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User's Guide to Program MIX: An Interactive Program for Fitting

1. 9 bete tpe or 9 4 GPO WEN CUEV o ie Nests tn cis 9 2023 Constraints O rE i aaea 10 pee ARE 10 8 fixed neni nn 10 2 Fixed coefficient OF yabdtbort dae ree Hbri oneri eater uento 10 3 Constant coefficient of variation nennen 11 4 Sipinas equal osss aca tive ec rae SU e t rte iate eem Redes 11 2 3 N mencal DEGBISIDILS poscit meto aue qun statt acis elotiut tesi met e rures 11 3 How torun MIX ico mii oem dE 12 Option Jast or 12 Option 1 Read new set of data ee ee 12 Option 2 Read a full set of parameter 13 Option 3 Revise specified parameter 13 iii Option 4 Estimate proportions for fixed means 14 Option 5 Estimate means sigmas for fixed 14 Option 6 Estimate proportions means 5 16 Option 7 Restore parameters to values from pr
2. 27 Standard licence agreement for MIX 0 000000 001001 60 iv User s Guide to Program MIX 1 INTRODUCTION 1 1 MIX An interactive program for fitting mixtures of distributions MIX analyzes histograms as mixtures of statistical distributions that is by finding a set of overlapping component distributions that gives the best fit to the histogram The components can be normal lognormal exponential or gamma distributions An example is shown in Figure 1 there are five component lognormal distributions with different weights and their sum shown as a thick line matches the shape of the histogram as closely as possible The statistical method used to fit the mixture distribution to the data is maximum likelihood estimation for grouped data MIX will fit up to fifteen components with the data grouped over as many as eighty grouping intervals This is the best way to analyze samples from mixed populations Size frequency distributions in animal populations with distinct age groups times to failure in a mixture of good and defective items and the distribution of some diagnostic measure in a mixed population of patients some of whom have a given disease and some of whom do not are all examples of mixed populations MIX can also be used in a more general descriptive way to analyze multimodal and other irregularly shaped histograms A Plot 001 Data Heming Lake Pike 1965 Components Logn
3. 4 19 75 10 21 75 Two errors will be put in deliberately so that editing 21 23 75 can 6e demonstrated ater 11 25 75 14 25 75 lt This right boundary should have been 27 75 31 29 75 39 31 75 70 33 15 71 35 15 44 37 15 42 39 15 36 41 75 23 43 15 22 45 75 17 41 15 12 49 75 12 51 75 11 53 75 8 55 75 3 57 15 6 59 75 6 61 75 3 63 75 2 65 75 Enter count 2 lt This count should have been 5 INTERVAL OBSERVED COUNT RIGHT BOUNDARY Heming Lake Pike 1965 25 MIX 2 3 Any errors to correct Y N Which interval is incorrect 25 BWN S 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 11 44 12 0000 10 21 0000 0000 0000 14 31 39 70 71 0000 0000 0000 0000 0000 0000 42 36 23 22 17 0000 0000 0000 0000 0000 0000 12 11 0000 0000 0000 0000 0000 0000 0000 0000 0000 Enter correct count 5 Any errors to correct Y N N ERROR INTERVAL OBSERVED COUNT RIGHT BOUNDARY 21 11 14 3 4 5 19 21 23 25 25 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 7500 Let MIX detect the incorrect boundary INTERVAL BOUNDARIES OUT OF ORDER 0000 000
4. Means 22 3050 31 8944 38 8973 49 7120 59 1185 Sigmas 2 0000 3 0000 4 0000 5 0000 6 0000 FIXED FIXED FIXED FIXED FIXED Degrees of freedom 19 Chi squared 29 8516 Option number 0 for list of options 1 to STOP 4 Now revise the proportions again to adjust for the new means Note that the iterations of Option 4 are very fast and they usually converge Here the fit is substantially improved by Option 4 ESTIMATE PROPORTIONS FOR FIXED MEANS SIGMAS Distribution selected is Normal Enter iteration limit 20 Number of iterations 6 Fitting Normal components Proportions and their standard errors 08376 37744 36832 11748 05299 01292 02833 03091 02159 01383 Means ALL HELD FIXED 22 3050 31 8944 38 8973 49 7120 59 1185 Sigmas ALL HELD FIXED 2 0000 3 0000 4 0000 5 0000 6 0000 Degrees of freedom 20 Chi squared 16 9645 Option number 0 for list of options 1 to STOP 11 32 User s Guide Even though we have not started to fit the sigmas the fit is looking very good Plot 004 Data Heming Lake 1965 Components Normal Option number 0 for list of options 1 to STOP 6 We now go for the final fit in this sequence using Option 6 with all proportions free all means free and a constant coefficient of variation ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Normal Enter iteration
5. User s Guide the histogram The axes are not labeled but the positions of the means p are indicated with triangles The abcissa is scaled so that no component extends off either side of the graph If lognormal or gamma distributions are being fitted the abcissa line begins at zero The leftmost and rightmost grouping intervals are shown extending to their respective ends of the abcissa line If the graph extends off the top of the screen you can have it re drawn with reduced vertical scale The plots done by Options 10 and 11 are numbered sequentially during the session beginning at Plot 001 Apple Macintosh users can copy a plot to the clipboard or save it as a MacDraw file The graphs in this User s Guide were produced in this manner Macintosh users can also elect to create an ultra high resolution plot with a 4x magnification factor this plot will appear the usual size on the screen but if it is saved on the clipboard or as a file and opened with a graphics program such as MacDraw it will be seen at its full size It can be then be reduced when it is printed to give publication quality results An example is shown in Figure 5 IBM PC users can send the plot to a printer by pressing Y or y when the graph is displayed although this may require additional software as explained in 1 4 Pressing almost any other key will clear the screen and bring the next prompt for an Option number Mainframe users may choose to send the plo
6. Sigmas 2 0000 3 0000 4 0000 5 0000 6 0000 Option number 0 for list of options 1 to STOP 11 Plot a graph now to check that the means and sigmas are reasonable Remember that the proportions have not yet been fitted A A A A A Plot 002 Data Heming Lake Pi ke 1965 Components Normal Option number 0 for list of options 1 to STOP 4 Now improve on the starting values for the proportions ESTIMATE PROPORTIONS FOR FIXED MEANS SIGMAS Distribution selected is Normal Enter iteration limit 20 Number of iterations 9 Fitting Normal components 28 User s Guide Proportions and their standard errors 05057 36771 43737 09155 05279 01024 02413 02743 02083 01313 Means ALL HELD FIXED 20 0000 30 0000 40 0000 50 0000 60 0000 Sigmas ALL HELD FIXED 2 0000 3 0000 4 0000 5 0000 6 0000 Degrees of freedom 20 Chi squared 104 174 Option number 0 for list of options 1 to STOP 11 Plot graph now to see how well we have done The means are not quite right but the proportions and sigmas look good A A A A A Plot 003 Data Heming Lake ke 1965 Components Normal Option number 0 for list of options 1 to STOP 6 We will attempt to fit all proportions all means and a constant coefficient of variation using Option 6 The attempt fails because we are trying to estimate too many parameters at once when the initial values of the parameters are too far from the true fit This should ha
7. 15 16 6000 17 0000 47 7500 16 13 7440 12 0000 49 7500 17 11 4795 12 0000 51 7500 18 9 4116 11 0000 53 7500 19 7 5849 8 0000 55 7500 20 6 1474 3 0000 57 7500 21 5 0830 6 0000 59 7500 22 4 2318 6 0000 61 7500 37 MIX 2 3 23 3 4404 24 2 6553 25 4 8736 3 0000 2 0000 5 0000 63 7500 65 7500 Variance covariance matrix for parameter estimates pi 5 and all fixed parameters are excluded 0 2182 03 0 7915E 04 0 9921 02 0 1501 03 0 4258 02 0 4598 02 0 1111 03 0 4480 02 0 3267 03 0 3545 02 0 1978 02 0 9162 02 0 5659 02 0 4348 02 0 2214 02 0 7246 01 0 3583 01 0 3096 01 0 5781 02 0 2607E 00 0 6190E 01 0 1527E 00 gt gt 0 5517 02 0 2552E 00 0 5088E 01 0 1892E 00 gt gt 0 1915E 02 0 2286E 02 0 8024E 01 0 6137E 01 0 4548E 01 gt gt 0 1093E 02 0 1176E 02 0 2113E 03 0 1467E 01 0 8373E 02 0 5545E 02 gt gt 0 2464 00 0 7937 02 0 5278 01 Fitting Gamma components Proportions and their standard errors 11729 05039 05954 03311 09684 49464 24084 01477 09961 06781 Means their standard errors 22 9483 33 3271 40 4566 4424 8025 2 9075 Sigmas CONSTANT COEF OF VAR 2 2777 3 3079 4 0155 2297 Degrees of freedom 14 Chi squared 11 7257 Option number 0 for list of options 11 1957 00 1089 00 2766 00 2449 00 7506 01 3175E 01 49 2933 60 3671 4 3763 3 4292 6440E 00 1898 01 1715 01 45
8. January 1988 Do you want to see a list of Options Y N LIST OPTIONS 0 List of options 1 Read a new set of data 2 Read a full set of parameter values 3 Revise specified parameter values 4 Estimate proportions for fixed means sigmas 5 Estimate means sigmas for fixed proportions by constrained search 6 Estimate proportions means sigmas with or without constraints and or give diagnostic displays 7 Restore parameters to values from previous step 8 Regroup data or restore to original grouping 9 Choose a distribution 10 Plot histogram 11 Plot histogram and fitted components 12 Toggle to echo all I O to I O log 1 STOP Option number 0 for list of options 1 to STOP 12 Open disk file to keep a record of this session OPENING FILE FOR I O LOG 24 User s Guide Enter file name in single quotes PIKE65 LOG Creating I O file PIKE65 LOG Option number 0 for list of options 1 to STOP 1 Reading the 1965 Heming Lake Pike data from the keyboard If the file PIKE6S is available respond N to the next prompt to read the data from the file instead of from the keyboard you will be prompted for the file name READ A NEW SET OF DATA Do you want to enter data from keyboard Y N Y Enter title 1 25 characters Heming Lake Pike 1965 Enter the number of intervals NOTE Must be at least 2 at most 80 25 Enter count and right boundary 24 times
9. T1 1 so there are only A 1 free proportions Suitable constraints for the means and standard deviations will depend on the application It may be that for some component i u and 0 are known from other data and can be held fixed at those given values In some applications it may be reasonable to assume that the standard deviations are all equal Op or that the coefficients of variation are all equal 0 u oy uj These and other constraints allowed by MIX are discussed in 52 2 MIX assumes that the data are grouped in the form of numbers of observations over successive intervals Data often come grouped as a histogram or can be grouped with very little loss of information Grouping greatly simplifies the calculation of maximum likelihood estimates Macdonald and Pitcher 1979 The grouping intervals are specified by their right hand boundaries The first leftmost and last rightmost intervals are open ended that is if there are m intervals the first interval includes everything up to the interval boundary the second everything from x the which includes everything above x 1 Thus it is only necessary to specify 1 boundaries choice of boundaries is discussed in 55 and in Macdonald and Pitcher 1979 to x and so on to the m 1 interval which includes everything from 2 to x MIX can be used if percent mass or something other than a sample count is given for
10. and thereby impairing our ability to develop new software A copy of the standard Licence Agreement is shown on page 60 8 UPGRADES Each time a new Release is announced licensed users will be offered the upgrade for a nominal charge Any licensed user who suggests a worthwhile improvement to MIX will be sent the next upgrade free of charge Any licensed user who succeeds in crashing MIX so that control is involuntarily returned from MIX to the operating system should send us details of the computer and operating system being used and a disk containing a copy of the data file and the complete input output log for that session In return the user will receive the next upgrade free of charge REFERENCES Cassie 1954 Some uses of probability paper in the analysis of size frequency distributions Australian Journal of Marine and Freshwater Research 5 513 522 Everitt B S and D J Hand 1981 Finite Mixture Distributions Chapman and Hall London xi 143 Macdonald P D M and T J Pitcher 1979 Age groups from size frequency data a versatile and efficient method of analysing distribution mixtures Journal of the Fisheries Research Board of Canada 36 987 1001 Macdonald P D M 1987 Analysis of length frequency distributions Jn R C Summerfelt and Hall editors Age and Growth of Fish Iowa State University Press Ames Iowa 371 384 McLachlan G J Basford 1988 Mixture Mode
11. considered as a poor approximation If the data give percents mass or anything other than counts over the grouping intervals then the P value will have no meaning User s Guide although a reduced chi square value will still indicate an improved fit relative to another fit to the same data If the data can be fitted with and without a certain constraint the validity of that constraint can be tested Removing the constraint will in general reduce the chi square and the degrees of freedom the reduction in chi square is itself a chi square statistic with degrees of freedom equal to the reduction in degrees of freedom Rao 1965 p 350 This is only valid if the data give actual counts over intervals and if most counts are 5 or greater In this way it is possible to test whether or not the proportions of the mixture are all equal whether or not the means lie on a growth curve or whether or not the data came from a mixture of exponential distributions to give just a few examples The test for exponential distributions is done by fitting gamma distributions first with the constraint that the coefficient of variation be fixed at 1 then without that constraint In the Example in the Appendix the hypothesis that the means lie on a growth curve assuming lognormal distributions and a constant coefficient of variation can be tested by a chi square statistic of 12 4566 11 9477 0 5089 on 16 14 2 degrees of freedom The fits used
12. constraints of the previous attempt It may even turn out that the standard errors cannot be computed because the information matrix is singular Macdonald and Pitcher 1979 This will happen if there is no information in the data for one or more of the parameters an extreme case being where the user assigns a zero proportion to one component and then attempts to estimate mean and standard deviation of that component This will also happen if the current parameter values are so far from their true values that the observed and fitted histograms bear no resemblance to each other In either case inspection of the plot from Option 11 inspection of the current parameter values and consideration of what the solution ought to be should suggest a revision of the starting values and or constraints that will be more successful on the next attempt In the event that it is still not evident how to adjust the starting values try Option 5 It is always better to choose initial values for the standard deviations that are too small rather than too large Large standard deviations cause the components to overlap more than is necessary obscuring the resolution of the means It is often possible to get good estimates of the means while holding the standard deviations fixed at values slightly less than their true values The Example in the Appendix illustrates a strategy that will often work In this Example we did have the advantage of knowing ahead of time that there
13. each interval but the standard errors of the estimates and the goodness of fit tests will not be valid in such cases except in a relative sense within the analysis of a given data set MIX can also be used to test the goodness of fit of the model to the data and in some cases it be used to test the validity of certain constraints These tests depend on the chi square approximation to the likelihood ratio statistic Rao 1965 and will be valid as long as most of the intervals have expected counts of 5 or greater The goodness of fit chi square statistic is printed after each fitting step The degrees of freedom are computed as the number of grouping intervals minus 1 minus the number of parameters estimated Note that MIX does not count parameters that were held fixed during an estimation step as parameters estimated if in fact they had been adjusted to fit the data at an earlier step in the session they have in a sense been estimated and the degrees of freedom computed by MIX should be reduced by at most 1 for each such parameter After a successful fit MIX will compute a significance level P value for the goodness of fit test see Option 6 in 53 In the situation just described where some fixed parameters had been estimated at earlier steps the P value should be re calculated from a table of chi square using the reduced degrees of freedom If the counts in most intervals are small most less than 5 say then the P value given should be
14. each Option is described in detail below Option 0 List of options Display the list of the Options on the screen Option 1 Read a new set of data Input a new data set either from a prepared file or from the keyboard The data may then be edited and written onto a file If you are entering data from the keyboard MIX will prompt for a title 1 to 25 characters and the number of grouping intervals A maximum of 80 grouping intervals is allowed If there are m grouping intervals in the data there will be m 1 right hand boundaries to enter see 2 1 so MIX will first prompt for m 1 counts and right boundaries The right boundaries must be in strictly ascending order Enter each count and right boundary pair on a new line enter the count first and separate it from the right boundary by a space or acomma When all m 1 pairs have been entered MIX will prompt for the count in the last interval After it has been entered the data will then be displayed on the screen for verification There is then provision to re enter any count and right boundary or in the case of the rightmost interval just the count if required MIX checks to see if 11 MIX 2 3 all the boundaries in strictly ascending order and will not proceed to the next step until any exceptions have been corrected Data files can be created beforehand using a text editor Write the title in columns 1 to 25 on the first line and write the number of intervals on the
15. force the means to lie along a von Bertalanffy growth curve Option 6 fails here Option 6 will not usually succeed in fitting a growth curve unless the means are already very close to the best fit especially when proportions and sigmas are also being estimated ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Lognormal Enter iteration limit 0 gives displays with current parameters 50 Display observed and expected counts as a table Y N N Display observed and expected counts as a graph Y N N Display variance covariance matrix Y N N Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 0 Constraints means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE 42 User s Guide Enter choice 4 Is Kth mean different Y N N Constraints on sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N N Do you want to abort Y N N PARAMETERS OUT OF RANGE AFTER 1 ITERATIONS RESTORE PARAMETERS TO VALUES FROM PREVIOUS STEP Proportions Means Sigmas 09967 51889 22677 10710 04757 23 0735 33 6069 41 1029 49 8826 60 4670 2 3722 3 4551 4 2258 5 1284 6 2166 Option number 0 for list of options 1 to STOP 5
16. from the previous step 14 User s Guide The iterations will not always converge especially if insufficient constraints are imposed or if the initial parameter values are not good This is discussed in 4 1 For diagnostic purposes the maximum number of iterations may be set to 0 in this case the parameter values will not be changed but any of the tables or displays may be obtained In cases where the proportions are not being changed by Option 6 such as when the iteration limit is set to 0 or when all the proportions are held fixed the proportions must be non negative and sum to 1 otherwise the goodness of fit chi square computed by MIX will be meaningless it may even be negative The proportions can be prepared using Option 3 The first prompt is for the maximum number of iterations to be allowed In most cases convergence will come after about 20 or 30 iterations if it comes at all but some pathological cases will not converge until about 60 iterations Enter 0 to get diagnostic displays without changing any of the parameter values The table of observed and expected counts is useful especially to see where any small expected counts occur 4 3 The graph of observed and expected counts is not as useful as the high resolution graphs plotted by Options 10 and 11 It is a histogram if the grouping intervals are of equal width but it is not re scaled if they are not It is useful only as a graphical representation of the table
17. in this test are found on pages 50 and 44 Since P 0 78 the hypothesis that the means lie on a growth curve cannot be rejected The goodness of fit test only indicates how well the mixture distribution g x fits the histogram overall If the components overlap extensively the test is not very sensitive to features that are obscured by the overlapping such as skewness of the component distributions Hence we cannot conclude from the analyses in the Appendix whether the component distributions in the pike data are really normal lognormal or gamma each fit is about as good as the other Similarly the test shown above to determine whether or not the means lie on a growth curve has very low power 2 2 Constraints on the parameters The constraints on the parameters are explained below under the headings that will appear on the screen as prompts 2 2 1 Constraints on proportions 0 none Only the natural constraint z7 1 is imposed MIX does not constrain the proportions to be non negative Negative values can occur in some pathological situations and suggestions for handling them are given in 4 2 1 Specified proportions fixed In addition to the natural constraint zt 1 any or all of the proportions may be held fixed while other parameters are being estimated If a is the number of proportions held fixed in this way the number of free proportions 15 k a 1 where the 1 accounts for the natural constrai
18. limit 0 gives displays with current parameters 30 Display observed and expected counts as a table Y N Y Display observed and expected counts as a graph Y N Y Display variance covariance matrix Y N Y Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 0 Constraints on means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE 33 MIX 2 3 Enter choice 0 Constraints sigmas 0 NONE 2 FIXED COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N N Do you want to abort Y N N Number of iterations 11 INTERVAL EXPECTED COUNT OBSERVED COUNT RIGHT BOUNDARY 1 1 OY 4 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 19 75 21 75 23 75 25 75 27 75 29 75 31 75 33 75 35 75 37 75 34 4 0295 11 5477 17 4449 13 7595 12 8211 26 5010 49 7208 66 0656 64 2171 51 3314 40 2333 33 1477 26 6248 20 4391 16 2169 13 8090 11 8220 9 5720 7 4102 5 8451 4 9222 4 3083 3 6669 2 8831 4 6610 EEEEEEEEEEEEEEE 1 SPECIFIED SIGMAS FIXED 3 CONSTANT COEF OF VARIATION 4 0000 10 0000 21 0000 11 0000 14 0000 31 0000 39 0000 70 0000 71 0000 44 0000 42 0000 36 0000 23 0000 22 0000 1
19. of observed and expected counts The symbols used are O Marks the observed count E Marks the fitted or expected count and used to shade in the column X Used when an O and an E are superimposed Used when the columns of E s goes off the page I Used when an O and a are superimposed If the variance covariance matrix for the estimates is requested it will appear as a lower triangular matrix The sequence of variables is the free proportions in order the directly estimated or free means in order the directly estimated or free standard deviations in order Parameters are tabulated with their standard errors Parameters that were held fixed are indicated by the word FIXED being displayed in place of a standard error Parameters which by reason of some other constraint were not estimated directly have no standard errors given If too many components have been assumed or too few constraints have been imposed or if the initial values are too far from the true values either the information matrix will become singular or the parameters being estimated will iterate out of the admissible range In either case a message will be displayed and Option 7 will be called automatically to restore the parameters to values from the previous step See 4 1 for a discussion of what to do next Option 7 Restore parameters to values from previous step Restore parameters to their values from previous step No input is required The restored
20. suitable and you may have to experiment to find a better value according to the relative magnitude of the chi square The final prompt of Option 5 asks if you want to abort this is the only provision for escape if you realize that your input is not appropriate Response Y leads back to the prompt to choose an Option response N begins the direct search The output is self explanatory The choice of constraints is indicated by acronyms FIXED fixed MEQ means equal EQSP means equally spaced GCRV means on a growth curve FCOV fixed coefficient of variation CCOV constant coefficient of variation and SEQ standard deviations equal under those parameters which by reason of the constraints are not being estimated directly It may be that the initial values lie outside the region of admissible values defined by the upper and lower limits on the means and standard deviations and the iterations never penetrate the admissible region This will be flagged by an error message and the final value of chi square will be 100000 17 In extremely pathological cases the means and standard deviations will lie within the upper and lower bounds specified but are inadmissible for some other reason For example they may specify a mixture that is nowhere near the observed histogram This will be flagged by an error message and the final value of chi square will be 100000 16 Option 5 should not be used if the proportions do not sum to 1 or
21. the leftmost first and rightmost intervals are always open ended 2 1 on the histogram the first interval is shown as being twice the width of the second and the is shown as being twice the width of the 15 The first and m 1 right boundaries are marked and labeled on the abcissa MIX also looks for three boundaries in between them that are as close as possible to being equally spaced and marks and labels them on the abcissa The plots done by Options 10 and 11 are numbered sequentially during the session beginning at Plot 001 Apple Macintosh users can copy the plot to the clipboard or save it as a MacDraw file The graph from Option 10 shown on page 30 was produced in this manner IBM PC users can send the plot to a printer by pressing Y or y when the graph is displayed although this may require additional software as explained in 1 4 Pressing almost any other key will clear the screen and bring the next prompt for an Option number Mainframe users may choose to send the plot to an off line plotter this prompt comes before the plot is displayed on the screen Mainframe screen graphics are in text mode Option 11 Plot histogram and fitted components A high resolution graph of the histogram of the current data will be displayed The weighted component distributions ny f x and the mixture distribution g x n 10 n f x are computed from the current parameter values and superimposed on 16
22. values are displayed 15 MIX 2 3 Option 8 Regroup data or restore to original grouping Regroup the data or restore the original grouping This option facilitates the removal or re insertion of interval boundaries Restoration of the original grouping is the only way to re insert interval boundaries Boundaries can be removed one at a time by entering a boundary at the prompt the two intervals on either side will then become one and the two counts will be summed You can use Option 8 to write the data to a file This is useful if you forgot to create a file in Option 1 or if you have regrouped the data and want to save it in its regrouped form To display the current data use Option 8 without restoring to the original grouping or removing a boundary Option 9 Choose a distribution Select a distribution The choice is between normal lognormal or gamma distributions By default the normal distribution is selected when execution begins Because the lognormal and gamma distributions are defined only for positive valued random variables the distribution will be reset to the normal distribution and a message will be displayed if the first right boundary is negative or if a mean is negative when the lognormal or gamma distribution is chosen This can happen during Option 9 or after any one of Options 1 2 3 7 or 8 Option 10 Plot histogram A high resolution graph of the histogram of the current data will be displayed Although
23. were exactly five components present and that the coefficients of variation could be assumed to be constant Ways to handle length frequency distributions and other applications where the number of components is large and unknown are discussed in 5 The main steps in the Example are as follows the data were entered and displayed on a histogram then starting values were given for the parameters The starting values of the proportions did not have to be chosen carefully because Option 4 succeeded in finding good values Option 5 restricted to about 200 iterations was then used to improve the means while holding the standard deviations fixed At this point it is best to have some constraints on the standard deviations holding most of them fixed if none of the other constraints offered seem to be applicable Option 4 was used to revise the proportions in light of the new means and the final fit with constant coefficient of variation was then found by Option 6 It was not possible to relax the coefficient of variation constraint but alternatives could be tried such as equal standard deviations Macdonald 1987 or holding some standard deviations fixed In this Example the fits with normal lognormal and gamma distributions are almost identical the normal is the fastest to compute so fits were first done using the normal distribution The distribution was then switched to the gamma and the fit was adjusted by Option 6 The lognormal distribution
24. 0 0000 23 25 25 7500 7500 7500 Heming Lake Pike 1965 Do you want to abandon this data set Y N N Which interval is incorrect 5 Enter correct count and right boundary 14 Any errors to correct Y N N 27 75 26 User s Guide Do you want to store these data on a file Y N Y Enter file name in single quotes PIKE65 Writing to file PIKE65 Do you want to display the data again Y N N Option number 0 for list of options 1 to STOP 10 19 75 31 75 41 75 53 75 65 75 Plot 001 Data Heming Lake Pi ke 1965 Option number 0 for list of options 1 to STOP 2 Read in starting values for all the parameters Starting values for the means and sigmas should be as good as possible they can be found inspecting the histogram from Option 10 above and from Knowing something about the population being studied Starting values for the proportions are less critical since they can usually be improved very efficiently by Option 4 READ A FULL SET OF PARAMETER VALUES How many components Must be at least 1 at most 15 5 Enter the 5 proportions 11111 The proportions need not sum to 1 since they can be re scaled Enter the 5 means 20 30 40 50 60 27 MIX 2 3 Enter the 5 sigmas 23 45 6 Proportions do not sum to 1 Do you want to re scale Y N Y Proportions 20000 20000 20000 20000 20000 Means 20 0000 30 0000 40 0000 50 0000 60 0000
25. 0 3 0000 57 7500 21 6 0000 59 7500 22 6 0000 61 7500 23 10 0000 Do you want to store these data on a file Y N N Option number 0 for list of options 1 to STOP 2 Initialize parameters for a 4 component fit Components 1 to 3 correspond to components 1 to 3 in the previous fits but component 4 now corresponds to the previous components 4 and 5 combined We can begin with equal proportions trusting Option 4 to improve them on the next step READ A FULL SET OF PARAMETER VALUES How many components Must be at least 1 at most 15 Enter the 4 proportions 1111 Enter the 4 means 23 34 45 60 Enter the 4 sigmas 2346 Proportions do not sum to 1 Do you want to re scale Y N Y 53 MIX 2 3 Proportions 25000 25000 25000 25000 Means 23 0000 34 0000 45 0000 60 0000 Sigmas 2 0000 3 0000 4 0000 6 0000 Option number 0 for list of options 1 to STOP 4 Revise the proportions to adjust for the new means and sigmas ESTIMATE PROPORTIONS FOR FIXED MEANS SIGMAS Distribution selected is Lognormal Enter iteration limit 20 Number of iterations 6 Fitting Lognormal components Proportions and their standard errors 10473 57831 24672 07024 01380 02356 02167 01239 Means ALL HELD FIXED 23 0000 34 0000 45 0000 60 0000 Sigmas ALL HELD FIXED 2 0000 3 0000 4 0000 6 0000 Degrees of freedom 19 Chi squared 36 8714 Option number 0 for list of options 1 to STOP 6 Us
26. 0 3388E 00 0 2002 02 0 1009 00 0 3493 01 0 5324 01 0 8489E 01 0 7211E 00 0 2872E 01 gt gt 0 1833E 02 0 7862E 01 0 1913E 01 0 6068E 01 0 6932E 01 0 5055E 00 0 2923E 01 gt gt 0 5895E 01 0 1278 02 0 2249 01 0 2166 01 0 8162 02 0 3875 01 0 1136E 00 0 1140E 01 gt gt 0 3486E 01 0 5131E 01 0 1277 03 0 5636 02 0 4071 02 0 1417 02 0 1156 01 0 5456 01 0 8525 01 gt gt 0 2467 01 0 4775 01 0 3276 01 Fitting Normal components Proportions and their standard errors 09200 46711 25858 12793 05438 01439 07080 05497 03991 02184 Means and their standard errors 22 7487 32 9675 39 7800 48 6285 60 1262 4317 5821 1 6946 2 4280 2 2651 Sigmas CONSTANT COEF OF VAR 2 1718 3 1474 3 1810 Degrees of freedom 14 Chi squared 11 2852 7978 P 4 6425 6635 0955 and standard error 5 7402 35 MIX 2 3 Option number 0 for list of options 1 to STOP 11 The chi square test above and the plot below both indicate an excellent fit We will not get a significantly better fit to these data A A A A amp Plot 005 Data Heming Lake Pi ke 1965 Components Normal Option number 0 for list of options 1 to STOP 9 Change from fitting mixtures of Normal distributions to fitting mixtures of Gamma distributions Fitting Gamma distributions can take a lot of computer time SELECT A DISTRIBUTION Enter 1 Normal 2 Lognormal or 3 Gamma 3 Distribution s
27. 1450 22268 06024 01596 03787 03495 02297 Means EQUALLY SPACED and standard errors 23 2958 34 4244 45 5531 59 3126 4837 4478 2 4084 Sigmas CONSTANT COEF OF VAR 1133 and standard error 2 6398 3 9009 5 1619 5 5745 1908 2 2103 Degrees of freedom 14 Chi squared 12 8913 5351 Option number 0 for list of options 1 to STOP 11 User s Guide This fit is more satisfactory than the one previously obtained with 5 equally spaced means Plot 009 The present fit finds almost exactly the same values for the parameters of components 1 to 3 but treats age groups 4 and 5 as a single component Plot 010 Data Heming Lake Pi ke 1965 Components Option number 0 for list of options 1 to STOP 1 Execution of MIX terminated Lognormal 57 MIX 2 3 Standard licence agreement for MIX users This Licence Agreement is made and entered into this day of 19___ BETWEEN ICHTHUS DATA SYSTEMS a duly registered partnership under the laws of the Province of Ontario Canada OF THE FIRST PART and hereinafter called the Licensee OF THE SECOND PART WHEREAS the Licensor has developed a program relating to the statistical analysis of mixtures of distributions AND WHEREAS the Licensee is desirous of obtaining from the Licensor a licence to use the said program and the Licensor is desirous of granting a licence to the Licensee to allow its use of the program NOW THEREFORE THIS AGREEMENT WITNES
28. 7 0000 12 0000 12 0000 11 0000 8 0000 3 0000 6 0000 6 0000 3 0000 2 0000 5 0000 19 7500 21 7500 23 7500 25 7500 27 7500 29 7500 31 7500 33 7500 35 7500 37 7500 39 7500 41 7500 43 7500 45 7500 47 7500 49 7500 51 7500 53 7500 55 7500 57 7500 59 7500 61 7500 63 7500 65 7500 EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEOEEEEEEEEE EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEOEEEEEE 0 0 User s Guide 39 41 43 45 47 49 51 53 55 57 59 61 63 65 75 75 75 75 75 75 75 75 75 75 75 75 75 75 EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEO EEEEEEEEEEEEEEEEEEEEEEEEEEEE O EEEEEEEEEEEEEEEEEEOEEE EEEEEEEEEEEEEEEEE O EEEEEEEEEEEEEX EEEEEEEEEOEE EEEEEEEEEX EEEEEEEEO EEEEEEO EEOEE EEEEO EEEEO EEX EX EEEX The variance covariance matrix is seldom useful but it can be displayed Option 6 if required the 10 rows and similarly 2 3 the 10 columns correspond to the 10 free parameters 1 4 H1 2 M4 MS O1 Variance covariance matrix for parameter estimates pi 5 and all fixed parameters are excluded 0 2069E 03 0 4251E 04 0 5013E 02 0 9522E 04 0 2793E 02 0 3022E 02 0 5008E 04 0 1763E 02 0 6488E 04 0 1593E 02 0 1840E 02 0 2142 02 0 2438 02 0 1199 02 0 1864E 00 0 1103E 02 0 3452 01 0 2208 01 0 1128 01 0 4927 01
29. 86E 00 1329 00 0993 and standard error 4 8926 5 9917 6283 1 to STOP 8454E 01 1009E 02 3987 01 3358 00 Because the components the Pike data have relatively small coefficients of variation the Gamma fit is not very different from the Normal fit The fitted components have slight positive skewness 38 User s Guide dh Plot 006 Data Heming Lake 1965 Components Gamma Option number 0 for list of options 1 to STOP 9 Change from fitting mixtures of Gamma distributions to fitting mixtures of Lognormal distributions Fitting Lognormal distributions is about as fast as fitting Normal distributions Lognormal distributions are positively skewed bell shaped curves Lognormals cannot take a wide range of shapes like Gamma distributions but will work as well as Gammas in an application ike this SELECT A DISTRIBUTION Enter 1 Normal 2 Lognormal or 3 Gamma 2 Distribution selected is Lognormal Option number 0 for list of options 1 to STOP 6 Repeat the previous fit assuming Lognormal distributions for the components ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Lognormal Enter iteration limit 0 gives displays with current parameters 50 Display observed and expected counts as a table Y N Y 39 MIX 2 3 Display observed and expected cou
30. 92 02 0 2099E 03 0 7950E 02 0 1273E 03 0 6533E 02 0 2305E 02 0 1767E 01 0 7453E 02 0 9688E 02 0 2160E 00 0 3509 02 0 1086E 00 0 3795E 01 0 5707E 01 0 1810E 00 0 9370E 00 0 1182 01 0 4658E 00 0 4134E 01 0 3117E 00 0 5967E 00 0 3447E 01 0 1765E 02 gt gt 0 1321 01 0 5433E 00 0 1744E 00 0 4279E 00 0 6286E 00 0 3759E 01 0 2441E 02 gt gt 0 4888E 02 0 4617 02 0 1718E 00 0 1524 00 0 1129 00 0 1721E 00 0 1043E 01 0 9481E 01 gt gt 0 2649 02 0 2358 02 0 5770 03 0 2442 01 0 1051 01 0 1178 01 0 5445 01 0 2148E 00 0 7053 00 gt gt 0 6700E 00 0 7747 01 0 7617 01 Fitting Lognormal components Proportions and their standard errors 09967 51889 22677 10710 04757 01518 12045 07676 08083 04847 Means and their standard errors 23 0735 33 6069 41 1029 49 8826 60 4670 4647 9680 4 2013 6 9918 4 8562 Sigmas CONSTANT COEF OF VAR 2 3722 3 4551 4 2258 5 1284 6 2166 2760 Degrees of freedom 14 Chi squared 11 9477 6105 Option number 0 for list of options 1 to STOP 11 1028 and standard error Because the components in the Pike data have relatively small coefficients of variation the Lognormal fit is not very different from the Normal or Gamma fit The fitted components have slight positive skewness 41 MIX 2 3 A A A Plot 007 Data Heming Lake 1965 Components Lognormal Option number 0 for list of options 1 to STOP 6 Repeat the previous fit but
31. ED or N FREE YYYYY 45 MIX 2 3 Constraints means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 4 Is Kth mean different Y N N Constraints on sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N N Do you want to abort Y N N Number of iterations 7 Fitting Lognormal components Proportions and their standard errors 09880 53898 24467 05141 06613 FIXED FIXED FIXED FIXED FIXED Means ON A GROWTH CURVE and standard errors Linf 109 042 1 0 1 8226 k 131107 s e 22 213 1690 039767 23 1774 33 7281 42 9824 51 0996 58 2194 4285 3083 4531 Sigmas CONSTANT COEF OF VAR 1052 and standard error 2 4374 3 5469 4 5201 5 3737 6 1225 1676 Degrees of freedom 20 Chi squared 13 6676 8469 Option number 0 for list of options 1 to STOP 6 Use Option 6 to go for the final fit with proportions all free means on a growth curve and constant coefficient of variation ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Lognormal Enter iteration limit 0 gives displays with current parameters 46 User s Guide 30 Display observed and expected counts as a table Y N N Display observed and expected coun
32. Force the means to on a growth curve using direct search optimization to adjust the means while the proportions and sigmas are all held fixed Option 6 will also work here if all the proportions and sigmas are held fixed ESTIMATE SPECIFIED MEANS SIGMAS FOR FIXED PROPORTIONS Distribution selected is Lognormal Constraints on means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 4 Is Kth mean different Y N N Constraints on sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 1 43 MIX 2 3 Which of the 5 sigmas are fixed For each in order enter Y FIXED or N FREE YYYYY Enter lower and upper limits for mean 20 65 Enter lower and upper limits for sigma 17 Enter initial steps for the first three means 1141 Enter initial steps for the 5 sigmas 5 5 5 5 5 Enter iteration limit 100 Enter convergence check frequency 10 Enter accuracy index 1 Do you want to abort Y N N Number of function evaluations 120 Number of restarts 4 Required standard deviation of vertex values 0 100E 01 CONVERGENCE CRITERION NOT SATISFIED Fitting Lognormal components Proportions ALL HELD FIXED 09967 51889 22677 10710 04757 Linf 125 882 1 0 1 9536 k 104495 23 2442 33 4280 42 6014 50 8645 58 3078 GCRV GCR
33. IX computes standard errors for these parameters Linear growth is permitted by constraining the means to be equally spaced If the rightmost component represents all the oldest age groups lumped together you may choose to estimate its mean separately or hold it fixed while constraining the remaining means to lie on a growth curve or to be equally spaced 1 3 Computer requirements Versions of MIX are available for the IBM PC and PC compatibles Apple Macintosh and mainframes Some steps of the fitting process require heavy iterative calculation On a mainframe a Macintosh II or PC AT or COMPAQ 386 with floating point coprocessor most steps will be completed within a few seconds A Macintosh Plus or an IBM PC XT with an 8087 coprocessor will give quite acceptable execution speeds but some steps may take a few minutes to complete An IBM PC XT without a coprocessor may take a few minutes to complete certain steps and may sometimes take an hour or more All microcomputer versions display an iteration counter to show how quickly the iterations are progressing and beep when the iterations are completed An IBM PC or a PC compatible should have at least 512K RAM and run MS DOS 2 1 or higher An 8087 floating point coprocessor while not required is highly recommended as it speeds up calculation by about a factor of 10 High resolution graphics require either a CGA EGA or Hercules graphics card but if one of these is not available M
34. IX will produce rough screen graphics in text mode Figure 2 One disk drive is sufficient MIX 15 supplied as an executable file The Apple Macintosh version will work with a 512K Macintosh but a Macintosh Plus SE or II is preferred One disk drive is sufficient MIX is supplied as a stand alone application in two versions One will run on a Macintosh 512K Plus or SE The other requires the MC68020 processor and MC68881 coprocessor on a Macintosh or upgraded SE and gives incredibly high execution speeds mainframe version is supplied as ANSI Standard FORTRAN 77 source code It has been compiled and tested on many systems including VAX VMS VAX UNIX Pyramid UNIX and Prime Code to drive an off line CALCOMP plotter is included and this code can be adapted to other plotters Screen graphics are in text mode Figure 2 User s Guide 1 4 Screen graphics for the IBM PC MIX 2 3 will produce high resolution monochrome screen graphics with either a CGA EGA or Hercules graphics card You must have the correct version of MIX 2 3 there is one version for CGA and EGA and another version for Hercules The CGA card gives a resolution of 640x200 pixels the EGA card gives either 640x200 pixels or 640x350 pixels and the Hercules card gives 720x348 pixels The IBM PC versions of MIX 2 3 are linked with subroutines from the GRAFMATIC library a product of Microcompatibles Inc 301 Prelude Drive Silver Spring MD 20901 U S A It is a
35. MIX and initiating execution will depend upon your computer and operating system If special instructions are needed for your version of MIX special 10 User s Guide documentation will be provided either on a separate sheet of instructions or in a file called README on the program disk When execution starts you will be prompted to respond with Y if you wish to see the List of Options displayed or N if you wish to proceed directly to the prompt for a choice of Option If you type Y the following will appear on the screen LIST OF OPTIONS List of options Read a new set of data Read a full set of parameter values Revise specified parameter values Estimate proportions for fixed means sigmas Estimate means sigmas for fixed proportions by constrained search Estimate proportions means sigmas with or without constraints and or give diagnostic displays 7 Restore parameters to values from previous step 8 Regroup data or restore to original grouping 9 Choose a distribution 10 Plot histogram 11 Plot histogram and fitted components 12 Toggle to echo all I O to I O log 1 STOP MIX is designed so that any option may be chosen at any step Illogical choices such as attempting to do a fit before data have been read or attempting to estimate proportions when only one component is being fitted will be skipped over after an explanatory message is displayed The use of
36. Machines Corporation Macintosh is a trademark licensed to Apple Computer Inc MacDraw is a trademark of Apple Computer Inc VAX and VMS are trademarks of the Digital Equipment Corporation UNIX is a trademark of Bell Laboratories il TABLE CONTENTS 1 trod ctioi erni ss 1 1 1 MIX An interactive program for fitting mixtures of distributions ssssse 1 1 2 Special features for length frequency 1 515 4 1 3 Computer ne terae uf ade e dpa E Mix E 4 1 4 Screen graphics for the IBM PC ss epe te eee dca tons Phe eene Re Posen pecie 5 2 Statistical and numerical methods sse eene eene 6 2 1 Fitting a mixture distribution to grouped data by maximum likelihood 6 2 2 Constraints on the parameters e c reo IRURE 8 2 2 1 Constraints on proportions e anres 8 OMONE 8 1 Specified proportions 8 2 2 2 Constraints on Te ABS EL e ep Pih 8 0 CHOHE os 8 means sei stants 8 2 CN TC ANG HS CUAL mats
37. S THAT in consideration of the mutual covenants conditions and terms hereinafter set forth and for other good and valuable consideration The Licensor hereby grants to the Licensee the nonexclusive right to use the program which is known as MIX subject to the terms and conditions of this Agreement and the Licensee hereby accepts such licence solely upon such terms and conditions 1 The distribution fee for this licence of one physical copy of the program is 225 00 Canadian Distribution fees are due and payable in advance Distribution fees do not include local provincial state or federal taxes or any governmental taxes or duties whatsoever and the Licensee hereby agrees to pay all such taxes and or charge which may be imposed upon the the Licensee or Licensor with respect to the distribution possession and use of the program pursuant to this agreement 2 The Licensee agrees that the program is and at all times the property of the Licensor the Licensee shall have no right or interest therein except as expressly set forth in this Agreement 3 The Licensee may i use the program on a single machine ii copy the program into any machine readable or printed form for backup or modification purposes only iii modify the program and or merge it into another program for use on a single machine the terms of this Agreement shall continue to apply to the portion of the program used 4 The Licensee covenants and agrees not to t
38. User s Guide to Program MIX An Interactive Program for Fitting Mixtures of Distributions Release 2 3 January 1988 by P D M Macdonald and P E J Green ICHTHUS DATA SYSTEMS 59 Arkell Street Hamilton Ontario Canada L8S 1N6 Copyright 1988 ICHTHUS DATA SYSTEMS ISBN 0 9692305 1 6 Printed in Canada by Guenther Printing 66 Pleasant Avenue Hamilton Ontario Canada L9C 4 7 This publication is documentation for the computer program MIX MIX is proprietary software ICHTHUS DATA SYSTEMS has the sole and exclusive right to distribute MIX and to grant licences If you wish a licence to use MIX please contact Peter Macdonald at ICHTHUS DATA SYSTEMS 59 Arkell St Hamilton Ontario Canada L8S 1N6 telephone 416 527 5262 A copy of the standard licence agreement form is shown on page 60 Please respect the terms of the licence agreement Because users have paid for MIX we are able to upgrade MIX and improve its documentation Licensed users of MIX are offered upgrades to subsequent releases at a much reduced price The run time library in the Apple Macintosh version of MIX 2 3 is Copyright Absoft Corporation 1987 The run time library in the IBM PC versions of MIX 2 3 is Copyright Microsoft Corporation 1982 1988 The IBM PC versions of MIX 2 3 include graphics routines from the GRAF MATIC Library Copyright Microcompatibles Inc 1984 IBM PC is a registered trademark of the International Business
39. V Sigmas 2 3722 3 4551 4 2258 5 1284 6 2166 FIXED FIXED FIXED FIXED FIXED Degrees of freedom 21 Chi squared 19 6888 Option number 0 for list of options 1 to STOP 4 Revise the proportions to adjust for the new means 44 User s Guide ESTIMATE PROPORTIONS FOR FIXED MEANS SIGMAS Distribution selected is Lognormal Enter iteration limit 20 Number of iterations 6 Fitting Lognormal components Proportions and their standard errors 09880 53898 24467 05141 06613 01417 02688 03046 02742 01785 Means ALL HELD FIXED 23 2442 33 4280 42 6014 50 8645 58 3078 Sigmas ALL HELD FIXED 2 3722 3 4551 4 2258 5 1284 6 2166 Degrees of freedom 20 Chi squared 14 9367 Option number 0 for list of options 1 to STOP 6 Use Option 6 to find the best fit with the means on a growth curve constant coefficient of variation while holding all proportions fixed ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Lognormal Enter iteration limit 0 gives displays with current parameters 30 Display observed and expected counts as a table Y N N Display observed and expected counts as a graph Y N N Display variance covariance matrix Y N N Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 1 Which of the 5 proportions are fixed For each in order enter Y FIX
40. ach other Note that if the components are gamma distributions fixing the coefficient of variation at 1 will force them to be exponential distributions since for the gamma distribution o u p 2 where p is the shape parameter Rao 1965 p 133 and a gamma distribution with p 1 is an exponential distribution If there are three or more components MIX gives the option to make the K rightmost component different while constraining components 1 to k 1 to have a fixed coefficient of variation can then be held fixed or estimated separately MIX 2 3 3 Constant coefficient of variation This constraint assumes that 01 1 05 u5 Oyu and MIX attempts to estimate the common value The common value is initialized MIX estimates and computes the other standard deviations from the relation 0 Hj 01 2 42 k This constraint is allowed if there are at least two components and all of the means are positive and different from each other If there are three or more components MIX gives the option to make the K rightmost component different while constraining components 1 to k 1 to have a constant coefficient of variation o can then be held fixed or estimated separately 4 Sigmas equal This constraint assumes that MIX attempts to estimate the common value The common value is initialized at This constraint is allowed if there are at least two components an
41. ail of the histogram is an ill defined smear comprising several sparse old age groups For this example we will treat ages 4 and 5 as a single component and show that we can still get excellent estimates of the parameters of the first three age groups Option number 0 for list of options 1 to STOP 8 We begin combining the rightmost intervals of the histogram This is a good idea if the right tail ends in a sparse scatter of very large individuals because putting them all into one grouping interval will make the fit less sensitive to them This is not necessary for the pike data but we will demonstrate it anyway REGROUP DATA OR RESTORE TO ORIGINAL GROUPING Need to restore data to original grouping Y N N Do you want to combine two adjacent classes Y N Y Which boundary is to be deleted 65 75 Do you want to combine two adjacent classes Y N Y Which boundary is to be deleted 63 75 22 User s Guide Do you want to combine two adjacent classes Y N N Regrouped data INTERVAL OBSERVED COUNT RIGHT BOUNDARY Heming Lake Pike 1965 1 4 0000 19 7500 2 10 0000 21 7500 3 21 0000 23 7500 4 11 0000 25 7500 5 14 0000 27 7500 6 31 0000 29 7500 7 39 0000 31 7500 8 70 0000 33 7500 9 71 0000 35 7500 10 44 0000 37 7500 11 42 0000 39 7500 12 36 0000 41 7500 13 23 0000 43 7500 14 22 0000 45 7500 15 17 0000 47 7500 16 12 0000 49 7500 17 12 0000 51 7500 18 11 0000 53 7500 19 8 0000 55 7500 2
42. d data by maximum likelihood A finite mixture distribution arises when samples are drawn from a population that is a mixture of k component populations Letting represent the proportion of the total population that the jth component population constitutes and letting f x represent the probability density function for some variable characteristic X within the i component population then B x 1 fi x HAC is the probability density function for X in the mixed population MIX assumes that the components can be described by either normal lognormal or gamma probability distributions These are two parameter distributions and without loss of generality the parameters can be taken to be the mean and standard deviation Let u represent the mean and the standard deviation of the i component density f x The objective of fitting the mixture to MIX 2 3 data is to estimate as many as possible of the parameters Op The component standard deviations are referred to as the sigmas in output from MIX For theoretical and practical reasons it will not always be possible to estimate all of the parameters particularly when the components overlap and obscure one another This is discussed by Macdonald and Pitcher 1979 Thus it is often desirable to reduce the number of parameters by assuming constraints The proportions are of course already subject to the constraint
43. d the means are all different from each other If there are three or more components MIX gives the option to make the K rightmost component different while imposing the constraint you can then hold o fixed or estimate it separately 2 3 Numerical precision Accuracy of the final estimates to four significant digits is adequate for most practical applications The estimates will be accurate to at least five significant digits because the normal and gamma probability integrals computed by MIX are generally accurate to at least seven digits Iterations in Option 4 and Option 6 continue until the absolute difference from the previous iteration is less than 0 0000005 for each parameter Absolute rather than relative difference was used on the assumption that measurement units would be chosen to keep the order of magnitude of the means and standard deviations more or less in the range of 1 to 100 The Nelder Mead optimization in Option 5 is the step most sensitive to imprecision because large changes in parameter values may only affect the least significant digits of the chi square being minimized For this reason all versions of MIX use DOUBLE PRECISION arithmetic throughout The subroutine for computing the gamma probability integral includes code for computing the derivative with respect to the shape parameter We have not seen this calculation in any other statistical software 3 HOW TO RUN MIX The method of opening
44. e Option 6 with proportions free means equally spaced and constant coefficient of variation while holding fixed the mean and sigma of the 4th component Because the 4th component does not represent a single age group it should not have to satisfy the same constraints as the first three components ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Lognormal Enter iteration limit 0 gives displays with current parameters 30 54 User s Guide Display observed and expected counts as a table Y N N Display observed and expected counts as a graph Y N N Display variance covariance matrix Y N N Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 0 Constraints means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 3 Is Kth mean different Y N Hold mean fixed Y N Constraints sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N Hold Kth sigma fixed Y N Do you want to abort Y N N Number of iterations 10 Fitting Lognormal components Proportions and their standard errors 10239 61921 22138 05703 01594 03682 03255 01353 Means EQUALLY SPACED and s
45. e previous fit have very large standard errors their coefficients of variation are about 50 This suggests that there is not enough evidence in the data to support von Gertalanffy growth curve model and that linear growth with equally spaced means might fit almost as well We will begin fitting this model with Option 6 holding the proportions and sigmas fixed while fitting the means ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Lognormal Enter iteration limit 0 gives displays with current parameters 30 Display observed and expected counts as a table Y N N Display observed and expected counts as a graph Y N N 48 User s Guide Display variance covariance matrix Y N N Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 1 Which of the 5 proportions are fixed For each in order enter Y FIXED or N FREE YYYYY Constraints on means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 3 Is Kth mean different Y N N Constraints on sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 1 Which of the 5 sigmas are fixed For each in order enter Y FIXED or N FREE YYYYY Do you want to abort Y N N Number of iteratio
46. ed if there are at least four components It cannot be used unless H2 lt uo H1 if this does not hold Option 3 can be used to increase until it does hold If there are five or more components MIX gives the option to let the k rightmost component be different while constraining u4 Uz to lie on a growth curve can then be held fixed or estimated separately 2 2 3 Constraints on sigmas 0 None MIX will attempt to estimate all k standard deviations o If all the proportions and all the means are also being estimated this choice is not likely to work unless the k components show as k clear modes in the histogram 1 Specified sigmas fixed Specified standard deviations will be held fixed while MIX attempts to estimate the remaining standard deviations If for example k 5 and you want the third and fifth standard deviations to be held fixed enter NNYNY at the prompt without separators between the characters 2 Fixed coefficient of variation The coefficients of variation will all be held at the same fixed value MIX will display the current value of oj u and give the option to use that value as the fixed value or input a new value Because the coefficient of variation and the means completely determine the standard deviations the standard deviations do not count as estimated parameters This constraint is allowed if all of the means are positive and different from e
47. elected is Gamma Option number 0 for list of options 1 to STOP 6 Repeat the previous fit assuming Gamma distributions for the components ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Gamma Enter iteration limit 0 gives displays with current parameters 30 Display observed and expected counts as a table Y N 36 User s Guide Y Display observed and expected counts as a graph Y N N Display variance covariance matrix Y N Y Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 0 Constraints on means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 0 Constraints on sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N N Do you want to abort Y N N Number of iterations 17 INTERVAL EXPECTED COUNT OBSERVED COUNT RIGHT BOUNDARY 1 3 7829 4 0000 19 7500 2 11 8123 10 0000 21 7500 3 17 4242 21 0000 23 7500 4 13 7666 11 0000 25 7500 5 12 9711 14 0000 27 7500 6 26 6764 31 0000 29 7500 7 49 8157 39 0000 31 7500 8 65 4037 70 0000 33 7500 9 63 8074 71 0000 35 7500 10 52 0518 44 0000 37 7500 11 40 7189 42 0000 39 7500 12 32 6402 36 0000 41 7500 13 26 1791 23 0000 43 7500 14 20 6984 22 0000 45 7500
48. enient interactive style a choice between extremely rapid quasi Newton optimization or slower but more fool proof Nelder Mead simplex optimization extensive error checks and excellent high resolution screen graphics With screen graphics the user can often get very close to the optimal solution by simple visual steps then use numerical optimization to finish off the fitting process MIX computes standard errors for all estimates and a goodness of fit test of the final fit 1 2 Special features for length frequency analysis Most length frequency applications can be handled by constraining either the component standard deviations or the component coefficients of variation to be equal Macdonald 1987 However in many applications there is an ill defined smear of older age groups with relatively 3 MIX 2 3 small numbers in the right hand tail of the distribution These age groups are sometimes best lumped into a single component but its standard deviation may then be relatively large When fitting three or more components MIX allows you to estimate the standard deviation of the rightmost component separately or hold it fixed while constraining the remaining components to have equal standard deviations or equal coefficients of variation When four or more components are being fitted the means can be constrained to lie along a von Bertalanffy growth curve 2 2 2 The usual growth curve parameters L k and are computed M
49. evious step 17 Option 8 Regroup data or restore to original grouping 17 Option 9 oboe pr e rn tr i ot ue elus a REG 18 Option Plot BIS t OBTAITI io odo Dt credo CO ON a ti e ta etae def 18 Option 11 Plot histogram and fitted components 18 Option 12 Toggle to echo all to I O logis iie ti eet ure e eoe eae 20 Options STOP o edis ectypo bead cc ir err 20 4 Stratedies for difficult Case Siinse a Ro att nu Da e ker aout eae Deae epis 20 4 1 What to do when iterations will not converge 2 2 20 4 2 What to do when proportions go negative or do not sum to 1 2 23 4 3 What to do when there are small expected counts 23 5 The analysis of fisheries length frequency distributions eese 24 6 Technical Support or MIX caos heo Doe t prts e e tee at es 25 sace vec dece Ce tto D eI Rd vido thc timidi e das 25 RA CHO RTT SA E 26 TR Be TN M M 26 PRS OS TIL ou 21 Example An analysis of Heming Lake pike
50. eye and revise the means using Option 3 We could try Option 6 with all proportions and all sigmas held fixed Here we demonstrate the slowest but safest method using Option 5 to adjust the means by direct search while holding the proportions fixed We hold all the sigmas fixed but we could have fitted a constant coefficient of variation and obtained equally good results ESTIMATE SPECIFIED MEANS SIGMAS FOR FIXED PROPORTIONS Distribution selected is Normal Constraints on means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 0 Constraints sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 1 Which of the 5 sigmas are fixed For each in order enter Y FIXED or N FREE YYYYY Enter lower and upper limits for mean 19 65 Enter lower and upper limits for sigma 1 10 Enter initial steps for the 5 means 11111 Enter initial steps for the 5 sigmas 5 5 5 5 5 Enter iteration limit 200 Enter convergence check frequency 10 Enter accuracy index 1 Do you want to abort Y N N Number of function evaluations 222 31 MIX 2 3 Number of restarts 8 Required standard deviation of vertex values 0 100E 01 CONVERGENCE CRITERION NOT SATISFIED Fitting Normal components Proportions ALL HELD FIXED 05057 36771 43737 09155 05279
51. f linear growth This constraint is allowed if there are at least three components If there are four or more components MIX gives the option to let the K rightmost component be different while constraining uj to be equally spaced can then be held fixed or estimated separately 4 Growth curve This constraint forces the means to lie along a von Bertalanffy growth curve of the form Lj Lo 1 t 9 where the components are assumed to be age groups spaced exactly one year apart is the mean size of individuals in the i age group age is measured in years fg is the hypothetical age at zero size f is the actual age of the i age group L is the hypothetical ultimate mean size of individuals in that population and x is the growth parameter Only the first three means are estimated Subsequent means are computed from the relation E A di Se 1 4 k MUM Hy H2 Ho H4 It can be shown that 8 User s Guide u5 Lo u 2 o ty u3 K logs t H3 H2 p2 H1 3 _1 ti to Ke MIX computes and displays these three values their standard errors but it must remembered that they are very unreliable when estimated from data Schnute and Fournier 1980 The fitted values of are much more interpretable The growth curve constraint is allow
52. forced according to the laws of the Province of Ontario If the Licensee is located outside Canada the parties hereto agree that any dispute arising in connection with this agreement shall nonetheless be determined by the Ontario Court System 10 This Agreement contains all the agreements representations and understandings of the parties hereto and supersedes any previous agreements and or commitments oral or written 11 Each of the undersigned warrants that he she has the authority to bind to this Agreement the party which he she represents IN WITNESS WHEREOF the parties hereto have executed this Agreement as of the day and year first above written ICHTHUS DATA SYSTEMS Licensor per Date Licensee per Date
53. he mean and standard deviation may be held fixed or estimated separately while imposing growth curve linear growth fixed coefficient of variation constant coefficient of variation or equal standard deviation constraints on the younger age groups 2 2 2 2 2 3 It might be worthwhile to repeat the fit several times changing the rightmost grouping interval to see how sensitive the estimates are to this choice 6 TECHNICAL SUPPORT FOR MIX MIX is special purpose software intended to solve problems that are inherently difficult Licensed users who encounter problems in applying MIX to their data may send ICHTHUS DATA SYSTEMS a disk containing a copy of the data file and a copy of the complete input output log we will do our best to find a solution You may also telephone us at 416 527 5262 9 00 am to 9 00 pm Eastern Time 7 LICENCE AGREEMENT We ask all users to sign a Licence Agreement acknowledging that the Licence Fee gives them the right to run MIX on a single machine and make copies for back up purposes only We ask all users to respect the terms of this agreement our capacity to improve MIX and its documentation depends on it You may believe that you are doing your colleagues a favour by handing out copies 22 User s Guide of MIX or running MIX on several computers in your laboratory when you have only paid for a single machine licence but you are depriving ICHTHUS DATA SYSTEMS of revenue to which we are legally entitled
54. if there is a negative proportion as the likelihood ratio chi square being minimized will be meaningless it may even be negative If necessary use Option 3 to prepare the proportions before entering Option 5 On the microcomputer versions of MIX during the direct search a counter displays the number of function evaluations When the iterations finish a short beep indicates convergence a long beep indicates that the limit of function evaluations was reached a double beep indicates that the parameter values were inadmissible Option 6 Estimate proportions means sigmas with or without constraints Use efficient scoring iterations Macdonald and Pitcher 1979 to estimate the parameters under specific constraints variance covariance matrix for the estimates is computed and standard deviations are given for the estimates The observed and expected counts for each interval may be tabulated or graphed The final prompt asks if you want to abort this is the only provision for escape if you realize that your input is not appropriate Response Y leads back to the prompt to choose an Option response N begins the iterations On the microcomputer versions of MIX a counter displays the number of iterations When the iterations finish a short beep indicates convergence a long beep indicates that the limit of iterations was reached a double beep indicates that the iterations failed and the parameters have been restored to their values
55. ifying that there is just one component This is not a mixture so no proportion is entered Option 3 Revise specified parameter values Change any of the parameters If you change a proportion and the proportions no longer sum to 1 you will be given the option to re scale them however you may not always wish to do so since re scaling will change the value you just assigned to a proportion unless the value was zero Even if the proportions do not sum to 1 the iterations of Option 4 or Option 6 may converge to proportions that do sum to 1 To display the current values of the parameters use Option 3 and quit without making any changes Note however that this will cause the values saved from the previous step to be overwritten by the current values so they cannot be recovered by Option 7 Option 4 Estimate proportions for fixed means sigmas Estimate all of the proportions while keeping all other parameters fixed This step is very fast and usually converges In any application where the proportions are being estimated try Option 4 immediately after Option 2 Ifa negative proportion results see 54 2 12 User s Guide You will be prompted for the number of iterations Usually 10 or 20 will be more than adequate Entering 0 will abort this Option without changing any of the parameter values On the microcomputer versions of MIX a counter displays the number of iterations When the iterations finish a short beep indicates con
56. imates are sensible Zero expected counts will also arise when the parameter values are appropriate but the data include a number of intervals with zero counts at the extreme left or right of the histogram or between two well separated peaks This is undesirable for two reasons the empty intervals increase computation time but add essentially no information to the data and they render the chi square goodness of fit test invalid by making the degrees of freedom and hence the computed P value higher than is warranted This situation can be avoided by combining with adjacent intervals any intervals that have small or zero observed counts either while preparing the data file or later with Option 8 MIX will calculate standard errors and compute the iterations in Option 6 even when there are intervals with expected counts of zero Minimizing the likelihood ratio chi square is still a valid estimation procedure in these cases However the goodness of fit test will not be valid if there are too many intervals with small expected values Most textbooks say that all expected counts must be gt 5 or that a few expected counts be as small as 2 if all others are gt 5 MIX will print a warning message if more than 2 expected counts are lt 1 In general users are advised to inspect the Table of Observed and Expected Counts produced by Option 6 before they attempt to interpret the P value computed for the chi square test If there are many
57. intervals with very small expected values or if there are intervals with zero expected values then it might be advisable to re group the data using Option 8 and repeat the fit by Option 6 before interpreting the goodness of fit test 5 THE ANALYSIS OF FISHERIES LENGTH FREQUENCY DISTRIBUTIONS The pike data analyzed in the Appendix is an example of a length frequency distribution The five components correspond to groups of fish aged one to five all older fish having been eliminated from the sample In many applications the data will be more difficult to analyze because an indeterminate number of age groups are present A few fish will live much longer than most growing more slowly as they age so even if the fast growing younger age groups define clear modes at the left of the histogram the right hand tail will be a smear of many components with very small proportions 21 MIX 2 3 One approach would be to determine the age of each of the older fish by reading rings on scales or otoliths and remove all fish beyond a certain age from the sample as was done with the pike data Another approach would be to obtain samples from the older age groups either stratifying by age and sampling lengths or stratifying by length and sampling ages Determine the mean and standard deviation of the lengths in each age group from these samples and estimate only the proportions from the mixed sample This approach is reviewed in Macdonald 1987 A future
58. irect search optimization typically requires up to 100 function evaluations per parameter being estimated However experience has shown that a total of as few as 100 or 200 function evaluations will often suffice to get the values close enough for Option 6 to converge on the next step Option 5 is expensive on mainframe computers and very time consuming on microcomputers especially if gamma distributions are being fitted so avoid requesting more than 200 function evaluations It is often faster to experiment with Option 6 adding constraints until convergence is achieved then gradually lifting the constraints than it is to wait for Option 5 to find a fit A convergence check frequency of 10 or 20 is recommended this is the number of function evaluations done between checks to see if the accuracy index is satisfied Very roughly if an accuracy index of 1 is attained the value of the chi square will have been minimized down to the units digit if an accuracy index of 0 1 is attained it will have been minimized down to the first decimal place An accuracy index of 1 or 0 1 is recommended but even if that accuracy is not attained before the limit of iterations is reached CONVERGENCE CRITERION NOT SATISFIED the parameter estimates may still be good enough for Option 6 to work on the next step 13 MIX 2 3 If the data give percents mass or anything other than counts over the grouping intervals an accuracy index of 1 or 0 1 may not be
59. ling will return a negative value for a proportion What to do if this happens is discussed in 4 2 4 2 What to do when proportions go negative or do not sum to 1 It is possible to leave Option 2 or Option 3 with proportions that do not sum to 1 If the proportions are then revised by Option 4 or Option 6 the new values will sum to 1 and there is no problem If they are not revised by Options 4 or 6 they should be re scaled either before leaving Option 2 or Option 3 or by choosing Option 3 quitting it and accepting the offer to re scale If this is not done any histograms or goodness of fit chi square values will be nonsensical MIX will also in some cases tolerate negative proportions and Option 4 or Option 6 may give a negative estimate for a proportion A warning is displayed when this happens If Option 4 or Option 6 gives a negative proportion it is probably an indication that you are trying to fit too many components It is also possible that there really is a component there but the current value of its mean places it too far into one of the neighbouring components There are then several strategies to choose from use Option 11 to plot the current fit and see if any components are obviously misplaced go back to Option 2 and re enter the parameters assuming fewer components use Option 3 to set the offending proportion to a small positive value and hold it fixed at that value for at least the next few steps use Option 3
60. ls Inference and Applications to Clustering Marcel Dekker New York 253 O Neill R 1971 Algorithm AS 47 Function minimization using a simplex procedure Applied Statistics 20 338 345 1965 Linear statistical inference and its applications Wiley New York 522 Schnute J and D Fournier 1980 A new approach to length frequency analysis growth structure Canadian Journal of Fisheries and Aquatic Sciences 37 1337 1351 Titterington D M A F M Smith and U E Makov 1985 Statistical Analysis of Finite Mixture Distributions Wiley New York x 243 pp 23 MIX 2 3 APPENDIX Example An analysis of Heming Lake pike data The data are described in Macdonald and Pitcher 1979 The mixture was known to consist of exactly five components The five components correspond to the five age groups present in the sample all fish more than five years old having been removed from the sample Results of other analyses of the same data are given in Macdonald 1987 The following pages show the input output log of an interactive session Data entered by the user are shown in bold type Explanatory remarks have been added in bold script either in boxes or on the right hand side of the page All else is output from MIX MACDONALD amp PITCHER MIXTURE ANALYSIS Reference J Fish Res Board Can 36 987 1001 Program MIX copyright 1985 1986 1987 1988 by ICHTHUS DATA SYSTEMS Release 2 3
61. n excellent collection of primitive and advanced graphics routines that can be linked with FORTRAN or PASCAL programs ICHTHUS DATA SYSTEMS is licensed to distribute executable code linked with GRAF MATIC object modules Your ability to get a hard copy print out of the screen in graphics mode will depend on what combination of graphics card printer and operating system utilities you have If you find that you are unable to print the screen we recommend GRAFPLUS by Jewell Technologies GRAFPLUS can be purchased from Microcompatibles for U S 50 00 1987 price subject to change When the GRAFPLUS or GRAFLASR command is executed from DOS you specify the graphics card and printer you are using From then on until the system is re booted the print screen function key or an equivalent software command will dump screen graphics to the printer You can also use GRAFPLUS to save screen graphics to a file to be retrieved and printed later If you do not have a graphics card if your graphics card is not sufficiently compatible with CGA EGA or Hercules or if you are running MIX 2 3 on a machine with the wrong graphics card you have the option of text mode graphics instead of high resolution graphics just respond with N at the prompt asking if the correct graphics card is installed This prompt comes the first time you use Option 10 or Option 11 to draw a graph 2 STATISTICAL AND NUMERICAL METHODS 2 1 Fitting a mixture distribution to groupe
62. ns 10 Fitting Lognormal components Proportions and their standard errors 10137 58091 21581 04628 05564 FIXED FIXED FIXED FIXED FIXED Means EQUALLY SPACED and standard errors 23 7755 33 8757 43 9760 54 0762 64 1765 3706 2729 Sigmas their standard errors 2 5300 3 7228 4 7597 5 6611 6 4446 FIXED FIXED FIXED FIXED FIXED Degrees of freedom 22 Chi squared 23 9032 3523 Option number 0 for list of options 1 to STOP 49 MIX 2 3 4 Revise the proportions to adjust for the new means ESTIMATE PROPORTIONS FOR FIXED MEANS SIGMAS Distribution selected is Lognormal Enter iteration limit 20 Number of iterations 6 Fitting Lognormal components Proportions and their standard errors 10385 57470 22737 07187 02220 01491 02694 02813 02061 01034 Means ALL HELD FIXED 23 7755 33 8757 43 9760 54 0762 64 1765 Sigmas ALL HELD FIXED 2 5300 3 7228 4 7597 5 6611 6 4446 Degrees of freedom 20 Chi squared 16 6046 Option number 0 for list of options 1 to STOP 6 Go for the final fit with all proportions free means equally spaced and constant coefficient of variation ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Lognormal Enter iteration limit 0 gives displays with current parameters 30 Display observed and expected counts a table Y N N Display observed and expected cou
63. nt If for example k 5 and you want the third and fifth proportions to be held fixed enter NNYNY at the prompt without separators between the characters MIX does not constrain the proportions to be non negative Negative values can occur in some pathological situations and suggestions for handling them are given in 4 2 MIX 2 3 To constrain the proportions to be equal hold each one fixed at 1 k 2 2 2 Constraints on means 0 none MIX will attempt to estimate all k means 1 Specified means fixed Specified means may be held fixed while MIX attempts to estimate the remaining means If for example k 5 and you want the third and fifth means to be held fixed enter NNYNY at the prompt without separators between the characters 2 Means equal This constraint assumes that uy MIX attempts to estimate their common value The common value is initialized at This constraint is allowed if there are at least two components and the standard deviations are all different from each other such a mixture 15 called a scale mixture Figure 4 3 Equally spaced This constraint assumes that u2 Ug Only two means u and are estimated directly Subsequent means are computed from the relation i73 In size frequency applications where the are mean sizes in successive age groups this constraint corresponds to the assumption o
64. nts as a graph Y N N Display variance covariance matrix Y N Y Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 0 Constraints on means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 0 Constraints on sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N N Do you want to abort Y N N Number of iterations 36 INTERVAL EXPECTED COUNT OBSERVED COUNT RIGHT BOUNDARY 1 3 7216 4 0000 19 7500 2 11 9134 10 0000 21 7500 3 17 3461 21 0000 23 7500 4 13 7970 11 0000 25 7500 5 13 0945 14 0000 27 7500 6 26 7151 31 0000 29 7500 7 49 7509 39 0000 31 7500 8 65 1425 70 0000 33 7500 9 63 7451 71 0000 35 7500 10 52 3416 44 0000 37 7500 11 40 8507 42 0000 39 7500 12 32 4418 36 0000 41 7500 13 26 0357 23 0000 43 7500 14 20 8043 22 0000 45 7500 15 16 7411 17 0000 47 7500 16 13 7185 12 0000 49 7500 17 11 3487 12 0000 51 7500 18 9 3475 11 0000 53 7500 19 7 6530 8 0000 55 7500 20 6 2733 3 0000 57 7500 21 5 1579 6 0000 59 7500 22 4 2075 6 0000 61 7500 23 3 3436 3 0000 63 7500 24 2 5456 2 0000 65 7500 40 User s Guide 25 4 9630 5 0000 Variance covariance matrix for parameter estimates pi 5 and all fixed parameters are excluded 0 2305E 03 0 2229 03 0 1451 01 0 1720 03 0 4170 02 0 58
65. nts as a graph Y N N Display variance covariance matrix Y N N Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 0 Constraints means 50 User s Guide 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 3 Is Kth mean different Y N N Constraints on sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N N Do you want to abort Y N N Number of iterations 14 Fitting Lognormal components Proportions and their standard errors 10303 62186 20236 06616 00659 01611 03650 03087 02163 00996 Means EQUALLY SPACED and standard errors 23 3395 34 4864 45 6333 56 7802 67 9271 4942 4344 Sigmas CONSTANT COEF OF VAR 1141 and standard error 2 6636 3 9357 5 2078 6 4799 7 7520 1926 Degrees of freedom 17 Chi squared 13 6015 6951 Option number 0 for list of options 1 to STOP 11 Although the chi square test above indicates an excellent fit the equal means constraint has pushed the Sth component almost off the histogram 51 MIX 2 3 A A A A A Plot 009 Data Heming Lake Pi ke 1965 Components Lognormal We will finish this session by demonstrating how to lump all the oldest age groups into a single component This is useful if the right t
66. onditions the iterations converge extremely quickly and standard errors of the estimates are computed in the process If however the starting values and constraints are not well chosen the iterations will soon diverge an error message will be given and the parameters will automatically be restored to the values they had before the iterations began For the inexperienced user finding the right starting values and constraints to achieve convergence can be a frustrating experience if a good strategy is not adopted What is happening in these difficult cases 15 that there is a very broad range of parameter values giving more or less equally good fits to the data and there is no one set of values that 15 clearly a best fit Option 6 attempting to find a maximum of the likelihood surface fails because the surface is too flat Alternative methods of calculation such as direct search optimization Option 5 or the EM algorithm will respond differently to this situation wandering over the plateau of the likelihood surface for an excessively large number of iterations and eventually stopping at a point that may be rather arbitrarily chosen Macdonald 1987 It would of course be more satisfying to summarize the data by defining a region of acceptable parameter values but this is not easy to do when dealing with more than two or three parameters at a time The strategy recommended for MIX 15 to take advantage of the good features of scoring iteration
67. ormal Figure 1 An example of fisheries length frequency analysis shown with high resolution graphics The five components correspond to the five age groups in the population the thick line is their sum the mixture distribution The abcissa unit is length in cm The triangles mark the mean lengths of the age groups The prototype of MIX was developed by Macdonald and Pitcher 1979 for the analysis of fisheries length frequency data and this remains an important application Macdonald 1987 Figures 1 and 2 show an example of length frequency analysis MIX 2 3 x XX x X X Xx XXX X XXX XX XXXX XXXX x Kk Kk XXXX X Kk kk kk kv kk XXXXXXXX kkkkkkc kk ck kk ck ckck ck ckck ck ck ck ckckckockckckckck kc k k Kk YYXXXXXXXXXXXXXXXXXXXXXXXX Plot 002 Data Heming Lake Pike 1965 Components Lognormal Figure 2 An example of text mode graphics This is the same fit as shown in Figure 1 A A A Plot 003 Data Three Exponentials Components Gamma Figure 3 gamma distributions with unit coefficient of variation see 2 2 3 A mixture of three exponential distributions fitted by MIX Exponential distributions are fitted as MIX can also handle many other mixture distribution applications such as mixtures of exponential distributions for time to failure studies Figure 3 and scale mixtures with equal means for non normal error analysis Figu
68. ransfer the program and licence to any other party should possession of the program or a copy thereof be transferred to another party this licence is automatically terminated and the Licensee shall be liable to pay to the Licensor any damages suffered by the Licensor as a result of the Licensee having breached this Agreement 58 5 This licence is effective until terminated It shall be terminated i by the licensee destroying its copy of the program together with any copies modifications and or merged portions of the program or ii by the Licensee breaching any of the terms and conditions of this Agreement 6 THE LICENSOR DOES NOT MAKE ANY WARRANTIES EITHER EXPRESSED OR IMPLIED AS TO ANY MATTER WHATSOEVER INCLUDING WITHOUT LIMITATION THE MERCHANTABILITY AND FITNESS OF THE PROGRAM FOR ANY PARTICULAR PURPOSE SHOULD THE PROGRAM PROVE DEFECTIVE THE LICENSEE AND NOT THE LICENSOR SHALL ASSUME THE ENTIRE COST OF ANY REPAIR SERVICING AND OR CORRECTION 7 Should the Licensor be required to take any legal proceedings to enforce this Agreement its full cost of doing so shall be paid by the Licensee 8 If any part term or provision of the Agreement shall be held illegal unenforceable or in conflict with any law of a federal provincial state or Government whatsoever having jurisdiction over this Agreement the remaining portions or provisions shall not be affected thereby 9 This Agreement shall be construed and en
69. re 4 Titterington et al 1985 describe many applications of mixtures where the current version of MIX will give useful results 2 User s Guide Plot 004 Data Means Equal Components Normal Figure 4 A scale mixture of three normal distributions fitted by MIX A scale mixture has equal means different standard deviations see 2 2 2 Estimating the parameters of a mixture distribution is difficult when the components are heavily overlapped because the overlapping obscures information about individual components The mixture can only be resolved by bringing additional information to the problem This information could be from additional samples or from some form of prior information about the parameters and the relations between them MIX allows the user to impose constraints on the parameters for example holding some parameters fixed or constraining all the components to have the same coefficient of variation The user can start with as many constraints on the parameters as necessary and work interactively towards a solution which has as few constraints as possible and makes sense in terms of the application A future release of MIX will allow the user to incorporate additional data in the analysis in the form of stratified sub samples in length frequency applications sub samples for age determination would be taken at specific lengths and analysed jointly with the overall length frequency distribution MIX features a conv
70. s while imposing enough constraints to prevent the iterations from diverging As MIX is guided towards the solution the constraints may be lifted gradually In cases where all the components are not well defined in the histogram it may not be possible to relax all of the constraints If the constraints used for the final fit seem arbitrary the fitting process can be repeated with an alternative choice of constraints to see how much the goodness of fit and the estimates depend on that choice Some users will routinely begin by using Option 5 to improve on the starting values of the means and standard deviations Others with experience will prefer to avoid the rather long calculation time of Option 5 and begin by using Option 6 at first with lots of constraints for example holding all of the proportions and all of the standard deviations fixed 18 User s Guide If Option 6 fails and it is not clear what to do next use Option 11 to plot a graph to see how well the starting values fit the histogram Then use Option 6 for diagnostic purposes specifying a limit of 0 iterations impose the same constraints as were imposed on the trial that failed It will usually turn out that one or more of the parameters have exceedingly large standard errors associated with them an indication that there is not enough information to estimate those parameters The next step would be to try Option 6 again holding those parameters fixed as well as imposing the
71. second line Then write the pairs of counts and right boundaries starting each pair on a new line and separating the count from the right boundary by a comma or space End with the count from the rightmost interval again on a new line Place an empty line at the end of the file Make sure that your text editor saves the file as an ASCII file with no extraneous control characters Boundaries must be in strictly ascending order The provisions for editing and checking described above will be used when MIX reads the data file There is a limit of 80 grouping intervals If the data on a file exceed that limit counts after the 80 interval will be added into the count for the 80 interval Option 2 Read a full set of parameter values Read in the number of components in the mixture and a complete set of parameter values The maximum number of components allowed is 15 The components must be indexed so that the means in non decreasing order lt lt lt If any two consecutive means are equal the corresponding standard deviations must be in strictly ascending order That is pj is allowed only if o lt MIX will not accept values unless these requirements are satisfied If the proportions do not sum to 1 you will be given the option to re scale them so that they do If any proportion is negative a warning will be displayed You can use MIX to fit a single normal lognormal or gamma distribution by spec
72. t to an off line plotter this prompt comes before the plot is displayed on the screen Mainframe screen graphics are in text mode A A A A A Plot 001 Data Heming Lake Pike 1965 Components Lognormal Figure 5 An example of ultra high resolution graphics from the Apple Macintosh version This is the same fit as shown in Figures and 2 Option 12 Toggle to echo all I O to I O log The first time Option 12 is chosen a file is opened to record all input and output Plots from Options 10 and 11 are written to this file in text mode Choosing Option 12 when the I O file is open suspends the I O log choosing Option 12 when the I O log is suspended re opens it 17 MIX 2 3 Option 1 STOP Terminate execution of MIX 4 STRATEGIES FOR DIFFICULT CASES 4 1 What to do when iterations will not converge Difficult cases arise when the components are extensively overlapped and the histogram does not show well defined modes The more the number of components exceeds the number of clear modes the more difficult the data are to analyze If each component shows as a clear mode in the histogram then starting values for the iterative calculations of Option 6 can easily be found by visual inspection of the histogram from Option 10 and these starting values will probably give convergence on the first attempt MIX uses scoring a quasi Newton iterative procedure to compute the best fitting parameter values in Options 4 and 6 Under the right c
73. tandard errors 23 2660 34 4931 45 7202 60 0000 4670 4222 FIXED Sigmas CONSTANT COEF OF VAR 1144 and standard error 2 6619 3 9465 5 2310 6 0000 1870 FIXED 55 MIX 2 3 Degrees of freedom 16 Chi squared 13 1304 6632 Option number 0 for list of options 1 to STOP 6 Repeat the previous fit letting the mean and sigma of the 4th component go free ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS Distribution selected is Lognormal Enter iteration limit 0 gives displays with current parameters 50 Display observed and expected counts as a table Y N N Display observed and expected counts as a graph Y N N Display variance covariance matrix Y N N Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 0 Constraints means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 3 Is Kth mean different Y N Hold mean fixed Y N N Constraints on sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N Hold Kth sigma fixed Y N 56 N Do you want to abort Y N N Number of iterations 13 Fitting Lognormal components Proportions and their standard errors 10257 6
74. to set the proportion to zero and hold it and the corresponding mean and standard deviation fixed unless they are constrained in some other way for at least the next few steps Remember that if there are for example 500 individuals in the sample and one component comprises 2 of the population it will be represented by only about 10 individuals in the sample If furthermore these individuals overlap with individuals from neighbouring component groups it should be evident that there will be very little information from the mixed data to estimate anything 20 User s Guide about that component In some cases the only solution will be to say that the component is negligible and set its proportion to zero 4 3 What to do when there are small expected counts Zero and near zero expected counts will arise in some of the grouping intervals when the initial parameter values are so inappropriate that the assumed distributions do not cover all of the data for example if the component standard deviations are all extremely small and the means lie nowhere near the observed histogram In this case the standard errors computed by Option 6 will be meaningless because they are only valid conditionally upon the parameter values being close to their true values and the iterations of Option 6 will generally diverge This situation should be avoided by using Option 11 to plot a graph before you begin the fitting process to make sure that the initial est
75. ts as a graph Y N N Display variance covariance matrix Y N N Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 0 Constraints means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 4 Is Kth mean different Y N N Constraints on sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N N Do you want to abort Y N N Number of iterations 9 Fitting Lognormal components Proportions and their standard errors 10137 58091 21581 04628 05564 01550 04290 04370 04322 04153 Means ON A GROWTH CURVE and standard errors Linf 106 840 1 0 1 7469 k 140071 s e 41 544 2260 076754 23 1899 34 1232 43 6276 51 8897 59 0719 4748 4681 9874 Sigmas CONSTANT COEF OF VAR 1091 and standard error 2 5300 3 7228 4 7597 5 6611 6 4446 1944 Degrees of freedom 16 47 MIX 2 3 Chi squared 12 4566 7120 Option number 0 for list of options 1 to STOP 11 Note that the means from left to right get progressively closer together in accordance with the von Bertalanffy growth curve assumption Plot 008 Data Heming Lake Pi ke 1965 Components Lognormal Option number 0 for list of options 1 to STOP 6 The growth curve parameters and K in th
76. ve been evident from Plot 003 In fact if we had looked carefully at the plots and then used Option 3 to revise the first three means to 23 33 and 43 respectively Option 6 would converge now ESTIMATE PROPORTIONS MEANS SIGMAS WITH OR WITHOUT CONSTRAINTS AND OR GIVE DIAGNOSTIC DISPLAYS 29 MIX 2 3 Distribution selected is Normal Enter iteration limit 0 gives displays with current parameters 30 Display observed and expected counts as table Y N N Display observed and expected counts as a graph Y N N Display variance covariance matrix Y N N Constraints on proportions 0 NONE 1 SPECIFIED PROPORTIONS FIXED Enter choice 0 Constraints on means 0 NONE 1 SPECIFIED MEANS FIXED 2 MEANS EQUAL 3 EQUALLY SPACED 4 GROWTH CURVE Enter choice 0 Constraints sigmas 0 NONE 1 SPECIFIED SIGMAS FIXED 2 FIXED COEF OF VARIATION 3 CONSTANT COEF OF VARIATION 4 SIGMAS EQUAL Enter choice 3 Is Kth sigma different Y N N Do you want to abort Y N N PARAMETERS OUT OF RANGE AFTER 5 ITERATIONS RESTORE PARAMETERS TO VALUES FROM PREVIOUS STEP Proportions Means Sigmas 05057 36771 43737 09155 05279 20 0000 30 0000 40 0000 50 0000 60 0000 2 0000 3 0000 4 0000 5 0000 6 0000 Option number 0 for list of options 1 to STOP 5 30 User s Guide We must improve the means now We could inspect the plots get better values by
77. vergence a long beep indicates that the limit of iterations was reached a double beep indicates that the iterations failed and the parameters have been restored to their values from the previous step Option 5 Estimate means sigmas for fixed proportions by constrained search While holding the proportions fixed use Nelder Mead direct search to fit the remaining parameters under the constraints chosen The algorithm is based on that of O Neill 1971 see Macdonald and Pitcher 1979 for additional references You will have to specify upper and lower limits for the means upper and lower limits for the sigmas an initial step size for each parameter being estimated the maximum number of function evaluations allowed the frequency of convergence checks and an accuracy index The accuracy index is your required standard deviation of vertex values that is the square root of the variable REQMIN discussed by O Neill 1971 Upper and lower limits on the means and sigmas make Option 5 more efficient by keeping the search within reasonable bounds The initial step sizes should reflect how far you think the initial values are from the true values if you think an initial value is within 2 units of the true value for example try a step size of 1 or 2 Note that if you are holding some or all of the values fixed you must enter step sizes for all of the parameters even though only those corresponding to free parameters will be used D
78. version of MIX will fit age at length data from length stratified sub samples simultaneously with the mixed length frequency data If no age determination can be done it may not be possible to estimate any parameters of the oldest age groups or to eliminate all of the older fish from the mixed sample You could try to use length at age data from another year or another location if they are available but the size distribution and population structure could be very different from that of the population you are trying to analyze You could try to constrain the means to lie along a growth curve in principle if the first few age groups show well defined modes they should suffice to define the growth curve but our own experience has been that there there is too much variation in growth patterns between age groups for this to be reliable If all the above suggestions fail use Option 8 to move the boundary of the rightmost grouping interval so that most of the older fish are included in the rightmost interval Treat all fish above a certain age as being in one component Its mean should be set near the boundary of the rightmost grouping interval Estimate the proportion and if possible the mean and standard deviation of this component along with the parameters of the younger age groups The mean and standard deviation will be artifacts of the grouping and hence will not have much biological significance but the estimated proportion will be meaningful T
79. was used for the rest of the Example 19 MIX 2 3 Constraining the means to a growth curve involves a major shift of the fit so this was done by using Option 5 with all of the standard deviations held fixed before getting the final growth curve fit with Option 6 This could also have been done by using Option 6 in two stages first with the standard deviations and proportions fixed then releasing the proportions and using the constant coefficient of variation constraint In the spirit of Cassie 1954 it has been suggested that first the parameters of the leftmost component be fitted while holding all others fixed then all parameters of the two leftmost components and so on until all have been fitted This strategy is not recommended for MIX It is preferable to adjust as many as possible of the components simultaneously on each step holding means and standard deviations fixed while estimating proportions then holding proportions fixed and constraining standard deviations while estimating means and so on until as many parameters as possible are estimated together Option 4 like Option 6 uses scoring iterations but is less likely to fail because the likelihood surface is quite well behaved when only the proportions are being estimated If it does fail it should be evident that very poor starting values were used or that too many components were assumed It may however happen that Option 4 or Option 6 while not actually fai

User's Guide to Program MIX: An Interactive Program for Fitting

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