User and methodological manual
Contents
1. 2 2 6 Reports Reports concerning general information about population and results of Bethel method are available EA Mauss R Project G Mauss proye Mauss Current version 1 File Definition Allocation Population Allocation results Current Version P i Multivariate Allocation of Units in Sampling Surveys R Figure 9 Report Menu Functions e Population Analysis of the population INFORMATION ON POPULATION DESCRIPTION COUNT DOMAIN TYPE DOMAIN POPULATION N OF STRATA Total Population 10001 DOM1 Al 1 0001 6 Popolation To Be Censused 0 DOM2 B1 8974 3 Popolation To Be Sampled 10001 DOM2 B2 1027 3 Number Of Variables 4 DOM3 C1 6526 3 Number Of Strata 6 DOM3 C2 3475 3 Strata To Be Censused 0 Strata To Be Sampled 6 Number Of Types Of Domain 3 b Figure 10 Information on population 24 In this window see fig 10 there are two tables containing information about population In the first there are general information as the number of population units the number of strata and of different domains The second is a table of population and number of strata by domain for every kind of domain These reports are written in Bethel Report Popl xls file e Allocation Results In this window see fig 11 there are three tables containing information about optimal allocation Comparison between alloca2. MAUSS R Multivariate Allocation of Units in Sampling Surveys Multivariate Allocation of Units in Sampling Surveys R User and methodological manual Teresa Buglielli Claudia De Vitiis Giulio Barcaroli Servizio Metodi Strumenti e Supporto metodologico Direzione Tecnologie e Supporto Metodologico Istituto Nazionale di Statistica AS A A AAN TAN A NN References Introduction Mauss is a tool for defining the sampling design for sample surveys on finite populations It guarantees optimality criteria flexibility and easy management for those who have the responsibility to design and conduct such surveys It enables the user once defined the objectives and the operational constraints of the survey to choose the best sampling design between those obtained by adopting different definitions of the key features of the survey such as the type of stratification the desired accuracy of the estimates the sample size the type of domains of study the variables of interest The use of this software also ensures transparency standardization and accuracy of the methods used The current version of Mauss is an evolution of previous applications developed in SAS The design and development of these first versions is due to methodologists and IT developers including among the first Marco Ballin Claudia De Vitiis Piero Demetrio Falorsi Germana Scepi in the second together with Marco Ballin and Piero Demetrio Falorsi
3. 0 0598 52 TOTAL 10001 569 572 570 DOM3 C202 0 06 0 0327 1 DOM3 C23 0 06 0 0559 1 DOM3 Cava 0 06 0 0464 1 4 gt Exit Figure 11 Allocation results Sensitivity Coefficient of variation and sensitivity In this table expected and actual coefficients of variations and sensitivity to a variation of 10 of the desired precision are reported This report is recorded in the Bethel Report2 xls file 25 2 3 Input data 2 3 1 Strata file File Format tab delimited txt Header The first line of the file must contain the names of the variables specified in the table in any order The file can also contain other variables Data A record for each stratum with at least the information listed in the table variables COST and CENS can be omitted Data may also be related to other variables not involved in calculating optimal allocation STRATO N Number of population units in the stratum DOMI DOM2 DOMp Domain codes 1 p S1 S2 Sn Standard deviation of n variables in the population COST Survey unitary cost for stratum Default 1 Indicator of stratum coverage N 1 stratum to be censused 0 stratum to be sampled Default 0 M1 M2 Mn Means of n variables in the population No A N A N N N 2 3 2 Constraints file File Format tab delimited txt Header The first line of the file must contain the names of the variables specified
4. Society B No 29 pp 115 125 KUHN H W TUCKER A W 1951 Nonlinear Programming Proceedings of II Berkley Symposium Mathematical Statistics and Probability SARNDAL C E SWENSSON B WRETMAN J 1992 Model Assisted Survey Sampling Springer Verlag New York SIGMAN R S MONSOUR N J 1995 Selecting Samples from List Frames of Businesses in Cox B G Binder D A Chinappa B N Christianson A Colledge M J Kott P S eds Business Survey Methods Willey New York 29 Appendix building input file strata for MAUSS In this appendix we show how it is possible using a function of the R package Mauss used by the software presented in this manual to build one of the inputs required by Mauss the one relating to the strata that characterise the frame of reference population To check the availability of the package Mauss in R environment you must run the command gt library mauss If the package has not been installed but it should as it is contextual to the MAUSS software installation you must install it as a priority To use the function buildStrataDF which allows for the construction of the input file strata required by MAUSS two options are given 1 the frame from which the sample will be selected contains information about the target variables the Y survey this is the case for example of frames containing census data or administrative data 2 the frame does not contai
5. useful for the design The population of interest must be defined on the basis of criteria that identify precisely the unit of analysis to be surveyed Examples of populations are the set of active enterprises in Italy with reference to a certain period of time the population of households living in Italy in a fixed point in time the babies born in Italy in a given calendar year The selection frame is the list of the units belonging to the population containing at least the information required to identify and contact them It may also contain auxiliary information useful for the design phase In some cases the frame identifies groups of units or clusters such as a list of families where the family is a cluster of individuals or the register of Italian municipalities in which the town is a cluster of households The variables to be collected may be qualitative qualitative answers to questions such as employment status or perception about a certain phenomenon or quantitative such as income production or sales Therefore the parameters to be estimated may be as in the first case the absolute or relative frequencies of response items or as in the second case averages or totals Anyway the software considers as parameters to be estimated the totals of these variables corresponding therefore to the absolute frequencies for qualitative items and to the totals for quantitative variables The domains of estimates are the sub populatio
6. 10 x File Definition Allocation Report Help Parameters Constraints OB Istat MAUSS Multivariate Allocation of Units in Sampling Surveys R Figure 5 Definition Menu Functions see fig 5 e Parameters Definition of parameters It is possible to modify the following parameters see fig 6 o Minimum number of units per strata default 2 o Maximum number of iterations default 25 of the general procedure This kind of iteration may be required by the fact that when in a stratum the number of allocated units is greater or equal to its population that stratum is set as census stratum and the whole procedure is re initialised o Maximum number of iteration in the algorithm of Chromy default 200 o Epsilon default le 11 this value is used to compare the difference in results from one iteration to the other if it it is lower than epsilon then the procedure stops 20 EB MAUSS R Parameter Definition x PARAMETERS Version 1 Minimum number of units per strata usually 2 lt n lt 4 Bo Maximum number of iterations 25 Maximum number of iterations Chromy Algorithm 200 Epsilon 1e 11 Figure 6 Parameter Definition e Constraints Definition of constraints This function allows see fig 7 o Choose the version of constraints o Modify the values of constraints in the table o Insert a new version of the file CONSTRAINTS ve
7. Survey Sampling Canadian Journal of Statistics Vol 12 pp 53 65 BETHEL J 1989 Sample Allocation in Multivariate Surveys Survey Methodology 15 pp 47 57 CAUSEY B D 1983 Computational Aspects of Optimal Allocation in Multivariate Stratified Sampling SIAM Journal of Scientific and Statistical Computing Vol 4 pp 322 329 CICCHITELLI G HERZEL A MONTANARI G E 1992 Il Campionamento Statistico Il Mulino COCHRAN W G 1977 Sampling Techniques 3 ed Wiley New York CHROMY J 1987 Design Optimization with Multiple Objectives Proceedings of the Survey Research Methods Section American Statistical Association pp 194 199 DAYAL S 1985 Allocation of Sample Using Values of Auxiliary Characteristic Journal of Statistical Planning and Inference Vol 11 pp 321 328 DI GIUSEPPE R GIAQUINTO P PAGLIUCA D 2004 MAUSS un software generalizzato per risolvere il problema dell allocazione campionaria nelle indagini Istat Istat Collana Contributi n 7 2004 FALORSI P D BALLIN M SCEPI G DE VITUS C 1998 Principi e metodi del software generalizzato per la definizione del disegno di campionamento nelle indagini sulle imprese condotte dal ISTAT Statistica Applicata Vol 10 n 2 KISH L 1965 Survey Sampling Wiley New York KOKAN A R KHAN S 1967 Optimum allocation in multivariate surveys an analytical solution Journal of the Royal Statistical
8. allocation by the method of Bethel for the current or for all versions of constraints file Report View the results and prints e Help Display the online help 16 2 2 3 Project definition In MAUSS R a project is individuated by the name of the folder in which all data files generated by the application will be located Other relevant information are the names of input files prepared by the user 1 the first one gives the population size and mean and variance for each variable of interest for each stratum 2 the second one includes for each domain the coefficients of variation for the estimates For a description of the two files see below the section Data file description Mauss R EEE le Definition Allocation Report Help New project Open project Close OB Istat MAUSS Multivariate Allocation of Units in Sampling Surveys R Figure 2 File menu Functions see fig 2 e New project Inserting a new project Choosing the item New project the window shown in Figure 3 will be open This window allows to choose the folder in which the result files will be written and the two input files 17 ES MAUSS R FILE SELECTION x INPUT FILE SELECTION Folder Browse Strata File Browse Constraints File Browse Figure 3 New project File names may be entered directly into the text box or can be selected using th
9. also Daniela Pagliuca Paolo Floris and Roberto Di Giuseppe The decision to migrate the SAS version to R was taken as part of a strategy that tends to reduce the dependence on proprietary software and to ensure full portability of the tools developed by ISTAT Moreover new functions have been added together with a more advanced interface The development of the version described in this manual is due to Teresa Buglielli Java interface for project management and execution modules Daniela Pagliuca implementation of the methodology in R and Giulio Barcaroli Chromy algorithm in Fortran 1 The methodology implemented in MAUSS 1 1 Definition of the methodological problem of allocation 1 1 1 The planning of the survey sampling design In designing a sample survey the phase of studying the sampling design and defining the sample size and its allocation among strata requires the specification of a set of parameters and information from which the construction of the input for the allocation procedure follows It is necessary to determine the population of interest the samplingunit the selection frame containing the unit of the population the variables of interest the parameters which are to be estimated the level at which the estimates have to be produced i e the domains of estimate the accuracy to be guaranteed for the estimates at the level of the different domains the auxiliary information
10. arameter of interest have to be provided for the total population In 11 general however sample surveys are intended to provide estimates not only for the entire population but also for subpopulations domains of study identified by a partition or domain type of the population under investigation Furthermore it is often necessary that the estimates are produced for more than one type of domain which identify alternative partitions of the same population In these cases the sample must be planned so as to ensure simultaneously the accuracy of the estimates at different required levels of detail and this can be achieved by generalizing the solution previously described To illustrate the method of multivariate allocation in the case of multi domain estimation we denote by d d 1 D the generic type of domain kg kg 1 Ka the generic domain of type d x the number of strata belonging to the domain kg The objective function 5 remains unchanged while the system of constraints can be redefined as follows ao A a DAS LS Vos PP Di kel Kd 10 h 1 h 1 where V5 k 15 the upper bound on the sampling variance of the estimate of the total of variable p for the domain ka Similarly to what was done in previous paragraph the 10 can be written as H L ap nn lt 1 p 1 P d 1 D k 1 Ka h 1 where DD F _ Ni pnd kah SR Is ss 11 NaS p no ka h Vok h 1 1 ifhek with 4 4 q 0 o
11. cause the coefficient of variation of this estimate was set at 10 CV1 0 10 this means that o to obtain a reduction in sample size of 567 units is necessary that the value of CV1 shifts from 0 10 to 0 11 which is equivalent to an increase of 10 of the expected sampling error o to obtain a reduction of 10 in the expected error 567 units should be added to the sample Using this tool the user is able to make the necessary adjustments to achieve the desired sample size or conversely to achieve the desired expected precision on target estimates 1 4 The multivariate and multi domain allocation methodology In general the determination of the sample size of the different strata is functional to the minimization of the sample variability of the estimates In the absence of specific information on the variability in the strata the objective is achieved through the proportional allocation conversely if this information is available it is possible to define more efficient allocations In the case of a single variable of interest being available an estimate of the variability one can refer to well known results for the optimal allocation in the univariate case Cochran 1977 these results are used to determine the sample size with the aim to minimize variance estimation for a fixed value of the cost function or conversely to minimize costs having previously established the level of accuracy of the estimates The univariate solutio
12. culated excluding NAs 31
13. e File Manager clicking on the Browse button After giving the confirmation OK the procedure checks the data entered and if everything is right prepares the environment sets the version number of the constraints to 1 and creates the BethV1 subdirectory of the work folder where copies the constraints file and where will write the results of the optimal allocation for the first version of constraints If an old project was defined in the chosen folder the system asks if you want create a new project If so it cleans the folder by moving all the results of prior process in a sub folder named backupNNNNNN where NNNNNN is a number that represents the system time in milliseconds Otherwise it closes the window without defining the project that can be opened using the Open Project function e Open project Opening an existing project In this case the user can choose the version of the file of the constraints and the project from a list of already defined projects The fields for the choice of the folder and of the two input files will automatically written and can not be changed 18 EA MAUSS R FILE SELECTION x INPUT FILE SELECTION Folder Browse Strata File h Browse Constraints File Browse Version in gt Figure 4 Open Project e Close Closing the current project e Exit Quit the application 2 2 4 Parameters and constraints definition EA Mauss R Project G Mauss proye Mauss Current version 1 E 2
14. em The system produces as output a the sample size per stratum b the expected sampling error for each target estimate in each domain of interest c some useful statistics for the improvement of the sampling plan The sample sizes for each stratum are added to the input dataset of strata while the expected sampling errors are reported both in the output dataset and in output tables 7 and 8 The statistics useful for adjusting the allocation solution are shown in Table 5 The system allows the user to choose the final solution by comparing the results of several tests obtained by defining the precision constraints in different ways Table 5 is the instrument at the user s disposal to evaluate how to modify input data particularly data in the second input file This table contains the information useful for the sensitivity analysis for each estimate and each type of domain is given the value of the additional sample size needed to achieve a decrease of 10 of the coefficient of variation of the corresponding estimate This number can also be interpreted in the opposite direction i e as it represents the decrease in sample size that would be achieved by increasing the error of the corresponding estimate of 10 at the level of that type of domain For example continuing the example regarding the survey on births suppose that the sensitivity of the estimate of the first variable in the first domain type region is equal to 567 units Be
15. h variable of interest and each type of domain 1 2 1 Strata file The first file have to contain one record for each stratum with the following variables rules on names and formats are given in chapter 2 of this manual O O O stratum identifier A h 1 H number of units of the population belonging to stratum h N domain code of type 1 type 2 type D to which the stratum h belongs population means calculated for each stratum and for each one of the P target variables that will be used to allocate the sample N Mph 7 L Yn hj la PBN ad 1a where Yp is the value of the variable Yp p 1 P in the j th unit of the population for qualitative variables you have to define a dichotomous variable for each response item and the mean of the variable corresponds to the relative frequency f of the value 1 of the dichotomous variable Yp F m fo 1b N h where Fp a is the absolute frequency of the item standard deviations of the P target variables in the population calculated for each stratum Sp Ar m E sa E 2s be for categorical variables the standard deviation will be calculated as Son SF NE ERA 2b indication on stratum to be sampled or taken all 0 to be sampled 1 otherwise fieldwork costs in the stratum cost per each interview For the construction of the first data set the main difficulty may arise from the obtaining of auxiliary informati
16. he data frame containing information about strata named strata txt structured as follows gt head strata strato N M1 M2 sl S2 cost cens DOM1 X1 X2 X3 1 1 1 156 623 4663 843 2696 469 921627 355 71351 1 0 bd J 1 1 2 68 1062 4884 867 4100 504 12793 366 40575 1 1 3 17 937 9182 905 4114 505 92665 327 92656 1 1 4 20 1377 0881 787 4087 359 69583 391 92049 1 1 5 3 1614 3787 660 2262 20 33451 250 12945 1 1 7 2 1809 0502 1324 6433 185 48919 86 84577 DU NA PRR PR OSOS NNNRR HRHHHHA PRR PR YO WNH 2 Availability of information from sources other than the frame other surveys Conversely if there is no information in the frame regarding the target variables you must build the data frame st rata from other sources such as from a previous round of the same survey or from other surveys In this case assuming the information available is contained in a file named samplePrev txt we need to read the data by running gt samp lt read delim samplePrev txt In addition to naming constraints introduced above this feature requires that a variable named weight is present in the data frame samp At this point you can perform the same function as already seen above gt buildStrataDF samp The result is much the same than the previous case the function writes out in the working directory the strata file named strata txt Note that in all cases for each target variable Y mean and standard deviation are cal
17. in the table in any order The file can also contain other variables Data Coefficients of variation for all domains A record for each type of domain that contains the information about the variables listed in the table Other not relevant variables can be present Type of domain code CV1 CV2 CVn Planned coefficient of variation for n N variables 26 2 4 Produced output OUTPUT FILE File Format tab delimited txt Filename Bethel campio txt Folder Sub folder in the working directory for the version It is a copy of the strata input file to which is appended the variable CAMP containing the result of Bethel s optimal allocation 27 2 5 Work datasets 2 5 1 List of projects File Format tab delimited txt Filename progetti txt Folder Sub folder HOME Mauss2 in the working directory for the version Variable name Description T A Name of strata file A Name of constraints file A versione corrente Progressive number of the last used N Peers e i version of the constraints HH MI 2 5 2 Parameters File format delimited file with delimiter character Filename savePar csv Folder Working directory Variable name Description minstrato Minimum number of units per stratum maxiter Maximum number of iterations maxiterChromy Maximum number of iterations Chromy Epsilon Format le 11 28 References BELLHOUSE D R 1984 A Review of Optimal Designs in
18. is a generalization of the method of Neyman known as a method of univariate optimal allocation and allows to minimize the sample size having established constraints on maximum expected sampling errors of target estimates for each type of domain we can define this approach as a multivariate and multi domain allocation The methodological aspects are described in detail in Section 5 It is important to add that some strata can be defined as take all strata on the basis of a decision of the responsible of the survey for example you may decide a priori to include in the sample all firms with more than 20 employees 1 2 Preparation of input files MAUSS requires that the user provides input data related to the characteristics of the population under investigation to the variables of interest for the estimates together with the constraints on the expected sampling error of the estimates As output the system produces the sample size per stratum the expected sampling errors of all estimates of interest and useful information to evaluate the solution found The input information must be provided to the software in two separate data files the first one contains the stratification of the population with the number of units within each stratum the indication of the domains of estimate and some estimates of the intensity and variability of the phenomena of interest the second one contains the constraints on sampling errors specified for eac
19. les 1 code of domain type d d 1 D 2 maximum allowable values of the expected coefficients of variation for each one of the total estimates of K variables of interest CV1 CVK The preparation of the second file requires the user to specify for each of the estimates of interest the maximum value of the coefficient of variation allowed for each type of domains 25 29 Piemonte It is worth noting that if for a certain estimate it is not needed to guarantee a limit for the sampling error for a certain type of domain it is possible to indicate a very high value of the coefficient of variation for that type of domain such as for example cv 1 Regarding the criteria used to set the level of error in the domain it is common practice to allocate the sample so that the level is approximately equal for all domains Sigman and Monsour 1995 1 2 3 Example of construction of input data sets The following example is based on the stratification adopted in ISTAT for the survey on births The target population consists of mothers of babies born in a given year stratified by age groups 5 and regions 21 the interview is conducted two years after the birth The estimation domains are region macro region age group and nation We assume for simplicity that the estimates of interest are only two the relative frequency of women who were employed before the birth but no longer employed at the time of the interview and the
20. n 7 0 or higher http cran r project org bin windows base The environment variable PATH must point at the programs java exe and r exe To change the variable PATH Start gt Settings gt Control Panel gt System gt Advanced gt Environment variables Now select the PATH variable and click on the Edit button Add here at the beginning of the string the path to the folder that contains the java exe file and the folder that contains r exe separated by For example PATH C Programmi Java jre1 6 0_03 bin C Programmi R R 2 7 1 bin C WINDOWS system32 C WINDOWS C WINDOWS System32 Wbem Installation To install the software you need to download the file setup MaussR exe on your PC and run it 14 2 2 Use of the software 2 2 1 Starting MAUSS From Windows Menu Start gt Programmi gt mauss gt MaussR From the desktop double click on the icon 15 2 2 2 Main menu The MAUSS R menu contains the following functions see fig 1 FA MAUSS R BETES File Definition Allocation Report Help OB Istat MAUSS Multivariate Allocation of Units in Sampling Surveys R Figure 1 Main Menu e File Definition of a project creation of a new project opening an existing project closure of the project in progress and quit the application e Definition Changing the parameters and the constraints used to compute the optimal allocation e Allocation Running the optimal
21. n is however not suitable for the design of most surveys which are usually characterized by a plurality of target estimates For these surveys therefore it is necessary to deal with the problem of optimal allocation under a multivariate approach The following is taken from Falorsi et al 1998 1 4 1 Multivariate allocation In a stratified sample with equal probabilities of selection of units and without replacement the variance of the estimator of the total of a generic variable of interest Yp p 1 P can be expressed as 2 V Vp V pls Tm 3 Pp Pp oP Aa ph a h p h where Fp Vpo Vp is the variance of variable p in stratum h and Vo is the part of variance not influenced by allocation We also define the following cost function H C Co C Co Comp 4 h 1 where Cy is the fixed cost of interviewing that does not depends on the sample size nor on the allocation C is the variable cost and C h 1 H the cost per sample unit in the stratum h It is possible to determine the number of units to be assigned to each stratum using two approaches Sigman and Monsour 1995 The first approach consists in minimizing the P product W C where W Y WpYp and W p 1 P are weights to be defined The p 1 solution is found by setting the value of W or C It is possible that this method does not work in concrete situations due to the difficulty of specifying non arbitrary weights In the second ap
22. n such data it will then need to calculate for each stratum the estimates for means and root mean square deviations of the Y s using different sources for example a previous round of the same survey or different surveys with proxy estimates In the following we examine both possibilities 1 Availability of information concerning Y s in the frame In the R environment a dataframe named frame contains the following information a unique identifier of the unit no restriction on the name may be cod the optional identifier of the stratum to which the unit belongs the values of m auxiliary variables named from X1 to Xm the values of p target variables named from Y1 to Yp the values of the domains of interest for which we want to produce estimates named domainvalue eee ho at For example gt frame lt read delim frame txt gt head frame cod domainvalue strato X1 X2 X3 Xi Y2 1 100 4 4solb4saul 2 4 1 3283 2128 1167 9092 2 200 4 4sola6saul 1 6 1 1997 4587 614 9569 3 300 4 4sola6saul 1 6 1 569 9164 1498 6392 4 400 4 Aselaisami 1 8 1 1786 9 91 1050 1127 5 900 4 4sola5saul 1 5 1 910 3036 808 0705 6 1200 4 4solblsau2 2 1 2 3273 3433 969 6291 30 If this information is available it is possible to use the function buildStrataDF in this way gt buildStrataDF frame The function takes as argument the name of the single frame and writes in the working directory t
23. ng we will show an example dealing with this particular situation The third situation happens when the user does not have any information at all on the variability of the phenomena of interest because the survey is planned for the first time In these cases it is possible to set the allocation procedure by establishing for each domain of estimation a set of typical frequency estimates in order to cover the range of variation for all estimates that the survey aims to produce For instance if the survey is used to produce estimates at national macro regional and regional levels you might desire that the sample is such as to guarantee a sufficient reliability for estimates at least of 1 at national level 3 at macro regional level and 5 at the regional level In this case the strata will coincide with the most disaggregated domain namely the region using three variables whose means will be constant for all strata FP 9 01 P 0 03 P 0 05 for each stratum A while the standard deviations can be obtained using the 2b The software will provide the overall sample size and its allocation among the strata in such a way that the constraints are respected with regards to sampling error of the estimates of typical frequencies at the level of the different specified domains 1 2 2 File of constraints on maximum expected sampling errors The second file should contain one record for each type of domain with the following variab
24. ns at the level of which the estimates of the parameters of interest have to be obtain These domains must be defined on the basis of variables available in the frame for each the unit of the population Examples of domains are the region the province the region cross classified with the economic activity for enterprises the age groups In addition it is often necessary that the estimates are produced for more than one type of domain or to alternative partitions of the same population The precision required for the estimates of interest represents the degree of reliability that the estimates have to guarantee It is expressed in terms of the coefficient of variation ratio between the standard error of the estimate and the estimate itself to be specified for each parameter and each type of domain For example it is possible to require that the estimate of the total turnover of the enterprises at level of region presents a coefficient of variation not exceeding 10 It is important to note that for a certain variable the coefficient of variation is the same when considering the estimation of the average and of the total for qualitative variables it is the same for the estimation of a relative frequency and the correspondent absolute frequency The auxiliary information useful for the planning of the design is generally contained in the frame or can be obtained from previous similar surveys or from a census The auxiliary variables necessar
25. on on the variables of interest There may be given different possible situations di means and standard deviations can be inferred from the sampling frame but are referred to a previous time reference and or to proxy variables of the variables investigated 4 means and standard deviations are obtained as estimates from a previous occasion of the same sample survey 5 means and variances are unknown In the first case which occurs for example in the situation of a business survey when the turnover or the number of employees is available from the frame a business register for each enterprise referred to a previous year it is immediate to calculate for each stratum the required quantities according to expressions 1a 1b 2a and 2b Often the variables available on the frame are proxy of the variables under investigation and if the correlation between the auxiliary variables and the variables of interest is high enough it is possible to ensure a good level of precision on the estimates of the variables of interest Cicchitelli et al 1992 In the second case it is possible to obtain the estimates of means and standard deviations in the population from sample data of a previous occasion of the same survey In this case it is necessary however to evaluate the reliability of these estimates and to use them at a higher level of aggregation than the stratum if they do not exhibit an acceptable accuracy In the followi
26. onal complexity of this algorithm especially in the case of multiple domains of study led to the use of the algorithm proposed by Chromy 1987 which is of more immediate implementation and seems to converge towards the optimal solution more quickly To illustrate this algorithm let A la p be the matrix of size R and H whose elements are defined by 11 and a be the r th row of A The Chromy algorithm is an iterative algorithm whose first step consists in computing the value of x according to 13 by setting each element of u equal to 1 R If this solution satisfies all the constraints the algorithm stops Otherwise the algorithm calculates xin correspondence of vector Yo whose generic element is provided by the following expression pP aP Vadut IP E Yar YIP sre 15 r 1 where xh 1 denotes the value of x obtained on the basis of 13 putting y y V 1 Since these algorithms do not ensure that in the optimal solution satisfies na lt Nn MAUSS contains a procedure for the iterative reallocation that sets as take all strata the strata in which gt Ny and recalculate the sample size under the changed conditions 13 2 MAUSS user manual 2 1 Installation Microsoft Windows The minimum hardware requirements for Mauss R are e RAM 512MB e Disk Space SMB Also need to be installed on your PC e Java 2 Runtime Environment version 6 or higher http java sun com javase downloads index jsp e R Environment versio
27. proach an upper bound Vy is set for each Vy and the cost function C is minimized under the constraints V lt Vy p 1 P 10 MAUSS uses the latter approach adopting a generalization of the solution proposed by Bethel 1989 that defines a constrained minimum problem with convex objective function and linear constraints In particular we reformulate the quantity C in 4 by defining np ifnp 21 o otherwise In this way the expression of the objective function to be minimized becomes H fo Y Chlxh 5 h 1 where x x7 XH The constraints Vy E Vy take the form H Lapa lt 1 p l P 6 being 2 NS ap h E 7 v Vop Since the minimization problem of 5 under constraints 6 satisfies the conditions of the theorem of Kokan and Khan 1967 then an optimal solution x exists Using the theorem of Kuhn Tucker 1951 Bethel demonstrates that there exist values e 2 0 so that the optimal solution takes the form JCh 8 P ji sal Ck L Ul pap p 1 where lp 4p LA p therefore yu pil 9 p 1 To determine simultaneously the optimal values x and y it is necessary to resort to y p h P Ty numerical algorithms such as those proposed in the work of Bethel which will be discussed in the following 1 4 2 Multivariate allocation for multiple domains and multiple domain types The solution described in the previous paragraph is related to the case when the estimates of the p
28. relative frequency of women whose children attend the nursery Information on the variables of interest in this case is drawn from the data of an previous survey case 2 in paragraph 1 2 1 This information was not reliable enough to be used at stratum level but only by considering domains defined as cross classification of macro regions and age groups The variable cost of each stratum is set equal to one because there is no difference in cost between the different strata The same for the variable that indicates the presence of strata to be taken all in this survey this indicator has always been set to zero The resulting file has the following structure Domain 1 Domain 2 Domain 3 Region Age 35 39 Sardegna Sardegna 35 39 Islands 40 gt Sardegna 40 e oltre Islands The second file containing the constraints on sampling errors has the following structure Domain type CVI CV2 Doml Region Dom2 Age group 005 Dom4 Ital 0 02 The values assigned to the coefficients of variation for the two estimates at level of the four types of domains are only examples but show how in general to the types of domains with a larger number of values is given a higher value of the coefficient of variation It can be noted that as for the second variable is not required to estimate the level of age groups DOM2 the bound was set equal to 1 1 3 How to use the output of the syst
29. rsion mi DOM CM cv2 Cv3 CVA DOMI 0 06 0 06 0 06 0 06 DOM2 0 12 0 12 0 12 0 12 DOM3 0 06 0 06 0 06 0 06 Figure 7 Definition of constraints You can change a coefficient of variation by double clicking the cell writing the new value and moving to the next box with the TAB key or the mouse 21 WARNING The change is recorded only if the cursor is positioned in a cell different from the cell that contains the changed value The Update button writes the constraint s table on the current version of the file The button Insert changes the version number of the constraints file creates a new sub folder of the working directory with the name BethVn where n is the new version number and inserts the data displayed in the table to a new file of constraints It is possible change the current version of the current constraints file using the list box Version 22 2 2 5 Allocation This menu runs the function that computes the multivariate optimal allocation for different domains of interest in a stratified sample design This function is an extension of Bethel methodology with Chromy Algorithm OB Istat MAUSS R E Multivariate Allocation of Units in Sampling Surveys R Figure 8 Allocation Menu Functions see fig 8 e Current version Calculating optimal allocation for the current version e All versions Calculating optimal allocation for all versions of the file of constraints 23
30. therwise By defining an index r whose values are in correspondence with the values found by lexicographically ordering the vector identified by three indices d ka p the system of constraints becomes H D Lane forr l R where R P Ka 12 h 1 d 1 i e a form totally equivalent to 6 Returning to the 8 and being the conditions of the theorems of Kokan and Khan Kuhn Tucker still satisfied the optimal solution that minimizes 5 under constraints 12 is 12 sui R x where ly 4 ri with Y u 1 14 r 1 r 1 R L Hran L a na 13 r 1 1 4 3 Resolution algorithms The algorithm proposed by Bethel for the calculation of the optimal multivariate allocation can be generalized to solve the same problem when there are multiple types of domains This algorithm comes to the optimal solution iteratively starting from an initial one v 1 which coincides with the optimal solution in the univariate case for the first variable on the first domain r 1 Typically with this solution the objective function assumes a very small value and the remaining constraints r 2 R are not satisfied In each of the following steps v 2 3 v the sample size is increased increasing the objective function f x al 1 in order to satisfy all the constraints Bethel shows that the algorithm converges and therefore u e x can be identified simultaneously so that 0 lt px DE flx The computati
31. tion results table for each stratum the sample size obtained by the optimal allocation of Bethel for the different versions of the file of constraints This report is written in Bethe Results xls file Allocation results table for each stratum the Bethel sample size computed with the current version of coefficient of variation is compared with the dimension of population and the values obtained with proportional and equal allocation This report is recorded in the Bethel_Report1 xls file 4 xl OPTIMAL ALLOCATION RESULTS version Mv Comparison between allocation results Sensitivity Current Version STRATUM ni n2 TYPE DOMAIN PLANNED ACTUAL CY SENSITIVI a 195 195 DOMI AN 0 06 0 0394 1 b 62 80 DOMI Ativ 2 0 06 0 0406 1 t 26 26 DOMI ANS 0 06 0 0297 1 d 222 152 DOMI Ativa 0 06 0 0346 1 e 179 51 DOM2 B1V1 0 25 0 0334 1 f 96 65 DOM2 B102 0 12 0 0453 1 TOT 780 569 DOM2 B1N3 0 12 0 0316 1 DOM2 Bia 0 12 0 0383 1 E DOM2 B21v1 0 25 0 2486 18 Allocation results Current Version DOM BIND 012 0 0201 1 STRATUM POPULATI BETHEL PROPOR EQUAL DOM2 B203 0 12 0 0879 1 a 5158 195 294 95 DOM2 B2iv4 0 12 0 0339 1 b 2632 80 150 95 DOM3 C1N1 0 06 0 0514 1 c 1184 26 68 95 DOM3 C1N2 0 06 0 0598 44 d 711 152 41 95 DOM3 C1v3 0 06 0 0345 1 e 184 51 11 95 DOM3 CINA 0 06 0 0469 1 f 132 65 8 95 DOM3 C2m 0 06
32. y for the allocation are stratification variables which are essential for defining strata and domains of estimate variables correlated with the ones of interest useful for the study of the variability of the variables of interest 1 1 2 One stage stratified sample design MAUSS allows to calculate the sample size and its allocation in the strata for a one stage stratified sample design To accomplish this sampling scheme the population should be divided into strata accordingly to one or more classification variables known a priori for all units in the frame In a standard stratification strata may be regarded as the minimum partition of the population that allows to obtain the domains of estimate as a union of strata planned domains In general finer strata produce an increase of sample size given the expected error this is due to the necessity to ensure at least one or two sample units per stratum In order to illustrate a standard procedure for the construction of the strata let s consider for example the case of a business survey aiming at producing estimates separately for classes of economic activity as identified by the first four digits of the classification of economic activities Nace and size classes of employees In this situation the strata are defined by the cross classification of economic activity and size class of employees The allocation of the sample size among strata is achieved following an approach which
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