Rank of Model is returned only when the model is not of full rank 61 5 4 3 6 Statistical Add Ins for Excel User Guide The Parameter Estimates consist of the four columns with headings giving Parameter Names Parameter Estimate Standard Error t value The variance covariance matrix of the parameters Cov is returned as an upper triangular matrix Remarks See Using the Add Ins and the Introduction to Generalized Linear Models for further information MM_GLM Set 1 MM_GLM computes the model matrix for a linear model specified by a formula It is for use with NORMAL_GLM BINOMIAL_GLM POISSON_GLM and GAMMA_GLM where it is called instead of providing the X array Parameters Formula Required The model formula that may consists of the names of the variables given in Group_ Names and Covariate_Names and the following symbols JE add term to formula interaction between two variables cross classify two terms e g a b a b a
108. ns The second column is the name of the objects in the order that is required for a dendrogram and the third is the height associated with that branch of the dendrogram The dendrogram can be constructed by drawing lines of the required height corresponding to each object in the dendrogram order and then connecting the top of each line with the next line to the right at that level Finally the first column is an index that relates dendrogram order to the order of the object in the original data Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information 74 5 5 3 4 5 5 3 5 Statistical Add Ins for Excel User Guide GROUPS_FROM_CLUSTER Set 1 GROUPS_FROM_CLUSTER NAG GO3EJF computes a cluster indicator variable from the results of a cluster analysis performed by CLUSTER Parameters Objects Required The object names in dendrogam order as output by CLUSTER in the second column of Dendrogram information Distances Required The distances as output by CLUSTER in the second column Dendrogram information Note EITHER No_of_clusters OR Cluster_distance must be provided No_of_clusters Optional The number of clusters required Cluster_distance Optional The distance at which clusters are to be produced Index Optional If present the output cluster indicator will be in the order of the original data rather than the dendrogram order Index should be the first column of t
109. observation is then allocated to the group with the highest posterior probability In the estimative approach to discrimination the parameters of the multivariate Normal distribution which appear in the posterior are replaced by their estimates calculated from the training set If it is assumed that the within group variance covariance matrices are equal then the linear discriminant function is obtained otherwise if it is assumed that the variance covariance matrices are unequal then the quadratic discriminant function is obtained In the Bayesian predictive approach a non informative prior distribution is used for the parameters giving the posterior distribution for the parameters from the training set A predictive distribution is then obtained by integrating over the parameter space This predictive distribution is then used in place of the posterior distribution In addition to allocating the objects to groups an atypicality index for each object and for each group can be computed This represents the probability of obtaining an observation more typical of the group than that observed A high value of the atypicality index for all groups indicates that the observation may in fact come from a group not represented in the training set The function DISCRIM_TEST computes a test statistic for the equality of the group variances and also computes the statistics required to carry out a discriminant analysis The information from DISCRIM_TEST can be pass
110. ohn 1987 The function ANOVA_ROW_COLUMN requires the data to be ordered by rows within columns within replicates if any Again the data can be arranged in a two way layout corresponding to rows and columns For ANOVA_ROW_COLUMN the full list of treatments must be provided Both ANOVA_BLOCK and ANOVA_ROW_COLUMN return a two way array of covariances and standard errors for the treatments The upper triangular part of this is the variance covariance matrix of the treatment means so the diagonal elements give the variances of the treatment means The lower triangle below the diagonal contains the standard error for the differences between pairs of means For balanced designs such as Randomized complete blocks and balanced incomplete blocks this value is constant and represents the standard error for the difference between any two treatments ANOVA_FACTORIAL requires the data to be ordered by the factors That is the data is sorted by Factor 1 then by Factor 2 then by Factor 3 etc If the data is not in the required order but the factors are available the Excel Sort macro can be used from the Data menu to re order the data The tables of means and effects are returned with the final factor defining the columns and the order of other factors defining the rows For example the A B C table of means would have the levels of factor C defining the columns and rows would be ordered by the levels of factor B within the levels of factor A as shown bel
111. on coefficients of skewness and kurtosis and the maximum and minimum values for a set of ungrouped data Weighting may be used 20 5 2 3 3 5 2 3 4 Statistical Add Ins for Excel User Guide Parameters X Required The observations upon which the calculations are to be performed Weights Optional If present Weights must contain the non negative weights associated with the observations Output Output as a NAGExtract format list Number of valid cases Mean Standard Deviation Coefficient of Skewness Coefficient of Kurtosis Minimum Maximum If weights are used then also Sum of Weights Note When weights are used the number of valid cases is the number of observations with non zero weights otherwise it is simply the number of observations Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information S5PT_SUMMARY Set 1 S5PT_SUMMARY NAG GO1ALF calculates a five point summary for a single sample Parameters X Required The sample there must be at least 5 observations Output Output as a NAGExtract format list Minimum Lower Hinge Median Upper Hinge Maximum Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information TWO_WAY_TABLE Set 1 TWO_WAY_TABLE NAG G11AAF computes chi squared statistics for a two way contingency table For a 2 x 2 table with 40 or fewer observations an exact probability is compu
112. on for further information STAND_RESIDS Set 1 STAND_RESIDS NAG G02FAF calculates two types of standardized residuals and two measures of influence for a linear regression Parameters Residuals Required The residuals from the regression Leverages Required The leverages from the regression RMS Required The residual mean square from the regression RDF Required The residual degrees of freedom from the regression Output Output as a NAGExtract format list Internally Standardized Residuals Externally Standardized Residuals Cook s D Atkinson s T 29 5 3 2 8 Statistical Add Ins for Excel User Guide Remarks See Using the Add Ins and the Introduction to Regression and Correlation for further information DUMMY_VARS Set 1 DUMMY_VARS NAG GO4EAF computes orthogonal polynomial or dummy variables for a factor or classification variable Parameters Factor Required The integer values of the factor which must be between 1 and Levels Levels Required The number of levels number of classes of the factor There must be at least two levels Form Optional The form of the dummy variable to be computed P an orthogonal Polynomial representation is computed H a Helmert matrix representation is computed F the contrasts relative to the first level are computed L the contrasts relative to the last level are computed C a complete set of dummy variables is computed Default cont
113. onal If present the distances from the group means to the points in the array X will be calculated otherwise the distances between the group means will be calculated X along with Select_x if present must be compatible with the input to DISCRIM_TEST Default between group mean distances calculated Select_x Optional Should only be present if X is present If Select_x gt O then the variable in the ith column of the array X is included in the analysis otherwise it is excluded This should be compatible with the input to DISCRIM_TEST Default all columns are included Output Output as a NAGExtract format list Distances Note If X is supplied then the distances are an array with rows for the observations and columns for the groups Otherwise if Equal U the between group mean distances are in a square array with blank diagonal cells the within group covariance matrix for the column group being used to standardize the distances If Equal E then only the lower triangle of symmetric between group distances is returned Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information DISCRIM_TEST Set 1 DISCRIM_TEST NAG GO3DAF computes a test statistic for the equality of within group covariance matrices and also computes information for use in discriminant analysis Parameters In_group Required The column that indicates to which group the observations Note that each
114. or a moving average model will cut off at lag q In the situation where the data has a cyclic or periodic nature for example days of the week a seasonal model may be appropriate For example Tuesday s observation may be related to both Monday s observation and the previous Tuesday s observation In seasonal models the terms are related to the seasonality s e g for days of the week s 7 The seasonal autoregressive and moving average models are 31 Statistical Add Ins for Excel User Guide Wi DW 5 OW_25 OpW_ps amp and W2 amp OjE15 Or amp _25 OQEt as These models can be combined to give a seasonal ARMA model Wi DW DW 25 DW ps ter OyOps O28425 OQ tas Et Q641 Q t2 Oplrp Et E1 Or amp 2 Ogg where e is an intermediate series for the purposes of defining the model In the situation where the series y is not stationary it may be possible to make it stationary by applying differencing or seasonal differencing First order differencing is given by Vt Yi Yi1 and first order seasonal differencing is given by V Yt Yi Yet Differencing can be combined with a constant term to give dy D _ V Vs Yao w Combining this with the ARMA model given above gives a seasonal ARIMA AutoRegressive Integrated Moving Average model which can be specified by the seven values p d q P D Q s Time Series Model Identification The basic tool
115. or more factors can be defined in a similar way For factorial experiments the ANOVA table will contain a row for each of the terms In the case of the two factor experiment with blocks and factor A having la levels and factor B having lp levels the partition will be 44 Source Degrees of Freedom Blocks b 1 Main effect of A la 1 Main effect of B Ib 1 AB interaction la 1 lb 1 Residual n b lalp 1 Total n 1 Statistical Add Ins for Excel User Guide In addition to computing a table of means for each treatment term in the ANOVA table a table of effects is also computed The table of effects measures the additional effect of the term over and above the previous terms For example the AB effect measures the effect of the AB interaction in addition to the A main effect and the B main effect Using the ANOVA Functions For the function ANOVA_BLOCK the data can be presented either in the default block order or across blocks using the Block_order option The data can also be arranged in a two way layout and the default will be to assume that rows correspond to blocks The treatments can be presented in full or in the case when the data has been ordered by treatments within blocks just the list of ordered treatments is required from which the full list can be generated As well as the analysis of variance and treatment means with standard errors the function also returns the canonical efficiency factor as described by J
116. order of seasonal differencing number of seasonal moving average terms the seasonality 2 ouye Variance Required The series sample variance calculated after appropriate differencing has been applied to the series See output of ACF ACF Required The autocorrelations starting at lag 1 which must have been calculated after the time series has been appropriately differenced See output of ACF 35 5 3 3 6 Statistical Add Ins for Excel User Guide Output Output as a NAGExtract format list Residual Variance Parameters Note The parameters are returned in the order autoregressive moving average seasonal autoregressive and seasonal moving average Remarks See Using the Add Ins and the Introduction to Time Series Methods for further information ARIMA FIT Set 1 ARIMA_FIT NAG G13BEF fits either a seasonal ARIMA model or a multi input transfer function model Parameters Y_series Required The output time series Model Required Model specifies the orders p d q P D Q s of the seasonal ARIMA model to be fitted to the output series p number of autoregressive terms d order of non seasonal differencing q number of moving average terms P number of seasonal autoregressive terms D order of seasonal differencing Q number of seasonal moving average terms s the seasonality Fix_constant Optional Indicates if the constant is to be fixed at its initial value or estima
117. ow Table A B C Factor C Factor A Factor B 1 2 1 1 1 2 2 1 2 2 45 5 4 2 2 Statistical Add Ins for Excel User Guide All three functions have an optional parameter called Df_adjustment This can be used to adjust the degrees of freedom when either analysing the data in several stages for example using ANOVA_ROW_COL followed by ANOVA_FACTORIAL or when estimating missing values Further Comments For experiments with missing values these values can be estimated by using the Healy and Westmacott procedure see John and Quenouille 1977 This procedure involves starting with initial estimates for the missing values and then making adjustments based on the residuals from the analysis The improved estimates are then used in further iterations of the process For designs that cannot be analysed by the above approach the function MULT_LIN_REG or NORMAL_GLM with the Identity link can be used to fit a general linear model ANOVA BLOCK Set 1 ANOVA_BLOCK NAG GO4BBF computes the analysis of variance for a block design with equal sized blocks or a completely Randomized design Parameters No_of_Blocks Required The number of blocks of equal size For a completely Randomized design set No_of_Blocks to zero Data Required The data to be analysed The data should be in block order i e Block 1 observations followed by Block 2 observations etc unless Block_Order A in which case
118. ower tail probability is returned S The two tail significance level is returned C The two tail confidence interval is returned Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information T_PPT Set 1 T_PPT NAG GO1FBF returns as a single value the deviate associated with the given tail probability of Student s f distribution Parameters Probability Required The probability of the required Student s t distribution as defined by tail Df Required The degrees of freedom of the Student s t distribution must be gt 1 0 Tail Required Indicates which tail the supplied probability represents U The upper tail probability L The lower tail probability S The two tail significance level probability C The two tail confidence interval probability Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information CHI_PROB Set 1 CHI_PROB NAG GO1ECF returns as a single value the lower or upper tail probability for the x distribution Parameters X Required The value of the y variate must be gt 0 Df Required The degrees of freedom of the x distribution must be gt 0 16 5 2 2 8 5 2 2 9 5 2 2 10 Statistical Add Ins for Excel User Guide Tail Required Indicates which tail the return probability should represent U The upper tail probability is returned L The lower tail probability is returned Rem
119. pendices Glossary array functions instead of returning just one scalar value as the result these functions return more than one value sometimes as a vector at other times as arrays The NAG Statistical Add In array functions always return results as a column vector See Array Functions function wizard a dialog box showing the list of available functions and their categories The wizard then helps you call the selected procedure by providing input boxes for the parameters See the diagram below NAGExtract format list the vector of results returned by any of the NAG Statistical Add Ins that return more than one scalar value reference a means of getting access to procedures in another workbook Nag menu Formula bar Function wizard X Microsoft Excel Book I File Edit view Insert F at Tools Data Window Nag Help BES o c 44 210121098 00 10 E 5 i858 EE O A 141 gt E OMA MMM 100 Statistical Add Ins for Excel User Guide 8 2 Algorithmic Association 8 2 1 Add In NAG Subroutine Correspondence Set 1 Modelling and Multivariate Methods Available Add In functions from Book 0 BETA_PPT GO1FEF BETA PROB GO1EEF CHI_ PPT GO1FCF CHI_PROB GO1ECF F_PPT GO1FDF F PROB GO1EDF GAMMA_PPT GO1FFF GAMMA_PROB GO1EFF MULTI_NORMAL_PROB GO1HBF NORMAL_PPT GO1FAF NORMAL_PROB GO1EAF S5PT_SUMMARY GO1ALF SUMMARY_STATS
120. phical displays for outlying and influential observations in regression Biometrika 68 13 20 Chatfield C 1976 Statistics for Technology Chapman and Hall Chatfield C 1982 The Analysis of Time Series an Introduction Chapman and Hall Cochran W G and Cox G M 1957 Experimental Designs Wiley Conover W J 1980 Practical Nonparametric Statistics Wiley Cook R D and Weisberg S 1982 Residuals and Influence in Regression Chapman and Hall Cox D R 1970 Analysis of Binary Data Chapman and Hall Davis O L 1978 The Design and Analysis of Industrial Experiments Longman Dobson A J 1990 An Introduction to Generalized Linear Models Chapman and Hall Draper N R and Smith H 1966 Applied Regression Analysis Wiley Everitt B S 1974 Cluster Analysis Heinemann Everitt B S 1977 The Analysis of Contingency Tables Chapman and Hall Hand D J Daly F Lunn A D McConway K J and Ostrowski E 1994 Small Data Sets Chapman and Hall John J A 1987 Cyclic Designs Chapman and Hall John J A and Quenouille M H 1977 Experiments Design and Analysis Griffin Krzanowski W J 1990 Principles of Multivariate Analysis Oxford University Press Lawley D N and Maxwell A E 1971 Factor Analysis as a Statistical Method Butterworths McCullagh P and Nelder J A 1989 Generalized Linear Models Chapman and Hall Morrison D F 1967 Multivariate Statistical Methods McGraw Hill Osborn J F 1979 Statistical Exercises in Medical Research Blackwell Scientific Publications S
121. principal component analysis the aim of factor analysis is to account for the covariances in these variables in terms of a smaller number of latent variables or factors These variables are assumed to be independent and to have unit variance In factor analysis the relationship between the variance covariance of the observed variables S and the factors is given by the model S LL y where L is the matrix of the factor loadings and wis a diagonal matrix of the unique variances for each variable The values 1 yj are known as the communalities The loadings relate the observed variables Y to the unobserved factors F Y LF e lf it is assumed that both the factors and the unique components e follow independent Normal distributions then the parameters of the model that is L and y can be estimated by maximum likelihood as described by Lawley and Maxwell 1971 The computation of the maximum likelihood estimates is an iterative procedure that involves computing the eigenvalues and eigenvectors of the matrix Ss Z ws y where S is the sample variance covariance matrix The use of maximum likelihood estimation means that likelihood ratio tests can be constructed to test for the number of factors required by comparing log likelihoods Having found the estimates of the parameters of the model the estimates of the values of the factors for the individuals the factor scores can be computed These involve the calculation of the factor score
122. ptional If the design is non orthogonal then the treatment means adjusted for blocks should be provided in Adj_means 48 5 4 2 5 Statistical Add Ins for Excel User Guide Output The results returned consist of one row for each contrast ANOVA Estimates Remarks See Using the Add Ins and the Introduction to the Analysis of Designed Experiments for further information MULT_COMPN Set 1 MULT_COMPN NAG G04DBF computes simultaneous confidence intervals for the differences between means following an analysis of variance Parameters Method Required Indicates which method is to be used the Tukey Kramer method is used the Bonferroni method is used the Dunn Sidak method is used the Fisher LSD method is used the Scheffe method is used OT gw Treatment_means Required The treatment means as computed by ANOVA_BLOCK and ANOVA_ROW_COLUMN rdf Required The residual degrees of freedom from the ANOVA C_array Required The array of covariances and standard errors as computed by ANOVA_BLOCK and ANOVA_ROW_COLUMN Confidence_level Required The required confidence level for the computed intervals such that 0 0 lt Confidence_level lt 1 0 Output The results are returned in upper triangular form with the first row corresponding to difference between the first treatment mean and the means of the other treatments Lower limits Lower limits Significance Note The significance is returned as
123. rasts relative to the first level X_values Optional If Form P the distinct values corresponding to the levels of the underlying variable for which the orthogonal polynomial is to be computed Output Output as a NAGExtract format list X Note The array of dummy variables returned X will have Levels 1 columns if Form is set to P H F or L and Levels columns if it is set to C Remarks DUMMY_VARS should be used when an orthogonal polynomial or a specific form for the dummy variables is required For general use the NAG MM_MLR function computes the dummy variables required for a regression model from the formula provided See Using the Add Ins and the Introduction to Regression and Correlation for further information 30 5 3 3 5 3 3 1 Statistical Add Ins for Excel User Guide Time Series Introduction Introduction to Time Series Methods Time Series Models The time series considered in this book are sequences of observations observed at equally spaced intervals for example the temperature of a process recorded every minute the daily price of shares or the monthly customer numbers For an introduction to time series see Chatfield 1982 The time series can be considered as having three basic components 1 Trend 2 Cyclic or periodic 3 Random or noise The trend represents the long term pattern e g increasing sales while the cyclic or periodic components follow patterns such as summ
124. riable contained in the jth column of array X is included in the model if Select_x j 0 it is excluded If Select_x is present then its elements gt 0 Default all X variables are included X_names Optional Names for the explanatory variables Default X_1 etc Offsets Optional Offsets for the model Default none Weights Optional User weights If a Weight is set to zero the observation is not included in the model in which case the effective number of observations is the number of observations with non zero weights If present Weights 0 0 Default no weights Mean Optional Indicates if a mean term is to be included M a mean term intercept is to be included in the model Z the model will pass through the origin zero point Default mean term included Power Optional If an exponent link is selected the power of the exponent Restriction Power 0 0 S Optional The scale parameter for the model o If omitted the scale parameter is estimated by the residual mean square If present s gt 0 0 Fitted_values Optional Indicates if fitted values etc are to be returned N No fitted values etc are returned Y Fitted values and residuals are returned A All of fitted values residuals leverages linear predictor variance function and working weights are returned Default Y Fitted values and residuals are returned Max_iterations Optional The maximum number of iterations for
125. riangular distance matrix This can either be packed by rows as output directly by DIST_MAT or as the lower triangular part of a two dimensional array with n 1 rows and n 1 columns i e without the diagonal cells Negative distances are not allowed Xx Optional The array of observation for the variable in columns Distance Optional Should only be present if X is present Distance indicates which type of distances are computed A absolute distances E Euclidean distances S Euclidean Squared distances Scaling Optional Should only be present if X is present Scaling indicates the standardization of the variables to be used S standard deviation R range U unscaled Names Optional The names of the objects to be clustered Default Obj 1 Obj 2 etc Output Output as a NAGExtract format list Cluster Steps Dendrogram information Notes The Cluster steps array consists of four columns If there are n objects then there are n 1 steps in the clustering process At a step two clusters are fused in which one cluster called the second cluster is fused into the first cluster The name of the cluster is then always the name of the first object to enter the cluster in its evolution For each of the n 1 clustering steps the following is returned Column 1 The step number Column 2 Name of first cluster Column 3 Name of second cluster Column 4 Distance The Dendrogram information array consists of three colum
126. rix of the treatment effects and the strictly lower part contains the standard errors of the difference between pairs of treatment effects Remarks If no treatments are specified then the rows of the ANOVA table corresponding to Replicates Rows Columns and Total are computed along with the means See Using the Add Ins and the Introduction to the Analysis of Designed Experiments for further information ANOVA_ROW_COLUMN Set 1 ANOVA_ROW_COLUMN NAG G04BCF computes the analysis of variance for a general row and column design such as a Latin square Parameters No_of_replicates Required The number of replicates of the square if only one replicate set No_of_replicates 1 No_of_rows Required The number of rows in the square No_of_columns Required The number of columns in the square Data Required The observations ordered by columns within rows within replicates Treatments Optional The treatments applied to the Data These would normally be present Tolerance Optional The tolerance value used in the computations to check for zero values If present Tolerance gt 0 0 Default 0 00001 Df_adjustment Optional This is the adjustment to the degrees of freedom for the residual and total The degrees of freedom for the total is set to number of observations Df_adjustment and the residual degrees of freedom adjusted accordingly If present Df_adjustment gt 0 Output The following
127. rrence should be extremely rare If it does occur then if the function has a tolerance parameter try making it larger Alternatively try using a simpler model If the problem persists with a range of problems please check the NAG Web site to see if there is a known problem with the function If it is not recorded please contact NAG preferably by email to report the problem Changes from Release 1 0 Users of Release 1 0 of the NAG Statistical Add Ins for Excel Modelling and Multivariate Methods should note that while the basic functionality has not been changed between Release 1 0 and subsequent releases there has been a major rewrite of the interfaces taking into account comments from users The interfaces of Release 1 0 are not compatible with the subsequent interfaces We hope users find the new interfaces easier to use and with the improved Help system will quickly adapt to them Any future releases of these Add Ins should not involve such a major change to the interfaces An additional functionality is provided by the new MM functions that can specify a linear model by means of a model formula There is also improved functionality in NAGExtract 3 9 Statistical Add Ins for Excel User Guide Further Technical Information The User Guide should provide you with all the information required to run the NAG Statistical Add Ins If you want further information on the algorithms used to compute the statistics please refer to the appropri
128. rs X Required The observations for sample one Y Required The observations for sample two Output Output as a NAGExtract format list Number of observations below the median for sample one Number of observations below the median for sample two Significance level The significance level is exact for number of observation lt 40 and uses a Normal approximation for larger numbers of observations Remarks Approximate 1 tail probabilities may be obtained by halving the returned significance level if the difference between the samples in the required direction See Using the Add Ins and the Introduction to Nonparametric Methods for further information TWO_SAMPLE_CI Set 2 TWO_SAMPLE_Cl NAG GO7EBF computes a rank based confidence interval for the difference in location of two samples of ordinal data Parameters Xx Required The observations from sample one Y Required The observations from sample two the interval is for the location of population two minus the location of population one 92 6 2 6 6 2 6 1 Statistical Add Ins for Excel User Guide Confidence Optional The required confidence for the interval in the range 0 to 1 Default 0 95 e a 95 confidence interval Method Optional Indicates which algorithm is used to compute the interval E exact algorithm A iterative algorithm Default exact algorithm Output Output as a NAGExtract format list Point estimate Lower
129. rther information MM_MLR Set 1 MM_MLR computes the model matrix for a general linear model specified by a formula It is for use with MULT_LIN_REG where it is called instead of providing the X array 27 Statistical Add Ins for Excel User Guide Parameters Formula Required The model formula that may consists of the names of the variables given in Group_Names and Covariate_Names and the following symbols Ot where the symbols represent the following actions add term to formula interaction between two variables cross classify two terms e g a b a b a b nest two terms e g a b a a b remove term from formula e g a gt a b at b power for covariates only The precedence of the operators is At least one of Groups or Covariates must be given along with the corresponding names Groups Optional The array whose columns contain the variables that are to be considered as grouping or classification variables Group_Names Optional The names of the variables in Groups Covariates Optional The array whose columns contain the variables that are to be considered as covariates i e not as classification variables Covariate_Names Optional The names of the variables in Covariates Interaction Optional The maximum number of terms to be included in an interaction gt 1 Default all terms included Output To avoid the restriction on the size of returned arrays the funct
130. s are returned A All of fitted values residuals leverages linear predictor variance function and working weights are returned Default Y Fitted values and residuals are returned Max_iterations Optional The maximum number of iterations for the iterative weighted least squares fitting If present Max_iterations gt 0 Default 10 56 Statistical Add Ins for Excel User Guide Tolerance Optional Indicates the accuracy required for the fit of the model If present Tolerance gt 0 0 Default 0 000001 Eps Optional The value of eps is used to decide if the independent variables are of full rank and if not the rank of the independent variables If present eps gt 0 0 Default 0 00000000001 Output Output as a NAGExtract format list Deviance df Rank of Model Parameter Estimates Cov If requested see Fitted_values the following are returned Fitted values Residuals Leverages Linear predictor Variances Working weights Notes The deviance and its degrees of freedom df are returned providing the degrees of freedom are not zero The Rank of Model is returned only when the model is not of full rank The Parameter Estimates consist of the four columns with headings giving Parameter Names Parameter Estimate Standard Error t value The variance covariance matrix of the parameters Cov is returned as an upper triangular matrix Remarks See Using the Add Ins and the Introduction to
131. s in the identification of a suitable time series model are data plots the autocorrelation function acf and the partial autocorrelation function pacf Plotting the data will indicate if the series is stationary If not the series can be transformed using standard functions such as LOG or SQRT and differencing can be applied using the function TIME_SERIES_DIFF Given a stationary time series the acf and pacf can be calculated using the functions ACF and PACF respectively By looking at the acf the order q of a possible MA model can be identified while the pacf for an AR p model will cut off at lag p The function PACF also gives approximate estimates of the parameters of the autoregressive model of order given by the number of partial autocorrelations requested and the predictor error variance ratio which is V var amp var w where var W is the variance of the stationary series and var is the estimated variance of the white noise when an AR has been fitted If neither approach is suitable then ARMA models can be considered The function ARIMA_APPROX_FIT gives an approximate fit for an ARIMA model This may be useful in model identification or for giving a set of initial values for the full fitting process 32 Statistical Add Ins for Excel User Guide Time Series Model Fitting and Forecasting The function ARIMA_FIT fits an ARIMA model using either maximum likelihood or least squares The model is specified by the
132. s through the origin zero point 26 5 3 2 5 Statistical Add Ins for Excel User Guide Weights Optional Contains the optional weights to be used in the regression A zero weight indicates that an observation is excluded from the analysis If present Weights gt 0 0 Fitted_values Optional Indicates whether fitted values residuals and leverages are to be returned Y Fitted values etc are returned N Fitted values etc are not returned Default Y Fitted values etc are returned Tolerance Optional The value of Tolerance is used to decide if the independent variables are of full rank and if not what is the rank of the independent variables If present Tolerance gt 0 0 Default 0 000001 Output Output as a NAGExtract format list Some or all of the following may be returned Rank of Model ANOVA R squared Parameter Estimates Fitted Values Residuals Leverages Covariance Matrix Notes The rank of the model is only returned if it is not of full rank The ANOVA table for the model is only returned if the residual degrees of freedom are not zero as is the Covariance Matrix and the standard errors etc for the parameters The Parameter Estimates array contains Parameter Names Estimates Standard Errors T values Significance The Fitted Values Residuals and Leverages are not returned if Fitted_values N Remarks See Using the Add Ins and the Introduction to Regression and Correlation for fu
133. section on Array Functions since the Add Ins make extensive use of this Excel facility Simple Usage From the Nag menu choose About A message box appears reflecting the personal details you gave on installation If you have the NAG Statistical Add Ins for Excel Modelling and Multivariate Methods product Set 1 select a cell on a worksheet and enter the formula CHI_PPT 0 5 1 A value of 0 454936 appears in the cell We can also call this NAG function from the function wizard To do this select a different cell on the worksheet and from the Toolbar click on the function wizard button and select the Statistical Category Under the Function name panel select CHI_PPT You will see that this requires two parameters There is also a brief statement of the routine s functionality In this case CHI_PPT returns the deviate associated with the given lower tail probability of the chi squared distribution with real degrees of freedom Click the Next button On the new dialog box select the first parameter and type 0 5 Do not press enter at this stage Select the second parameter and enter the value 1 Either click on the Finish button or press Enter We can refine this process somewhat by entering data in the cells Select a cell and type in the value 0 5 Now select another cell and enter the value 1 Select a different cell to contain the final result and as before enter the function wizard for CHI_PP
134. stribution 0 0 lt A lt 1046 B Required The second parameter of the required beta distribution 0 0 lt B lt 1046 Tolerance Optional The relative accuracy required in the result 18 5 2 2 13 5 2 2 14 Statistical Add Ins for Excel User Guide Default 10 times machine precision Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information GAMMA_PROB Set 1 GAMMA_PROB NAG G01EFF returns as a single value the lower or upper tail probability of the gamma distribution Parameters Xx Required The value of the gamma variate must be gt 0 0 Shape_param Required The shape parameter of the gamma distribution must be gt 0 0 Scale_param Required The scale parameter of the gamma distribution must be gt 0 0 Tail Required Indicates which tail the return probability should represent U The upper tail probability is returned L The lower tail probability is returned Remarks The parameterization of the gamma distribution used by this function is such that if the shape parameter is a and the scale parameter is b the mean is ab and the probability density function is exp a 1 log x b x b bI a See Using the Add Ins and the Introduction to Basic Statistics for further information GAMMA _PPT Set 1 GAMMA_PPT NAG GO1FFF returns as a single value the deviate associated with the given lower tail probability of the gamma distri
135. sually presented in an analysis of variance ANOVA table as shown below 43 Statistical Add Ins for Excel User Guide Source Degrees of Freedom Sum of Squares Mean Squares F ratio Significance Blocks b 1 SSB MS Treatment t 1 SST MST MST MSE Residual n t b 1 SSE MSe Total n 1 SStoT For row and column designs the model is Y Ut py yt Gt ej where is the effect of the th row and y is the effect of the jth column Usually the rows and columns are orthogonal In the analysis of variance the total variation is partitioned into rows columns treatments and residual For designs in which blocks or rows and columns are not orthogonal to the treatments for example in incomplete block designs the canonical efficiency factors measure the loss of information due to the non orthogonality see John 1987 In such cases the treatment means computed are means adjusted for blocks Once the significant treatment effects have been uncovered they can be further investigated by comparing the differences between the means with the appropriate standard error In the case of unstructured treatments these can be examined using a multiple comparison procedure or using simultaneous confidence intervals MULT_COMPN computes simultaneous confidence intervals for the differences between means with the choice of different methods such as the Tukey Kramer Bonferron and Dunn Sidak When the treatments have a structure suitable comparisons can be
136. t values change automatically Organising the Output We might be very pleased with the results of our computation but not require the output in contiguous locations In fact we might wish to place the 5 values in different cells on different worksheets We can use the Excel Paste Link facility to do this Select one cell of the output Let us suppose itis the 1 corresponding to the minimum of the set of values Copy this into the clipboard either by using the Edit Copy menu or the copy button on the toolbar Now select the new destination cell From the Edit Paste Special menu click the Paste Link button The value 1 of the original cell appears in the new destination cell If however the input data changes so that 1 is no longer the minimum value produced by S5PT_SUMMARY then both the original and 3 6 3 7 Statistical Add Ins for Excel User Guide the new destination cells change automatically This is because the new destination cell contains a reference to the original output cell not just a copy of the original value We will now use NAGExtract a macro provided by NAG in a separate Add In Ensure that the NAGExtract Add In is loaded and click on the NAGExtract menu from the Nag menu on the menu bar We need to tell NAGExtract where the raw output cells are Select the actual cells with the mouse on the worksheet The entry may read something like F 10 F 20 if these cells contain the raw output Now click the Search
137. tated or shifted without affecting the distances between the points If a one dimensional representation was required the best representation might give distances of 7 3 10 3 and 17 3 which may be an adequate representation If the distances were 3 4 and 8 then these distances could not be exactly represented in Euclidean space even in two dimensions the best representation being the three points in a straight line yielding distances 3 4 and 7 In practice the user of scaling methods has to decide upon the number of dimensions in which the data is to be represented The smaller the number the easier it will be to assimilate the information The chosen number of dimensions needs to give an adequate representation of the data but will often not give an exact representation because either the number of chosen dimensions is too small or the data cannot be represented in Euclidean space 71 Statistical Add Ins for Excel User Guide Two basic methods are available depending on the nature of the dissimilarities or distances being analysed If the distances can be assumed to satisfy the metric inequality dij gt dik Ay then the distances can be represented exactly by points in Euclidean space and the technique known as metric scaling classical scaling or principal co ordinate analysis can be used This technique involves the computing of the eigenvalues of a matrix derived from the distance matrix The eigenvectors corresponding to the
138. te 85 6 2 3 4 RUNS TEST S662 cnica ii ii ria 86 6 2 3 5 WILCOXON S6t 2 ives cree mr cesses ces r a E r dice ies es iaa noti 86 6 2 3 6 ONE SAMPLE C1 Set 2 ciencia i aa E a ERRE chase svi coadecescbivecessbavnedecoives 87 6 2 3 7 PROPORTION CF Set 2 csicsces inicios cases e a EE datada R 88 6 2 4 Matched Pairs Samples ccccccccccceccesesseesetseseecuseescnseeseusecseesecassecenseesensecsessecaeesecaaeeseeaesaeeeesaeeeeenaeeeeeaeeaes 88 6 2 4 1 MCNEMAR Set 2 cvies scscuicscesciavscoscnueh ses cnavecuscouvecs coupe cuicvucboescbivbcodenavbeus cbavbsudcbavbeus cbivbsusebavbeds is 88 6 2 4 2 SIGN PAIRS SED acorta ont ita ats cour T rte cias 89 6 2 4 3 WILCOXON PAIRS Set 2 nados ia atari aie 89 6 2 5 TWO AMPLIA A A A see aU eh E AE 90 6 2 5 1 CHAS OZ SCt2 nta 90 6 2 5 2 KOLMOGOROV ZASE diia ia AA 91 6 2 5 3 MANN WHITNEY Set 2 cto EAER 91 6 2 5 4 MEDIAN TEST Set ami ARa R 92 6 2 5 5 TWO SAMPLES GEO O Dania ai 92 6 26 KSamples nne haea A a a dise 93 6 2 6 1 CHIL SQ Ee iii indi E R 93 6 2 6 2 COCHRAN O Set iia E in iaa E 94 6 2 6 3 FRIEDMAN SEE korara i a EAE AEAT E E EN EEOAE O ATEO E N ate 94 6 2 6 4 KRUSKAE WALLIS IS 2 hea E O NEEE TOA E EONO 94 6 2 7 Correlation and A A 6 2 7 1 CONTINGENCY CAS ti E OEA ET A AAA A EONA 95 6 2 7 2 KENDALC CC Dosis E EE EENE E NENTA TAAA AE EAS EO E 95 6 2 7 3 KENDALC TAUS cie AO AEA EALE AE i E ei 96 6 2 7 4 SPEARMAN RHO Seta ti E AENEA AAEE ATAA i EOAR 96 7 TRO
139. ted 21 5 3 3 1 5 3 2 5 3 2 1 Statistical Add Ins for Excel User Guide Parameters Table Required The array containing the contingency table Margins should not be included Output Significance Chi squared Likelihood ratio Degrees of freedom Expected values Chi squared contributions Notes The significance returned is for the chi squared statistic except for a 2 x 2 table when the total number of observations is less than or equal to 40 in which case the probability from Fisher s exact test is returned The chi squared contributions are the values of observed expected expected for each cell which make up the chi squared statistic Remarks See Using the Add Ins and the Introduction to Basic Statistics for further information Book 1 Introduction This book contains materials for two areas e Introduction to Regression and Correlation e Introduction to Time Series Please note that a complete set of examples for this Add In are provided in the workbook Book1E xls This is installed in the Examples subdirectory of the Add Ins directory You might find it helpful to work through this to familiarize yourself with the use of the Book 1 Add Ins Regression and Correlation Introduction Introduction to Regression and Correlation Regression and correlation are both concerned with the relationship between variables In the case of correlation the variables are considered in pairs a
140. ted F fixed at initial values E estimated Default E estimated Parameters Optional Initial values of the model parameters in the order ARMA parameters autoregressive moving average seasonal autoregressive and seasonal moving average transfer function parameters omega parameters followed by delta parameters if present and finally the constant Default initial parameters are set to zero except for the constant which is set equal to the output series mean T_Model Optional For a multi input transfer function model the jth column of the array T_Model specifies the form of the model relating the jth input series to the output series The array must have 4 rows For each column Row 1 is the delay Row 2 is the number of omega parameters Row 3 is the number of delta parameters and 36 Statistical Add Ins for Excel User Guide Row 4 should take the value 1 for a simple input rows 1 2 and 3 are then ignored 2 for a transfer function input for which no allowance is to be made for pre observation period effects and 3 for a transfer function input for which pre observation period effects will be treated by estimation of appropriate nuisance parameters Note There will be at least one omega parameter Omega_0 for each input series in addition to any specified by Row 2 X_series Optional The columns of array X series must contain the original undifferenced values of each of the input series correspond
141. the data should be ordered across blocks i e first observation from Block 1 followed by first observation from Block 2 etc Treatments Optional The treatments applied to the Data These would normally be present A full set of treatments may be given but if the treatments are in a fixed order then only a list of the treatments in order is required and they will be expanded to give the full set Block_order Optional If the data is not given by blocks Block_order should be set to A to indicate that Data goes across blocks When Block_order A a list of treatments will be expanded so that the list of treatments is repeated within each block Tolerance Optional The tolerance value used in the computations to check for zero values If present Tolerance gt 0 0 Default 0 00001 Df_adjustment Optional This is the adjustment to the degrees of freedom for the residual and total The degrees of freedom for the total is set to number of observations Df_adjustment and the residual degrees of freedom adjusted accordingly If present Df_adjustment gt 0 Output The following are output when appropriate 46 5 4 2 3 Statistical Add Ins for Excel User Guide ANOVA table Treatment Means Covariances and Standard Errors Replication Efficiency Factors Grand Mean Block Means Residuals Note In the Covariances and Standard Errors array the upper triangular part contains the variance covariance mat
142. the iterative weighted least squares fitting If present Max_iterations gt 0 Default 10 54 5 4 3 3 Statistical Add Ins for Excel User Guide Tolerance Optional Indicates the accuracy required for the fit of the model If present Tolerance gt 0 0 Default 0 000001 Eps Optional The value of eps is used to decide if the independent variables are of full rank and if not the rank of the independent variables If present eps gt 0 0 Default 0 00000000001 Output Output as a NAGExtract format list RSS df Rank of Model Parameter Estimates Cov If requested see Fitted_values the following are returned Fitted values Residuals Leverages Linear predictor Working weights Notes The residual sum of squares RSS and its degrees of freedom df are returned providing the residual degrees of freedom are not zero The Rank of Model is returned only when the model is not of full rank The Parameter Estimates consist of the four columns with headings giving Parameter Names Parameter Estimate Standard Error t value The variance covariance matrix of the parameters Cov is returned as an upper triangular matrix Remarks See Using the Add Ins and the Introduction to Generalized Linear Models for further information BINOMIAL_GLM Set 1 BINOMIAL_GLM NAG G02GBF fits a generalized linear model with binomial errors Parameters Link Required Indicates which link function is to be use
143. their names 3 A formula where as described above the group variables are classification variables e g eye colour sex These variables can be either text e g red green blue or numerical e g 100 150 300 MM_MLR recodes these values first into an integer representation 1 2 then into the dummy variables required for the model fitting Covariates are ordinary variables e g weight temperature A formula uses the variable names and the symbols A Some examples of formulae are Group variables a b and c Covariates x and z a x fits the group variable a and covariate x that is regression model with different intercepts for the different groups x x2 fits the quadratic model in x a b fits the terms a b a b that is the main effects of a and b and the a b interaction that is a cross classification model a b fits a a b that is the nested model 24 5 3 2 2 5 3 2 3 Statistical Add Ins for Excel User Guide a b c fits a c plus b c that is a b c a c b c The notation is only available for use with covariates There is also an additional parameter Interaction which sets the maximum number of terms allowed in an interaction So for example a b c with Interaction 2 gives a b c a b a c b c with the three term interaction excluded Note that the model matrix produced by MM_MLR assumes that the mean is also fitted If the mean is to be omitted then DUMMY_VARS should be
144. time series Model Required Model specifies the orders p d q P D Q s of the seasonal ARIMA model to be fitted to the output series 38 Statistical Add Ins for Excel User Guide number of autoregressive terms order of non seasonal differencing number of moving average terms number of seasonal autoregressive terms order of seasonal differencing number of seasonal moving average terms the seasonality 2OVUVL AD T_Model Required For a multi input transfer function model the jth column of the array T_Model specifies the form of the model relating the jth input series to the output series The array must have 4 rows For each column Row 1 is the delay Row 2 is the number of omega parameters Row 3 is the number of delta parameters and Row 4 should take the value 1 for a simple input rows 1 2 and 3 are then ignored 2 for a transfer function input for which no allowance is to be made for pre observation period effects and 3 for a transfer function input for which pre observation period effects will be treated by estimation of appropriate nuisance parameters X_series Required The columns of array X series must contain the original undifferenced values of each of the input series corresponding to the model specified in T_Model Parameters Required Initial values of the model parameter in the order ARMA parameters autoregressive moving average seasonal autoregressive and seasonal moving a
145. tion PARTIAL_CORREL_MAT computes partial correlations In contrast to correlation regression looks at how one variable is linearly related to other variables This variable is known as the dependent or response variable and is often denoted by y The other variables are known as the independent or predictor variables and are often denoted by x In the case of one independent variable the model is Yy Po Bixte where is an independent error or noise term with mean zero and constant variance o This can be extended to many variables X4 X2 X3 etc for example y Po PiX B2X2 P3X3 P4X4 which is known as a multiple linear regression model A particular case of this is the polynomial regression model in which x x squared etc giving y Bo Bix Box BX In cases where instead of continuous variables there are classification or grouping variables the multiple linear regression model is known as the general linear model see Searle 1971 In the general linear model the classification variables are fitted by means of a number of dummy 0 1 variables which represent each class or level For example if the classification variable is colour and takes one of the values red green blue or brown the following dummy variables can be used Classification Dummy Dummy Dummy Dummy Variable Variable 1 Variable 2 Variable 3 Variable 4 Red 1 0 0 0 Green 0 1 0 0 Green 0 1 0 0 Blue 0 0 1 0 Brown 0 0 0 1 Re
146. tral estimates and confidence limits are required L logged U unlogged 40 5 4 5 4 1 5 4 2 5 4 2 1 Statistical Add Ins for Excel User Guide Default U unlogged Note If Order is set Division must also be set Order Optional The order of the fast Fourier transform FFT used to calculate the spectral estimates Order must be greater or equal to twice the length of the series It should be a multiple of small primes such as 24m The largest prime factor of Order must not exceed 19 and the total number of prime factors must not exceed 20 Default 2 n where n represents the smallest integer such that 24n gt 2xlength of series Division Optional If Order is set the frequency division of smoothed spectral estimates as 2x Division Division must be a factor of Order Default 2 n 2 where n represents the smallest integer such that 2 n gt 2xlength of series only if Order is not set Output Output as a NAGExtract format list Spectrum Lower Confidence Limit Upper Confidence Limit Bandwidth Degrees of Freedom Remarks See Using the Add Ins and the Introduction to Time Series Methods for further information Book 2 Introduction This book contains materials for two areas e Analysis of Experiments e Generalised Linear Models Please note that a complete set of examples for this Add In are provided in the workbook Book2E xls This is installed in the Examples subdirector
147. uccessfully installed The meaning of this output is explained later in the section on array functions 3 2 Statistical Add Ins for Excel User Guide Using the Add Ins Loading an Add In In order to use the NAG Add Ins they must first be loaded into Excel To do this use the Tools Add Ins menu and select the Books required The Add In NAG Add In Utilities or NAG _Library xla is referenced by the other NAG books and theoretically should not need to be explicitly loaded In practice we have often found it necessary for the user to do this You are advised to experiment with your system to determine whether it is necessary in your particular case If all has gone well you should see a Nag item appear on the menu bar If not then you will have obtained a message box indicating that either an error has occurred in the installation process or the system has been corrupted You should note that installation must be performed via the NAG setup exe routine It does not suffice to copy files from one machine to another Please note that a complete set of appropriate examples are provided in the Excel workbooks These are located in the Examples subdirectory of the Add Ins directory and on the CD or other media After you have read other sections in this chapter you might find it helpful to work through the examples to familiarize yourself with the use of the Add Ins In particular we recommend that you study closely the
148. uired The array containing strictly lower triangle of the distance matrix i e no diagonal elements Note that negative distances are not allowed Dimensions Optional The number of dimensions used to represent the data Default 2 Roots Optional Indicates if all the eigenvalues are to be computed or just the largest corresponding to the number of dimensions A all the eigenvalues are computed L only the largest eigenvalues are computed Default L Largest only Output Output as a NAGExtract format list Coordinates Eigenvalues Remarks See Using the Add Ins and the Introduction to Multivariate Methods for further information 79 80 Statistical Add Ins for Excel User Guide 6 2 6 2 1 Statistical Add Ins for Excel User Guide NAG Books Set 2 Nonparametric Methods Introduction There are 3 Add Ins provided with the NAG Statistical Add Ins for Excel Nonparametric Methods product NAG Add In Utilities A set of utilities used by the other NAG Books Book 4 Nonparametric statistics NAGExtract A utility to reposition output more conveniently on the worksheet Book 4 Introduction to Nonparametric Methods Nonparametric statistics have two advantages over distribution based statistics The first advantage is that they do not rely on the form of the distribution of the data A standard assumption is that the data has a Normal distribution Based on this assumption various statistica
149. vel are constructed from the clusters at a previous level There are two basic approaches to hierarchical cluster analysis agglomerative methods which build up clusters starting from individuals until there is only one cluster or divisive methods which start with a single cluster and split clusters until the individual level is reached This Book contains the more common agglomerative methods The stages in a hierarchical cluster analysis are usually as follows 1 Form a distance matrix 2 Use the selected criterion to form the hierarchy 3 Print the cluster information in the form of a dendrogram or use the information to form a set of clusters Given a distance matrix for the n individuals as described above an agglomerative clustering method produces a hierarchical tree by starting with n clusters each with a single individual and then at each of n 1 stages merging two clusters to form a larger cluster until all individuals are in a single cluster thus forming a tree At each stage the two clusters that are nearest are merged to form a new cluster and a new distance matrix is computed for the reduced number of clusters The methods differ as to how the distances between the new cluster and other clusters are computed Six methods are available 69 Statistical Add Ins for Excel User Guide Single link or nearest neighbour Complete link or furthest neighbour Group average Centroid Median Minimum variance DAARION
150. verage transfer function parameters omega parameters followed by delta parameters and finally the constant Default initial parameters are set to zero except for the constant which is set equal to the output series mean X forecasts Required The columns of the array X_forecasts must contain in the forecasted values for the input series in the same order as in X series Fix_constant Optional Indicates whether the constant is to be fixed at its initial value or estimated F fixed at initial values E estimated Default E estimated X_Model Optional The array X_Model may contain the ARMA model for each of the input series in the standard form of p d q P D Q s In the case of those inputs for which no ARIMA model is available the corresponding orders should be set to 0 RMSX Optional If X_Model is set RMSX must contain the estimated residual variance of the input series ARIMA models In the case of those inputs for which no ARIMA model is available or its effects are not to be excluded in the calculation of forecast standard errors the corresponding entry of RMSX should be set to 0 39 5 3 3 9 Statistical Add Ins for Excel User Guide ParamX Optional If X_Model is set ParamX must contain the values of the autoregressive moving average seasonal autoregressive and seasonal moving average parameters for each of the input series ARIMA models corresponding to the models specified in X_Model Cells
151. xample the McNemar test for paired data and the Cochran Q test for k related samples For data with more than two categories most of the statistics are based on the 2 test for contingency tables For more complex modelling of binary and general nominal data generalised linear models can be used in particular the logistic regression model and the log linear model Both of these models are available in Book 2 of the NAG Statistical Add Ins for Excel In addition to tests it is also possible to provide interval estimates confidence intervals for location parameters based on ranks rather than the t distribution These provide a robust alternative to the standard methods The table below gives a list of the nonparametric functions available in this Book The first row is for nominal data the second for ordinal data You are advised to consult a standard text book on nonparametric statistics for further information on these tests In particular see Siegel 1956 and Conover 1980 Both are available in later editions Most of the examples given in the Example Workbook are from Siegel 82 6 2 2 6 2 2 1 Table of available functions Statistical Add Ins for Excel User Guide Sample type Nominal Ordinal One sample Binomial test BINOMIAL_TEST Chi squared test for one sample CHI_SQ_1 Proportion confidence interval PROPORTION_Cl Cox Stuart test COX_STUART One sample confidence interval ONE_SAMPLE_C
152. y of the Add Ins directory You might find it helpful to work through the examples to familiarize yourself with the use of the Book 2 Add Ins Analysis of Experiments Introduction Introduction to the Analysis of Designed Experiments using the NAG Excel Add Ins Background An experimental design is the plan used by the experimenter when allocating a set of controlled conditions to the experimental material A designed experiment can be contrasted with an 41 Statistical Add Ins for Excel User Guide observational study in which the conditions are observed as they occur naturally The controlled conditions are known as treatments and the experimental material is divided up into units sometimes called plots each of which can receive a different treatment Two examples follow 1 In an experiment to examine the effects of different diets on the growth of chickens the chickens were kept in pens and a different diet was fed to the birds in each pen In this example the pens are the units and the different diets are the treatments 2 In an experiment to compare four materials for wear loss a sample from each of the materials is tested in a machine that simulates wear The machine can take four samples at a time and a number of runs are made In this experiment the treatments are the materials and the units are the samples from the materials In designing an experiment the following principles are important Randomisation Given the overa

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