: not H. S i. max and. min. Paper ## P911

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1 Paper ## P9 Usng A to Perform Robust -ample Analyss of Means Type Randomzaton Tests for Varances Peter Wludyka, Unversty of North Florda, Jacksonvlle, FL ABTRACT A A macro for performng Analyss of Means (ANOM) type randomzaton tests for testng the equalty of varances s presented. Randomzaton technques for testng statstcal hypotheses can be used when parametrc tests are napproprate. uppose that ndependent samples have been collected. Randomzaton tests are based on shuffles or rearrangements of the (combned) sample. Puttng each of the samples n a bowl forms the combned sample. Drawng samples from the bowl forms a shuffle. huffles can be made wth replacement (bootstrap shufflng) or wthout replacement (permutaton shufflng). The tests that are presented offer two advantages. They are robust to non-normalty and they allow the user to graphcally present the results va a decson chart smlar to a hewhart control chart. The decson chart facltates easy assessment of both statstcal and practcal sgnfcance. elected results from a Monte Carlo study used to dentfy robust randomzaton tests that exhbt excellent power when compared to other robust tests wll be presented. NTRODUCTON Often t s useful to test whether each of populatons have the same varance. Ths homogenety of varance (HOV) hypothess may be wrtten H 0 : σ σ () th where σ s the varance of the populaton. The alternatve hypothess s H A : not H 0. Ths paper s concerned wth s > the case where ; that s, where three or more populatons are beng compared. The focus wll be on one-way balanced desgns, but the dscussons extend to unbalanced and more complex desgns. RANDOMZATON TET There are two types of randomzaton tests: exact randomzaton tests and approxmate randomzaton tests. Randomzaton tests are based on shuffles (resamplngs or rearrangements) of the (combned) sample. uppose that samples of sze n have been selected. The combned sample s formed by puttng each of the samples n a bowl. Let X denote the combned sample. Drawng from the bowl forms shuffles. The shuffles can be made wth replacement (called a bootstrap shuffle) or wthout replacement (called a permutaton shuffle). ANALY OF MEAN TYPE RANDOMZATON TET FOR VARANCE Four ANOM type randomzaton tests for varances have been proposed by Bernard and Wludyka (to appear). They are randomzaton versons of the Analyss of Means for Varances (ANOMV), the normal based test proposed by Wludyka and Nelson (997A). These tests can be used when normalty cannot be safely assumed. Each test can be performed by plottng the sample varances (or standard devatons) on a decson chart or equvalently by evaluatng an emprcal p-value. Usng the decson chart, the HOV hypothess s rejected f any sample varance plots outsde the decson lnes. Ether permutaton shuffles or bootstrappng shuffles can be used. The tests are RANDANOMV-D, RANDANOMV-DD, RANDANOMV-R and RANDANOMV-RD. Each test can be classfed as ether a dfference test or a rato test. The dfference tests are RANDANOMV-D and RANDANOMV-DD, and the rato tests are RANDANOMV-R and RANDANOMV-RD. n the DD and RD tests, devatons from the sample mean are shuffled nstead of the orgnal observatons. ee Bernard (999) for Monte Carlo results whch show that randomzaton tests are as powerful and robust as typcal robust alternatves, such as Levene s test. n ths paper only RANDANOMV-R wll be consdered. For a thorough treatment of the other approaches see Bernard (999). Ths test can be used when there are k samples (populatons) of sze n. Ths test should not be used when the populatons have markedly dfferent means. ee Bernard and Wludyka (to appear) or Bernard (999) for alternatves. Note that f the means are known, the mean of each populaton can be subtracted from each value pror to applyng RANDANOMV-R. RANDANOMV-R RANDANOMV-R, whch s based on the rato of each sample varance to the sum of the sample varances, s the randomzaton verson of the Analyss of Means for Varances (ANOMV), presented by Wludyka and Nelson (997A). ANOMV s a test that s sutable when the populatons can safely be assumed to be normal. For a A macro to perform ths test see Wludyka (999). There are two equvalent methods for performng ths test. The p- value method and the decson chart method. Each requres that the followng steps be performed. The A program n ths paper uses permutaton shufflng. Ether that or bootstrap shufflng can be used. A Monte Carlo study has shown that, n general, permutaton shufflng produces a more powerful test.. Calculate the mean of each sample, x. Calculate zj xj x n zj j 3. Calculate and ( n ) 4. For the ntal sample, calculate and mn mn 5. Randomly shuffle the orgnal data x j some number of tmes, N

2 j ( n ) 6. After each shuffle, calculate the current shuffle. 7. Calculate q mn mn, where x and n z j and zj xj x are based on q and To use the p-value method: 8. f q > then ngmx ngmx + q < then ngmn ngmn + 9. f mn mn ngmx + < α 0. f p-value-hgh ( N + ) ngmn + < α or p-value-low ( N + ) hypothess () s rejected., then Note that ngmx and ngmn are used to count the number of tmes that the shuffled values are more extreme those from the ntal sample (the unshuffled data). To use the decson chart method: For level of sgnfcance α, the sample varances are plotted wth decson lnes α UDL () CL LDL α mn α α where the quantles and mn are found usng equatons (6) - (9). The equal varance hypothess s rejected whenever at least one sample varance plots outsde the decson lnes. ( α ) (3) (4) can be found by orderng the set { q N} q,..., Α. Denote the r th largest value ( [] r n A by α th ). Let be the N ( N ) + α largest value n set Α, where [ X ] s the greatest nteger n X. (5) That s ( ) [ M ] α (6) where ( N + ) α M N (7) ( α ) mn can be found by orderng the set Β { q N} q mn,...,. Denote the r th smallest value ( α ) [] r n B by mn. Let mn be the α N ( N + ) [ ] X s the smallest nteger n X. That s, ( α ) [ M ] mn mn th smallest value n set Β, where where α M N ( N + ) (9) THE %RANOMV MACRO The %RANOMV macro can be used to perform the RANOMV- R test for the equalty of k varances. The varable ns s used dentfy the number of shuffles --- typcally 000. Data Preparaton The nput data fle must be contan two varables: a classfcaton varable that dentfes each of the k populatons and a measurement varable. The basc dea s that ns permutaton shuffles of the data wll be made. Ths s easly acheved by readng the data set ns tmes and assgnng a shuffle varable (,,ns) and a unform random (pseudo) varable (n the data step n the program ths varable s denoted shufno) to each data pont. The data set s then sorted by the random number (shufno) wthn shuffle, so that there are then ns random permutaton samples. These samples are used to gauge how unusual the orgnal sample s wth regard to the rato of smallest varance to the sum of the varances and the rato of the largest varance to the sum of the varances. ee the NPUT DATA comment n the source code. %RANOMV Output The decson table can be used to decde the hypothess: f any of the TD0 (sample standard devatons) plot outsde the UDL (upper decson lne) or LDL (lower decson lne) then reject at pre-specfed alpha. ee Tables below. (8)

3 Table : Decson Table RANDANOMV decson table for equalty of 5 varances OB TOOL UDL CL LDL TD A decson chart s also qute easy to nterpret: snce the standard devaton for tool plots below the LDL the equal varance hypothess s rejected. ********************************/ %macro ranomv( k, /* the number of populatons beng compared */ n, /* the sample sze */ alpha, /* level of sgnfcance */ ds, /* the data set contanng the observatons */ varname, /* the varable name for the observatons */ classvar, /* the varable name for the populatons */ ns, /* the number of shufles */ tops); /* randanomv-r */ /*************************** Prnt Data et ***************************/ Fgure : Decson Chart One may also use the p-value method. The output n Table llustrates the dea. The key pont s that the HOV (homogenety of varance) hypothess s rejected whenever ether p-value s less than alpha/. nce PVALLOW (the lower p-value) s less than 0.05/ 0.05 the hypothess s rejected. Table : P-values Emprcal p-values should be compared to alpha/ OB PVALLOW PVALH %RANOMV OURCE CODE /* RANOMV A TET FOR THE EQUALTY OF K VARANCE BAED ON K NDEPENDENT AMPLE OF ZE N */ /******************************* NPUT DATA ********************************/ data data ; pops 7; samp 0; ns 000; do tool to pops; do j to samp; dam tool*rannor(-) + 00; do shuf to ns; shufno unform(-); output ; end; end; end; drop j ; data basedat; set data; f shuf > then delete; ttle 'basedat: orgnal data set'; proc prnt; var &classvar &varname; /******************************** DETERMNE RANOMV CRTCAL VALUE ********************************/ ttle 'ntal data set'; /* proc prnt data &ds;*/ data shufdat; set &ds; proc sort; by shuf shufno; ttle 'shuffled data'; /* proc prnt;*/ data shufdat; set shufdat; drop &classvar; ttle 'shuffled data wthout tool'; /*proc prnt;*/ data orgdat0; set data; proc sort; by shuf; ttle 'orgnal data wth dam'; /*proc prnt;*/ /******************************* DEFNE MACRO

4 data orgdat; set data; drop &varname; proc sort; by shuf; ttle 'orgnal data wthout dam'; /* proc prnt; */ ranks rmnrat; /* proc prnt data rkadmn; */ data shufdat3; merge shufdat orgdat; ttle 'merged data'; /* proc prnt;*/ proc means data shufdat3 noprnt; by shuf &classvar; var &varname ; output out stats var varx; ttle 'varance of each shuffled sample'; /*proc prnt data stats;*/ proc means datastats noprnt; by shuf; var varx; output out stats sum sumvarx; /*proc prnt data stats;*/ data vardat; merge stats stats; by shuf; varrat varx/sumvarx; ttle 'vardat: varances of shuffled samples along wth sums and ratos'; /* proc prnt data vardat; */ data crtlow; set rkadmn; ranklow &ns-floor((&ns+)*(- &alpha/)-)-; f rmnrat > ranklow then delete; f rmnrat < ranklow then delete; lowcrt mnrat; ttle 'crt low'; /*proc prnt data crtlow;*/ proc rank dataad outrkad; var rat ; ranks rrat; ttle 'rkad: ranked ad'; /*proc prnt data rkad; */ data crth; set rkad; rankh &nsfloor((&ns+)*(&alpha/)-); f rrat > rankh then delete; f rrat < rankh then delete; hcrt rat; ttle 'crth: crt hgh'; /*proc prnt;*/ data adwk; set vardat; proc means noprnt; by shuf; var varrat; output out ad rat; /* proc sort data ad ; by rat;*/ ttle 'ad: ad dstrbuton'; /* proc prnt data ad;*/ data admnwk; set vardat; proc means noprnt; by shuf; var varrat; output out admn mn mnrat; ttle 'admn: admn dstrbuton'; /* proc sort data admn; by mnrat; */ /*proc prnt data admn;*/ data crtdum; merge crth crtlow; codex ; ttle 'crtdum: crtcal values'; /*proc prnt;*/ data c;set crtdum;codex; data c3;set crtdum;codex3; data c4;set crtdum;codex4; data c5;set crtdum;codex5; data c6;set crtdum;codex6; data c7;set crtdum;codex7; data c8;set crtdum;codex8; data c9;set crtdum;codex9; data c0;set crtdum;codex0; data c;set crtdum;codex; data c;set crtdum;codex; data c3;set crtdum;codex3; data c4;set crtdum;codex4; data c5;set crtdum;codex5; data c6;set crtdum;codex6; data c7;set crtdum;codex7; data c8;set crtdum;codex8; data c9;set crtdum;codex9; data c0;set crtdum;codex0; proc rank dataadmn var mnrat; outrkadmn; data crtvals; set crtdum c c3 c4 c5 c6 c7 c8 c9

5 c0 c c c3 c4 c5 c6 c7 c8 c9 c0; f codex > &k then delete; ttle 'crtvals: crtcal values'; proc prnt datacrtvals; var rankh hcrt ranklow lowcrt ; /******************************** DETERMNE DECON LNE *********************************/ data basedat; set orgdat0; proc means data basedat noprnt; by shuf &classvar; var &varname ; output out stats4a var var0 std std0; ttle ' varances and standard devatons of orgnal data shuf replcated '; /*proc prnt data stats4a; */ proc means datastats4a noprnt; by shuf; var var0; output out stats5a sum sumvar0; /* proc prnt data stats5a; */ data vardat0; merge stats4a stats5a; by shuf; varrat0 var0/sumvar0; ttle 'varances of orgnal data shuf repl samples along wth sums and ratos'; /* proc prnt data vardat0;*/ data vardat0; set vardat0; f shuf > then delete; avgvar sumvar0/&k; ttle 'vardat0: varance data for ntal sample'; /* proc prnt data vardat0;*/ data vardat; merge vardat vardat0; by shuf &classvar; ttle 'varances of orgnal data + shuf data repl samples along wth sums and ratos'; /* proc prnt data vardat;*/ data admnwk0; set vardat; proc means noprnt; by shuf; var varrat0; output out admn0 mn mnrat0; data adwk0; set vardat; proc means noprnt; by shuf; var varrat0; output out ad0 rat0; data dldat; merge crtvals vardat0; UDLVAR sumvar0*hcrt; CLVAR avgvar; LDLVAR sumvar0*lowcrt; UDL sqrt(udlvar); CL sqrt(clvar); LDL sqrt(ldlvar); namet &classvar; ttle 'dldat: data to determne decson lnes'; /*proc prnt;*/ ttle "RANDANOMV decson table for equalty of &k varances"; proc prnt data dldat; var &classvar UDL CL LDL std0 ; /******************************** OUTPUT ANOMV DECON CHART *********************************/ proc gplot datadldat ; plot std0*&classvar4 ldl*&classvar cl*&classvar udl*&classvar3 /overlay haxsaxs /* annotatebars */ legend; symbol cblue,jon, l4, vnone; symbol cblue, jon, l, vnone; symbol3 cblue, jon, l vnone; symbol4 cblack, none, vstar; axs order( to &k by ) offset() label(h.5); ttle "RANDANOMV Decson Chart for &varname"; ttle "Alpha &alpha and &ns Permutaton huffles"; ttle3 "tandard Devaton Plotted"; /********************* Calculate p-values **********************/ data pvaldat; merge ad admn admn0 ad0; f rat > rat0 then sgh ; else sgh 0; f mnrat < mnrat0 then sglo ; else sglo 0; sg mn(sgh+sglo,); ttle 'pvalue data'; /*proc prnt datapvaldat; */ proc means data pvaldat noprnt; var sg; output out pvaldat mean emppval; /*ttle "p-value for test for equalty of &k varances"; ttle "&n observatons per group and &ns shuffles";*/ /*proc prnt data pvaldat; var emppval; */ proc means data pvaldat noprnt; var sgh; output out pvaldat mean pvalh; /*ttle "p-value for test for equalty of &k varances"; ttle "&n observatons per group and &ns shuffles";*/

6 /*proc prnt data pvaldat; var pvalh; */ proc means data pvaldat noprnt; var sglo; output out pvaldat3 mean pvallow; /*ttle "p-value for test for equalty of &k varances"; ttle "&n observatons per group and &ns shuffles";*/ /*proc prnt data pvaldat3; var pvallow;*/ data pvalout; merge pvaldat pvaldat3; ttle 'Emprcal p-values should should be compared to alpha/'; proc prnt datapvalout; var pvallow pvalh; %mend ranomv; %ranomv(k7,n0,alpha0.0,dsdata,varname dam,classvartool,ns000,tops); DOWNLONG A PROGRAM The source code can be downloaded from the Unversty of North Florda Center for Research and Consultng n tatstcs web page ( as techncal report # REFERENCE A. J. Bernard and P.. Wludyka, Robust -ample Analyss of Means Type Tests for Varances, Journal of tatstcal Computaton and mulaton, to appear. P.. Wludyka, Usng A to Perform the Analyss of Means for Varances Test, Conference Proceedngs outheast A Users Group, 999. P.. Wludyka and P. R. Nelson, An Analyss of Means Type Test for Varances from Normal Populatons, Technometrcs, 997. CONTACT NFORMATON Your comments and questons are valued and encouraged. Contact the author at: Peter Wludyka Unversty of North Florda Jacksonvlle, Florda Work Phone: Fax: Emal: pwludyka@unf.edu Web: