Comparison of the Efficiency of the Various Algorithms in Stratified Sampling when the Initial Solutions are Determined with Geometric Method

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1 International Journal of Statistics and Applications 0, (): -0 DOI: 0.9/j.statistics Comparison of te Efficiency of te Various Algoritms in Stratified Sampling wen te Initial Solutions are Determined wit Geometric Metod Şebnem Er Quantitative Metods Department, Istanbul University, Istanbul, 0, Turkey Abstract Te main aim of tis paper is to examine te efficiency of Genetic Algoritm (GA) of Keskintürk and Er (007)[], Kozak s (00) Random Searc[] and Lavallée and Hidiroglou s (9) Iterative Algoritm metod[] on determination of te stratum boundaries tat minimize te variance of te estimate. Initial starting boundaries of te mentioned algoritms are obtained randomly. Here, it is aimed to reac better results in a sorter period of time by utilizing te initial boundaries obtained from Gunning and Horgan s (00) geometric metod[] compared to te random initial boundaries. Tree algoritms are applied on various populations wit bot random and geometric initial boundaries and teir performances are compared. Wit te stratification of eterogenous populations tat ave different properties, iger variance of te estimates or infeasible solutions can be observed once te initial boundaries are obtained wit geometric metod. Keywords Stratified sampling, Stratum boundaries, Genetic algoritm, Random searc, Iterative metod. Introduction In stratified sampling,in order to gain more precision tan oter metods of sampling, a eterogeneous population is divided into subpopulations, eac of wic is internally omogeneous. As a result te main problem arising in stratified sampling is to obtain te optimum boundaries. Several numerical and computational metods ave been developed for tis purpose. Some apply to igly skewed populations and some apply to any kind of populations. An early and very simple metod is te cumulative square root of te frequency metod (cum f) of Dalenius & Hodges in 99[]. More recently Lavallée & Hidiroglou algoritm[] and Gunning & Horgan's (00) geometric metod[] ave been proposed for igly skewed populations wereas Kozak's (00) random searc metod[] and Keskinturk & Er's (007) genetic algoritm (GA) metod[] ave been proposed for even non-skewed populations. Very recently, Brito et.all[] proposed an exact algoritm for te stratification problem wit only proportional allocation based on te concept of minimum pat in graps and tey called teir metod StratPat. Moreover, developed an iterated local searc metod to solve te stratification problem of variables wit any distribution wit eyman allocation[7].all * Corresponding autor: er.sebnem@gmail.com (Şebnem Er) Publised online at ttp://journal.sapub.org/statistics Copyrigt 0 Scientific & Academic Publising. All Rigts Reserved tese metods aim to acieve te optimum boundaries tat maximise te level of precision or equivalently minimise te variance of te estimate or te sample size required to reac a level of precision and some of tem are available in te stratification package stratification for use wit te statistical programming environment R[]; freely available on te Compreensive R Arcive etwork (CRA) at ttp://cra.r-project.org/package=stratification. Te main aim of tis researc is to compare te efficiency ratios of te Lavallée ve Hidiroglou iterative metod, Kozak s random searc metod and Keskinturk and Er s genetic algoritm approac wen te initial boundaries are obtained eiter randomly or from te geometric metod of Gunning and Horgan, and to examine te performances of te tree metods. Te predetermined total sample size (n) is allocated using eyman[9] optimum allocation metod. Te paper is structured as follows: In te second section te exact solution of Dalenius[0] and te metods tat are developed in order to approximately solve te Dalenius equations are briefly explained. In te tird section, te results obtained wit Lavallée and Hidiroglou s iterative metod, Kozak s random searc metod and Keskintürk and Er s genetic algoritm are given wen te initial boundaries are obtained randomly or from te geometric metod of Gunning and Horgan and te performance of te algoritms are compared.. Dalenius (90) Exact Solution

2 Şebnem Er: Comparison of te Efficiency of te Various Algoritms in Stratified Sampling wen te Initial Solutions are Determined wit Geometric Metod Dalenius (90)[0] considers a density f ( x ) wit mean + µ = tf t dt () Te range (X max -X min ) of te stratification variable x is divided into L parts at points b <b <...<b L-, eac part corresponding to a stratum. Wen a sample of n= n observations is selected from f ( x ), te true mean L µ = W µ () = is estimated by Cocran as [] x L st = W µ () = were for te t stratum W, as follows[]: b ( ) µ, xst and are calculated W = f () t dt = () b x i i µ = n x i i x n = xst as a variance of = () =. () Te estimate of te mean σ n σ (7) L ( x ) st = W = n were te true variance is ( ) x i i µ = σ =. () If te sampling fractions n are negligible ten te variance could be written in sort, L σ σ ( x ) st = W (9) n = It is well-known tat tis variance of te estimate is minimum σ min L = ( x ) st = W n σ (0) wen total sample size n is allocated using eyman s optimum allocation metod [9]: σ =. () n n L σ = Terefore te variance of te estimate is a function of te boundaries b. As a result, it is very difficult to find te boundaries tat minimise te variance of te estimate. Dalenius (90)[0] as sown tat te variance of te estimate obtained wit eyman s optimum allocation metod is optimum or in oter words minimum, wen te stratum boundaries satisfy te following equations: σ + ( b µ ) σ + + ( b µ + ) = () σ σ+ It is very difficult to find te stratum boundaries b tat satisfy tese equations remembered as Dalenius equations since tese equations include σ and µ tat bot vary wit b stratum boundaries. As a result, tere ave been many approximations and algoritms proposed for solving Dalenius equations. Te widely known simple metod among te proposals is te cumulative square root frequency metod of Dalenius and Hodges (99) ( cum f ) []. Ten, in 9 Lavallée and Hidiroglou s iterative approac[], in 00 Gunning and Horgan s geometric metod [] and Kozak s random searc metod[], in 007 Keskintürk and Er s genetic algoritm metod[] are developed in order to find te stratum boundaries. Among tese metods, geometric metod is te simplest metod tat does not include any complex algoritms. Terefore, te main aim of tis researc paper is to set te initial boundaries of te proposed algoritms wit geometric metod and compare te efficiencies of te algoritms wen te boundaries are obtained wit or witout geometric metod since it is believed tat tese algoritms would reac te solution in a sorter period once tey start searcing te entire space at a reasonable point. Te details of te approaces and algoritms of tese metods could be obtained from te original papers of Dalenius and Hodges (99)[], Gunning and Horgan (00)[], Kozak (00)[] and Keskintürk and Er s (007)[]. All of tese metods could be applied in R statistical environment using stratification[] and GAstratification[] packages but te GA results given in tis studyare obtained in Matlab 7.0 since in te package tere is no option for setting te initial boundaries wit non-random results.. Application.. Populations for Stratification In tis paper, many populations are used for stratification wit different skewness, kurtosis, mean, standard deviation and size properties.tose populations tat are available in te R stratification[] and GAStratification[] packages are used for stratification. Eac of te populations are divided into,, and strata and te boundaries are obtained using Lavallée and Hidiroglou, Kozak and GA metods wit random and geometric initial boundaries. Pop: An accounting population of debtors in an Iris firm (Debtors). Pop: Te population in tousands of US cities in 90 (UScities). Pop: Te number of students in four-year US colleges in 9-9 (UScolleges). Pop: Te resources in millions of dollars of large commercial US banks (USbanks). Pop: umber of municipal employees of municipalities in Sweden in 9 (ME). Pop: Population in tousands of municipalities in Sweden in 97 (P7). Pop7: Real estate values in millions of kronor according to 9 assessment of municipalities in Sweden in 9 (REV) Pop: Simulated Data from te Montly Retail Trade

3 International Journal of Statistics and Applications 0, (): -0 Survey of Statistics Canada (MRTS) Pop9: Houseold income before taxes from te 00 Survey of Houseold Spending carried out by Statistics Canada (HHICTOT) Pop0: et sales data of 7 Turkis manufacturing firms among te largest 00 firms in 00 by Istanbul Camber of Industry (ICI) (iso00) Pop: et sales data of Turkis manufacturing firms among te largest 00 firms in 00 by Istanbul Camber of Industry (ICI) (iso00) Te boxplots of te populations are displayed between Figures and, and te summary statistics of te populations are given in Table. Referring te descriptive statistics in Table and boxplots in Figures -, we see tat te populations to be stratified are igly eterogenous wic makes stratified sampling efficient to use. For comparison, te initial boundaries are obtained wit bot random initial boundaries and wit geometric metod. Te populations are divided into,, and strata and te total sample size is determined as 00 for Pop-Pop. For genetic algoritm, te number of iterations is set to 0000, te GA population size to, te crossover rate to 0.99 and te mutation rate to 0.. For efficiency (efficiency eff) comparisons of te ratio of variance of te estimates or te ratios of squares of coefficient of variations (CV) are calculated and given in Appendix. Since Lavallée and Hidiroglou s (LH) metod is based on sampling all of te elements in te last stratum (take-all top stratum), te following efficiency ratios are calculated if GA and Kozak s metods provide a take-all top stratum solution: σ GA ( xst ) ( CVGA µ xst ) CV GA effga / Kozak = = = () σ x st CV µ CV Kozak ( ) ( st ) Kozak Kozak x CV GA effga / LH CV LH CV Kozak effkozak/ LH CV LH = () = () For tose situtations were some of te last stratum is sampled, only te efficiency ratio between GA and Kozak s metod ( eff GA/ Kozak ) is calculated. From te efficiency and te coefficient of variation ratios given in Table in Appendix and from te strata and sample sizes given in Table in Appendix, it can be seen tat te algoritms compared in tis paper provide very close results and tat te stratum boundaries are very close to eac oter wen te initial boundaries are set randomly.wen we look at te summary of te results given in Table, we see tat te number of cases were GA or Kozak is better tan te oter one does not differ muc and te gains in efficiencies are close to eac oter. Table. umber of Cases were te Cosen Algoritm Gives Better Results and te Range of te Efficiency Gain (Random Initials) H Better results wit GA Better Results wit Kozak Bot Same Total ( 0.-0.) none 7 ( 0.-7.) (.) ( 0.-%) (.-%7) ( 0.-%) ( 7.-%7) none On te oter and, te results are different wit iger coefficient of variations wen te initial boundaries are obtained wit geometric metod (Table ).Moreover, wen te initial boundaries are set to be found wit geometric metod, many infeasible or nonconverged results are obtained. For example, wen we look at Table were te initial boundaries are obtained wit geometric metod, we see tat te coefficient of variations for GA increases in cases among cases. Yet some of tese increases in te CVs result from a nonconverged or an infeasible solution. Only in cases tere is a gain in efficiency ranging in between 0.0 (CV falling from 0.07 to 0.0 for H= for Pop-UScolleges) and %0. (falling from 0.0 to for H= for Pop-MRTS), wic could be counted as a very minor gain. Te results for L&H and Kozak s are more or less te same wit te results obtained for GA. Wen te initial boundaries are obtained wit geometric metod, wit eac of Kozak s and L&H s metods tere is an efficiency gain in only cases, wic are again minor. For tese reasons, Lavallée and Hidiroglou s iterative metod, Kozak s random searc metod and Keskintürk and Er s genetic algoritms give more efficient results wen te initial boundaries are set randomly due to teir nature. As a result, it can be concluded tat starting wit geometric initial boundaries does not ave muc contribution on te efficiency ratios or on te stratum boundaries for te computational metods. As proposed by Horgan (0) [], in order to obtain feasible solutions in some data sets,some modifications sould be applied before utilising te geometric metod. Horgan (0) [] suggests tat te data sould be analysed before applying te stratified sampling sceme if tere are extreme outliers. In tis paper te revitised version of te geometric metod is not applied since te algoritms examined ere already give good results wit random initials. Furtermore, if any researcer wants to use te geometric initial boundaries for data sets wit extreme outliers, modified version of te geometric metod sould be used.

4 Şebnem Er: Comparison of te Efficiency of te Various Algoritms in Stratified Sampling wen te Initial Solutions are Determined wit Geometric Metod Figure. Boxplots of Pop-Pop Figure. Boxplots of Pop-Pop9 Figure. Boxplots of Pop0-Pop

5 International Journal of Statistics and Applications 0, (): -0 Table. Summary Statistics of te Populations Pop ame Range Skewness Kurtosis Mean StdDev. Pop Debtors Pop Uscities Pop UScolleges Pop USbanks Pop ME Pop P Pop 7 REV Pop MRTS Pop 9 HHICTOT i Pop 0 iso Pop iso Conclusions Stratified sampling is a sampling metodology used for eterogeneous populations in order to gain more precision tan oter metods of sampling. Tis paper examines te improvement in te efficiency ratios and stratum boundaries obtained wit Lavallée and Hidiroglou [], Kozak [] and Keskintürk and Er s (007) [] metods once te initial boundaries are obtained wit geometric metod. Wit te stratification of eterogenous populations tat ave different properties, iger variance of te estimates or infeasible solutions can be observed. As a result, researcers sould be muc more rigorous wen using geometric metod for te initial boundaries in algoritmic metods or else use te modified version of geometric metod once te data as very extreme values. ACKOWLEDGEMETS I would like to tank te reviewer of tis article wose comments and suggestions ave elped improve te paper. REFERECES [] Keskintürk, T., Er, Ş., A Genetic Algoritm Approac to Determine Stratum Boundaries and Sample Sizes of Eac Stratum in Stratified Sampling. Computational Statistics & Data Analysis,,, pp.-7, 007. [] Kozak, M., Optimal Stratification Using Random Searc Metod in Agricultural Surveys. Statistics in Transition,,, pp.797-0, 00. [] Gunning, P., Horgan, J.M., A ew Algoritm for te Construction of Stratum Boundaries in Skewed Populations. Survey Metodology, 0,, 00. [] Dalenius, Tore, Hodges, Josep L.Jr., Minimum Variance Stratification, Journal of te American Statistical Association, (), pp.-0, 99. [] Brito, J., Maculan,. Lila, M., Montenegro, F. An Exact Algoritm for te Stratication Problem wit Proportional Allocation. Optimization Letters,,, pp.-9, 00. [7] Brito, J., Oci, L., Montenegro, F., Maculan,. An Iterative Local Searc Approac Applied to te Optimal StratificationProblem. International Transactions in Operational Researc. 7,, pp.7-7, 00. [] R Development Core Team. R: A language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, (URL) ttp:// 00. [9] eyman, Jerzy. On te Two Different Aspects of te Representative Metod: Te Metod of Stratified Sampling and te Metod of Purposive Selection, Journal of te Royal Statistical Society, 97 (), pp.-, 9. [0] Dalenius, T. Te problem of optimum stratification, Skandinavisk Aktuarietidskrift, pp. 0-, 90. [] Cocran, W. G., Sampling Tecniques, rd ed., Jon Wiley & Sons, Inc. USA., 977. [] R: stratification. ttp://cra.r-project.org/package=stratification [] R: GAStratification.ttp://CRA.R-project.org/package=G Astratification [] Horgan, J.M., Geometric Stratification Revitised. ISI World Congress 0 Proceedings, 0. [] Lavallée, P., Hidiroglou, M., On te Stratication of Skewed Populations, Survey Metodology,,, pp.-, 9.

6 Şebnem Er: Comparison of te Efficiency of te Various Algoritms in Stratified Sampling wen te Initial Solutions are Determined wit Geometric Metod APPEDIX Table. Te efficiency and coefficient of variation ratios of LH, GA and Kozak s metods wen te initial boundaries are obtained randomly H CVLH CVGA CVKozak effga/kozak effga/lh effkozak/lh Pop: Debtors 0.090* * * * Pop: Uscities 0.07* * * * n= Pop: Uscolleges 0.00* * * * Pop: USbanks 0.09* * 0.070* 0.070* * 0.00* 0.00* * * 0.007* Pop: ME 0.09* 0.09* 0.09* * * * * 0.00* 0.00* * * 0.00* Pop: P7 0.0* 0.09* 0.09* * 0.009* 0.009* * 0.00* 0.007* * 0.00* 0.00* Pop7: REV 0.0* 0.007* 0.007* * 0.00* 0.00* * 0.00* 0.007* * 0.00* 0.007* Pop: MRTS 0.09* * * * 0.0* 0.0* Pop9: HHICTOT 0.00* * * * Pop0: iso * 0.09* 0.09* * 0.00* 0.00* * * 0.009* * * 0.00* Pop: iso00 0.0* 0.0* 0.0* * 0.0* 0.0* * * * * * 0.000* * Were tere is a take-all top stratum

7 International Journal of Statistics and Applications 0, (): -0 7 Table. Te coefficient of variation ratios of LH, GA and Kozak s metods wen te initial boundaries are obtained wit geometric metod H CVLH CVGA CVKozak Pop: Debtors Same Same Same Same Same Same Same Same Pop: UScities Same Same Same (n=00) Same Pop: UScolleges Same Same Same Same Same Same Pop: USbanks Same Same Same Same Same Pop: ME C. Same Same C C I.F C C.I.F C I.F. Pop: P C Same C I.F C., I.F I.F I.F. Pop7: REV Same C C., I.F I.F C., I.F Pop: MRTS Same Same Same Same Same Same Same Same Same Pop9: HHICTOT Same Same Same Same Same Same Same Same Pop0: iso C. Same Same C,.I.F I.F C., I.F I.F I.F C., I.F I.F Pop: iso Same Same C., I.F I.F I.F C., I.F I.F I.F C., I.F. OE I.F. : Infeasible;.C. : Algoritm did not converge; - : a decrease in CV; + : an increase in CV.

8 Şebnem Er: Comparison of te Efficiency of te Various Algoritms in Stratified Sampling wen te Initial Solutions are Determined wit Geometric Metod APPEDIX Table. Size of te strata () and te sample sizes (n) obtained fromlh, GA and Kozak s metods wen te initial boundaries are obtained randomly H LH GA Kozak Pop: Debtors n n n n Pop: Uscities n n n n Pop: UScolleges n n n n Pop: USbanks 0 n n n n Pop: ME n n n n Pop: P n n n n

9 International Journal of Statistics and Applications 0, (): -0 9 Table. Continues: Size of te strata () and te sample sizes (n) obtained fromlh, GA and Kozak s metods wen te initial boundaries are obtained randomly H LH GA Kozak Pop7: REV 9 n n n n Pop: MRTS n n n n Pop9: HHICTOT n n n n Pop0: iso n n n n Pop: iso n n

10 0 Şebnem Er: Comparison of te Efficiency of te Various Algoritms in Stratified Sampling wen te Initial Solutions are Determined wit Geometric Metod n n i Observations wit values of zero are excluded from te data since geometric metod could not be applied wit dataset including zeros as a minimum value.