BOOTSTRAP METHOD FOR TESTING THE EQUALITY OF MEANS: IN CASE OF HETEROSCEDASTICITY

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1 SAÜ Fen Edebyat Dergs(0-II) M.EKİZ, U.EKİZ BOOTSTRAP METHOD FOR TESTING THE EQUALITY OF MEANS: IN CASE OF HETEROSCEDASTICITY Meltem EKİZ, Ufuk EKİZ Gaz Unversty, Faculty of Scence, Department of Statstcs, Ankara, 06500, Turkey ABSTRACT In the statstcal analyss usng the F test for testng the equalty of means n a one-way ANOVA s msleadng under the volaton of the assumpton of equal populaton varances. To overcome the problems wth unequal varances Welch developed the so-called Welch test whle Chen and Chen developed the snglestage test for ANOVA. In ths paper, these two tests are compared by usng the bootstrap method wth the wrtten Matlab computer programme. Accordng to the results and the obtaned fgures; the Welch test s emprcal values were substantal better than the sngle stage test s. Key words: heteroscedastcty; bootstrap; the sngle-stage test; the Welch test ÖZET Tek-yönlü ANOVA da ortalamaların eştlğ test çn F testnn kullanıldığı statstksel analzlerde, yığın varyanslarının eştlğ varsayımının sağlanmadığı durumlarda yanıltıcı sonuçlar ortaya çıkar. ANOVA da varyansların eştszlğ problemlernn üstesnden geleblmek çn Welch, Welch testn, Chen ve Chen tek-aşamalı test önermşlerdr. Bu makalede Matlab paket programında bootstrap yöntemnn kullanılmasına dayalı yazılan programdan yararlanarak söz konusu testler karşılaştırılmıştır. Elde edlen şekller ve sonuçlara göre Welch testnn deneysel değerlernn tek-aşamalı testn deneysel değerlernden daha y olduğu görülmüştür. Anahtar kelmeler: varyansların eştszlğ, bootstrap, tek-aşamalı test, Welch test

2 M.EKİZ, U.EKİZ SAÜ Fen Edebyat Dergs(0-II). INTRODUCTION The procedure of testng the equalty of means n the conventonal analyss of one-way ANOVA s based on the assumptons of ndependence, normalty and equal varances. When populatons have dfferent varances, t s well-known that the results obtaned wth ths method msleads to wrong conclusons. Studes have shown that the dstrbuton of the F test statstc depends heavly on the unknown varances and t s not robust under the volaton of equal error varances n case of unequal sample szes. To overcome ths problem Cochran suggested a test method where the nversons of sample varances are used as weghts n the sum of squares explaned and he provded a ch-squared test for equal means based on a transformaton of ths statstc []. Weghtng the terms n the sum of squares explaned by estmates of the nverses of the varances of the respectve sample means have been proposed by James and Welch [, 3]. The Welch test of Welch s commonly used whch wll be descrbed n the next secton. Later, Brown and Forsythe compared the small sample behavor of four statstcs under heteroscedastcty [4]. The two-stage test developed by Dudewcz and Dalal s appled on varance analyss problems by Bshop and Dudewcz [5, 6]. Sngle-stage test for testng the equalty of means s compared wth the two-stage test by a smulaton study by Chen and Chen [7]. The smulaton results have shown that the power of sngle-stage test s better than the two-stage test when the ntal sample sze s small. Ekz and Gamgam have compared Welch and sngle-stage test n case of dfferent sample szes [8]. Also they are compared when the null hypothess s as ordered means by Ekz and Ekz [9].

3 SAÜ Fen Edebyat Dergs(0-II) M.EKİZ, U.EKİZ In ths study, after usng a bootstrap procedure wth the Monte Carlo smulaton method for the comparson of the Welch and sngle-stage test we have seen that the Welch test performs much better results when the number of populatons are 3. If the number of populatons and the sample szes are ncreased the sngle stage test could be prefered.. THE WELCH TEST Assume that Y j s, =,...,I, j=,..., n are ndependent samples wth observatons n populaton normally dstrbuted, Y ~ N, denoted by. Let the mean and varance of the -th sample be n n Y Y j / n, S Yj Y / n j j and the overall sample sze and mean by I n n, I n I Y Y j / n ny / n. j When the populaton varances are equal, 3... I, the classcal F test wth degrees of freedom I- and n-i s approprate for testng the null hypothess of the equalty of means,... I. Under the assumpton of unequal populaton varances the weghts w n /, =,...,I are suggested to use, Welch, for defnng the populaton characterstcs such as the weghted mean I I w / w [3]. The estmated weghts based on the sample varances for the - th populaton are calculated usng

4 M.EKİZ, U.EKİZ SAÜ Fen Edebyat Dergs(0-II) ŵ n /S. The weghted populaton mean would be estmated by the weghted sample mean I I Yw wy / w () f the approprate populaton weghts were known [3]. However, n practce, populaton weghts are not generally known; therefore, weghted sample mean s calculated from the followng equaton I I Yŵ wˆ Y / wˆ by substtutng the estmated ŵ weghts wth the unknown weghts n Equaton () [3]. For known weghts w n / w, the varablty from populaton to populaton would be measured by the weghted sum of squares explaned q I w w Y Yw (). When the weghts are unknown, q w s estmated by q ŵ q I wˆ wˆ Y Ywˆ, obtaned by replacng ŵ n /S wth w n Equaton () [3]. Let f n be the degrees of freedom n the -th sample and I I ˆ A w ˆ / w ˆ / f. Then the Welch test statstc q W ŵ I I ˆ I A 4

5 SAÜ Fen Edebyat Dergs(0-II) M.EKİZ, U.EKİZ has, under the equalty of means, an approxmate dstrbuton wth degrees of freedom I F, ˆ ˆ I / 3Aˆ []. and 3. THE SINGLE-STAGE TEST Under unequal varances the sngle-stage test s an alternatve for the Welch test n one-way ANOVA for testng the equalty of means. Let sample of sze wth unknown mean Y j, =,..., I, j=,..., n, be an ndependent random n n 3 drawn from the normal populaton Y and unknown and unequal varance 5. Employ the frst (or randomly chosen) n observatons to defne the usual sample mean and sample varance, respectvely, by n Y Y j / n j and n S Yj Y / n j. Let the weghts of the observatons be U S / S, n k n n V S / S n k n n where S k s the maxmum of the followng condtons: n U V, k n U V S / n S (3) S,..., S I and [7]. Let the fnal weghted sample mean be defned as U and V satsfyng

6 M.EKİZ, U.EKİZ SAÜ Fen Edebyat Dergs(0-II) n Y. wjyj j where U, for j n wj V, for j=n. When the sample varances sample mean and varance n w defned as t Y S. n wj j, S are gven, =,..., I, the weghted Y has a condtonal normal dstrbuton wth mean j j. Furthermore, gven S, the test statstc t has a condtonal normal dstrbuton wth mean zero and varance /S [7]. It s obvous that the condtonal normal dstrbutons of t are uncondtonal and ndependent Student s t dstrbutons wth n degrees of freedom. Usng the condton gven n Equaton (3), t can be wrtten as Y. t =,...,I. S k / n Ths statstc has an ndependent t dstrbuton wth n degrees of freedom [7]. When the sample szes drawn from the populatons are equal, say formulated by I Y.. Y. / I. n... n n, the overall weghted mean s then I In order to test the null hypothess of the equalty of the populaton means, under unequal varances, the test statstc F 6

7 SAÜ Fen Edebyat Dergs(0-II) M.EKİZ, U.EKİZ I Y. Y.. F S k / n s recommended [7]. Under the null hypothess, t s the sum of squares of ndependent Student s t varables each wth n- degrees of freedom. The percentage ponts of statstc F was obtaned from 0000 smulated data sets whch were generated for each of the equal sample szes n =, 3, 4, 5, 6, 8 and number of populatons I= 3, 4, 5, 6, 8. The crtcal values F,I,n of probablty P F F,I,n F were estmated by calculatng the for.5,.0,.05,.05,.0 [7]. Thus the null hypothess that the populaton means are equal were rejected f F F,I,n. 4. DATA COLLECTION: BOOTSTRAPPING The bootstrap method s used to resample the smulated data wth replacement, producng samples of sze n that can be consdered to have come from the same populaton as the orjnal smulated data. The orjnal smulated data s generated under equal means from the dstrbutons Y ~ N,, for sample szes n 5, 0, 30 and number of populatons I 3, 5. Then for each smulated data set the rebooted (B) data sets are obtaned and the values of the Welch and sngle-stage test statstcs are calculated by usng the Matlab computer programmng. Ths samplng procedure s repeated 000 tmes for each B= 50, 00, 500. The percentage of the rejectons of the null hypothess are summarzed n Table for the level of sgnfcance =.05,.0. Also some of the obtaned fgures based on 7

8 M.EKİZ, U.EKİZ SAÜ Fen Edebyat Dergs(0-II) the varatons of the two test s emprcal sgnfcance values are gven n Fg.,, 3, 4 for 0.05, I= 3, 5, n= 0, 30 and B= 50, COMPARISON OF THE TWO TESTS; USING BOOTSTRAPPED SIMULATED DATA After comparng the percentages of the rejectons of the two tests, our smulaton results ndcated that the Welch test has gven the best results for I= 3. If the number of populatons are ncreased as 5 the sngle-stage test would be an alternatve to the Welch test. It s known that the sngle-stage test s more usable n case of small sample szes. But, because of the bootstrappng prosedure, one shouldn t have to use the sngle-stage test, when I= 5 and n= 5. Furthermore, by consderng the results summarzed n the fgures, t s clear that the Welch test s stll an approprate choce for a researcher, however the sngle stage test s more applcable n case of small sample szes n the lterature. 8

9 SAÜ Fen Edebyat Dergs(0-II) M.EKİZ, U.EKİZ Table. The emprcal values of the reject ratos of the true null hypothess when I= 3,5, 0; n= 5, 0, 30; B= 50, 00, 500 ; = 0.05, 0.0. I n B Sngle stage Welch

10 M.EKİZ, U.EKİZ SAÜ Fen Edebyat Dergs(0-II) 6. DISCUSSION AND CONCLUSIONS The assumpton of equal varances n ANOVA s often n doubt. The sngle-stage test and the Welch test both gnore the necessty of such assumptons and these tests adresses the queston of whether sets of data are samples from the same populaton. Bootstrap s a commonly used computer-based nonparametrc method, whch requres no assumptons regardng the underlyng populaton and can be appled to a varety of stuatons. In ths study the bootstrap method s used to resample the smulated data. Accordng to the results and fgures, the Welch test s emprcal values have gven better results than the sngle-stage test s, especally when I= 3. Works (lke on medcne, dentstry) where researchers have to study wth small number of populatons, the Welch test could be a better test statstc for testng the equalty of the means n case of unequal varances. Ths test may be employed for many other analogous works. Fg.. Emprcal sgnfcance values when 0.05, I= 3, n= 0 0

11 SAÜ Fen Edebyat Dergs(0-II) M.EKİZ, U.EKİZ Fg.. Emprcal sgnfcance values when 0.05, I= 3, n= 30 Fg. 3. Emprcal sgnfcance values when 0.05, I= 5, n= 0 Fg. 4. Emprcal sgnfcance values when 0.05, I= 5, n= 30

12 M.EKİZ, U.EKİZ SAÜ Fen Edebyat Dergs(0-II) 7. REFERENCES []. Cochran, W.G. (937). Problems arsng n the analyss of a seres of smlar experments, Journal of the Royal Statstcal Socety, Suppl. 4, 0-8. []. James, G.S. (95). The comparson of several groups of observatons when the ratos of the populaton varances are unknown, Bometrka, 38, [3]. Welch, B.L. (95). On the comparson of several mean values: an alternatve approach, Bometrka, 38, [4]. Brown, M.B., Forsythe, A.B (974). The small-sample behavor of some statstcs whch test the equalty of several means, Technometrcs, 6, 9-3. [5]. Dudewcz, E.J. and Dalal, S.R. (975). Allocatons of observatons n rankng and selecton wth unequal varances, Sankhya, 37B, [6]. Bshop, T.A. and Dudewcz, E.J. (978). Exact analyss of varance wth unequal varances: test procedures and tables, Technometrcs, 0, [7]. Chen, S.Y., Chen H.J. (998). Sngle-stage analyss of varance under heteroscedastcty,commun.statst.-smula, 7(3), [8]. Ekz, M, Gamgam, H. (007). On the comparson of the Welch test and the sngle-stage test: wth a smulaton study, Communcatons, 56-, 5-6. [9]. Ekz, M, Ekz, U. (007). One-way analyss of varance under heteroscedastctyfor testng the equvalency of means aganst an ordered alternatve wth varyng ntal sample szes, Internatonal Journal of Pure and Appled Mathematcs, 4-6,