Robust Modifications of the Levene and O Brien Tests for Spread

Transcription

1 ourna of Modern Appied Statistica Methods Voume Issue Artice Robust Modifications of the Levene O Brien Tests for Spread Abdu R Othman Universiti Sains Maaysia The Sin Yan Universiti Sains Maaysia Georgetown Penang Maaysia H Keseman University of Manitoba Winnipeg Manitoba Canada R R Wicox University of Southern Caifornia Los Angees rwicox@uscedu ames Agina University of Forida Foow this additiona works at: Part of the Appied Statistics Commons Socia Behaviora Sciences Commons the Statistica Theory Commons Recommended Citation Othman Abdu R; Yan The Sin; Keseman H ; Wicox R R; Agina ames (0) "Robust Modifications of the Levene O Brien Tests for Spread" ourna of Modern Appied Statistica Methods: Vo : Iss Artice 5 DOI: 037/masm/ Avaiabe at: This Reguar Artice is brought to you for free open access by the Open Access ournas at DigitaCommons@WayneState It has been accepted for incusion in ourna of Modern Appied Statistica Methods by an authorized editor of DigitaCommons@WayneState

2 Robust Modifications of the Levene O Brien Tests for Spread Cover Page Footnote This research was supported by grants provided by the Socia Sciences Humanities Research Counci of Canada the Fundamenta Research Grant Scheme of Maaysia This reguar artice is avaiabe in ourna of Modern Appied Statistica Methods: iss/5

3 ourna of Modern Appied Statistica Methods Copyright 0 MASM Inc May 0 Vo No //$9500 Robust Modifications of the Levene O Brien Tests for Spread Abdu R Othman Teh Sin Yin H Keseman Universiti Sains Maaysia Georgetown Penang Maaysia University of Manitoba Winnipeg Manitoba Canada R R Wicox University of Southern Caifornia Los Angees CA ames Agina University of Forida Gainesvie FL Variants of Levene s O Brien s procedures not investigated by Keseman Wicox & Agina (008) were examined Simuations indicate that a new O Brien variant provides very good Type I error contro is simper for appied researchers to compute than the method recommended by Keseman et a Key words: Levene test of spread O Brien test of spread Type I error Introduction Keseman Wicox Agina et a (008) compared a number of tests for spread that were based on either east squares or trimmed estimates of centra tendency variabiity These estimators were based on either the origina data or transformations suggested by Levene (960) O Brien (98) The adaptive trimming estimators they used were defined by Reed Stark (996) estimators which rey on procedures that determine whether data shoud be trimmed symmetricay asymmetricay or not at a The transformed scores were used in an anaysis of variance (ANOVA) F-test a Wech (95) test a robust ANOVA test due to Lee Fung (985) Based on their extensive simuation study Keseman et a recommended a Levene-type Abdu R Othman is a Professor of Statistics Emai him at: arothman60@yahoocom Teh Sin Yin is a Senior Lecturer of Statistics Emai her at: syinteh@gmaicom H Keseman is a Professor of Psychoogy His research interests are in appied statistics Emai him at: kese@msumanitobaca R R Wicox is a Professor of Psychoogy Emai him at: rwicox@uscedu ames Agina is a Professor of Educationa Psychoogy Research Evauation Methodoogy at the University of Forida Emai him at: agina@ufedu transformation based on empiricay determined 0% asymmetric trimmed means invoving a particuar adaptive estimator where the transformed scores are then used with an ANOVA F test In their investigation Keseman et a ony examined a imited number of variations of the Levene (960) O Brien (979) methods variations where by-in-arge the transformed variabes were obtained via the appication of asymmetricay trimmed means invoving one of the seven hinge estimators defined by Reed Stark (996) However there are many other ways in which the transformed variabes may be created For exampe the transformed variabes may be based on symmetricay trimmed means then these transformed variabes may be symmetricay/asymmetricay transformed with one of the seven hinge estimators Thus the purpose of this study was to examine other variants of the Levene O Brien methods not examined by Keseman et a (008) Background As Keseman et a (008) others have noted the traditiona test for equaity of s variances eg F where s s is the usua unbiased sampe variance for the th group is affected adversey when the data in the groups are not normay distributed (ie it is sensitive to kurtosis) That is the actua eve of 54

4 OTHMAN YIN KESELMAN WILCO & ALGINA significance can differ substantiay from the nomina significance eve In addition power can be ow Levene (960) suggested an aternative test statistic that can be used to assess equaity of spread across independent treatment groups For the one-way ayout with mode μ +ε (i n ; ) where μ is the popuation mean for the th group ε is rom error Levene suggested that the scores coud be modified with the transformation z where is the th sampe mean then these scores can be used in an ANOVA test That is the test suggested by Levene is where n(z z ) ( ) 0 n i W (z z ) (n ) z z n n i z n n z () Critica vaues for W 0 are obtained from the F- distribution based on - (n ) degrees of freedom Another statistic reevant to this artice is W t This statistic repaces the group mean in obtaining the transformed Zs with the group trimmed mean Other methods have aso appeared in the iterature in addition to Levene s (960) procedure Lee Fung (985) presented a robust ANOVA F-test based on trimmed means Keseman et a (979) others (eg O Brien 98) have indicated that a Wech statistic can be adopted instead of the usua ANOVA F-test to assess spread across independent groups O Brien (979) aso suggested that a Wech test can be used with his transformation of the data namey (n 5)n ( ) 5s (n ) r (n )(n ) () Adaptive Trimming Methods Keseman et a (008) provided a detaied description of adaptive trimming methods Reed Stark (996) defined seven adaptive ocation estimators based on measures of tai-ength skewness for a set of n observations based on the work of Hogg (974 98) To define these estimators measures of tai-ength skewness must first be defined Using the notation of Hogg (974 98) Reed Stark (996) based on the ordered vaues et L α the mean of the smaest [ α n ] observations where [ α n ] denotes the greatest integer ess than α n U α the mean of the argest [ α n ] observations When α 05 L (05) is the mean of the smaest [005n ] observations B is the mean of the next argest 05n observations C is the mean of the next argest 030n observations D is the mean of the next argest 030n observations E the mean of the next argest 05n observations U(05) the mean of the argest 005n observations Tai-Length Measures Hogg (974) defined two measures of tai-ength Q Q where ( (05) (05)) (5) (5) Q U L U L ( ()) (5) (5) Q U L U L (3) Q Q are ocation free statistics are uncorreated with ocation statistics can be used to cassify symmetric distributions as ighttaied medium-taied or heavy-taied (Reed & Stark 996) According to Hogg (974) Reed Stark (996) vaues of Q < impy a ight-taied distribution 0 Q 6 a medium-taied distribution 6 Q 3 a heavy-taied distribution Q > 3 a very 55

5 ROBUST MODIFICATIONS OF THE LEVENE AND O BRIEN TESTS FOR SPREAD heavy-taied distribution The cutoffs for Q are: Q < 8 (ight-taied) 8 Q 87 (medium-taied) Q > 87 (heavy-taied) Hogg (98) introduced yet another measure of tai-ength: 3 ( U L 05 (05)) H ( E B) (4) With this measure vaues of H3 < 6 suggest that the tais of the distribution are simiar to a uniform distribution; vaues of 6 through 76 suggest a norma distribution vaues greater than 76 suggest the tais are simiar to those of a doube exponentia distribution Measures of Skewness Reed Stark (996) defined four measures of skewness as: ( 05 (5)) (5) (05) ( ) ( ) ( () ) ( ) Q U T T L H U D C L SK MD MD n 5 ( () ) ( n) SK M M (5) where MD is the median M is the arithmetic mean T (5) is the 05-trimmed mean ( T α ) () ( n) are the first ast ordered observations respectivey Reed (998) defined the α -trimmed mean as: n-k Tα i ( k n)( k n-k ) n ( + ) + α + α i k+ (6) In this definition a proportion α has been trimmed from each tai the accompanying Winsorized variance S is defined as: S ( i Tα) + k( k Tα) n k n α + k T i k+ n-k+ α (7) where k [ α n] + Based on the definitions of tai-ength skewness Reed Stark proposed a set of adaptive inear estimators that have the capabiity of asymmetric trimming (996 p 3) They defined a genera scheme for their approach as foows: Set the vaue for the tota amount of trimming from the sampe α Determine the proportion to be trimmed from the ower end of the sampe ( α ) by the foowing proportion: α α UW ( UW + LW) where UW LW are the numerator denominator of the defined seector statistics (ie tai-ength skewness) 3 The upper trimming proportion is: α α α Based on this genera schema Reed Stark (996) defined seven hinge estimators which are trimmed means as: HQ α α UWQ UWQ + LW Q HQ α α UWQ UWQ + LW Q 3 HH3 α α UW H UWH + LW H 4 HQ α α UWQ UWQ + LW Q 5 HH α α UWH UWH + LW H 6 HSK α α UWSK UW SK + LW SK 7 HSK5 α α UWSK ( UW ) 5 SK + LW 5 SK 5 (8) 56

6 OTHMAN YIN KESELMAN WILCO & ALGINA Keseman et a (008) investigating Type I error rates of procedures for testing spread examined the Reed Stark (996) procedure with various vaues for α because the iterature varies on the amount of recommended (symmetric) trimming Rosenberger Gasko (983) recommend 5% when sampe sizes are sma (athough they state that 0% generay suffices) Wicox (005) recommends 0% Mudhokar Mudhokar Srivastava (99) suggest 5% Ten percent has been considered by Hi Dixon (98) Huber (977) Stiger (977) Staudte Sheather (990); resuts reported by Keseman et a (00) aso support 0% trimming In addition Keseman et a (005) obtained good resuts with 5% symmetric trimming According to Keseman et a (007) Reed Stark s (996) tai-ength skewness measures may be modified for the muti-group probem appied to the modified muti-group measures to the hinge estimators In particuar they indicated that each of the measures can be modified by taking weighted averages in a manner anaogous to the modifications of tai-ength symmetry measures suggested by Babu Padmanaban Puri (999) of each numerator denominator term For exampe for the muti-group probem where n represents the number of observations in each group Q Q can be defined as: ( ()) n U L n Q n( U( 5) L( 5)) n n ( U( 05) T(5) ) n Q n ( T(5) L( 05)) n (9) The other measures woud be simiary modified it is these muti-group measures of taiength skewness that are appied to the genera scheme proposed by Reed Stark (996) treating the transformed Zs as the origina variabes One coud go a step further than merey appying the transformed Zs in a Wech test It is suggested that the transformed Zs be treated as the origina rom variabe in a test statistic that has been found to be generay insensitive to nonnormaity namey a Wech test based on trimmed means that is Yuen s (974) test) Thus consider the foowing The α trimmed means Winsorized variances can be defined in a number of different ways (see Hogg 974; Reed 998; Keseman et a 007; Wicox 003) Let Z() Z ( ) Z ( n ) represent the ordered observations associated with the th group Reed s (998) approach is based on the work of Hogg (974) Hogg defined the α -trimmed mean as: n-g () i m α h Z (0) i g+ where α is seected so that g nα h n g n [n α ] The stard error of m( α ) Hogg suggests is based on the works of Tukey McLaughin (963) Huber (970) is estimated by: ( α) ( ) SS S m α h h () where SS( α ) is the Winsorized sum of squares defined as: ( g+ ) Z m( α) g+ + Z m α g+ + + Z m ( α) n-g- + ( g+ ) Z m ( α) n-g () 57

7 ROBUST MODIFICATIONS OF THE LEVENE AND O BRIEN TESTS FOR SPREAD When aowing for different amounts of trimming in each tai of the distribution Hogg (974) defines the trimmed mean as: n g m ( α α) z () i h (3) i g + where g nα g nα h n-g-g Hogg suggests that the stard m α α can be estimated as: deviation of where SS ( ) SS α α S m ( ) α α hh α α can be cacuated as (4) ( g + ) Z m ( α α ) g + + Z m ( α α ) g Z m ( α α ) n-g- + ( g + ) Z m ( α α ) n g ( g) Z( g ) m ( α α ) + + ( g) Z m ( α α ) n g SS ( α α ) n Test Statistics Let n i i n i (5) μ ˆ Z /n s (Z Z ) /(n ) where ˆμ is the estimate of μ s is the unbiased estimate of the variance for popuation A heteroscedastic statistic (Wech 95) can be defined as: where w( μˆ μˆ) /( ) W ( ) ( w / W) + ( ) n F w ˆ /W μ ˆ μ W w a w n /s (6) The test statistic is approximatey distributed as an F variate is referred to the critica vaue F[( α);( ) ν W ] the ( α ) quantie of the F distribution where error degrees of freedom are obtained from w υ 3 ( w /W) n (7) A Robust ANOVA F-Test Lee Fung (985) defined an ANOVA F-test based on trimmed means Because the ANOVA F-test can be more powerfu than the Wech F-test this statistic was chosen for this investigation The Lee Fung (985) statistic is defined as: where F t h(mˆ t M) ˆ t /( ) SS (a a ) / (H ) H h Mˆ h m ˆ /H t t SS ( α α ) the ( α α ) (8) 58

8 OTHMAN YIN KESELMAN WILCO & ALGINA The Winsorized sum of squared deviations for the th group; h ˆm t are defined the same as previousy Note that when α α 0 Ft F (O Brien 979) O Brien (979) indicated that the rs can be used in the Wech test Accordingy the trimmed mean version is given by: t ( h 5) n( m ˆ t) 5sm( h ) ( h )( h ) r (9) where h n g g ˆm t the ( α α ) trimmed mean of the th group s m sm( α α) for group The rs rt s were aso used with the Lee Fung (985) test Methodoogy A tota of 70 new Levene (960) type procedures were created These procedures were (see Tabe for a summary of the Levene methods examined): (A) Let Z The Zs were then trimmed symmetricay α % the robust F-test F tα was computed There are 4 variants with this designation because there are four symmetric trimming percentages: 5% 0% 5% 0% For exampe A 5 signifies Z the Zs were trimmed symmetricay 5% Ft α was computed (A) Let Z the Zs were trimmed asymmetricay β % with a hinge estimator H the robust F-test F β computed A 0HSK 5 signifies t transformation Z these vaues were subected to 0% asymmetric trimming with hinge estimator HSK 5 before computing F tβ Because there are four asymmetrica trimming percentages seven hinge estimators there are 8 variants with this designation (A3) Let Z The Zs were used with the Wech test F W (B3) Let Z M The Zs were used with the Wech test F W (E) These variants are designated E α H where Z t α t α are group α% symmetric trimmed means The Zs were used with the usua F-test W hence variant E 0 signifies transformed vaues Z t 0 where t 0 are group 0% symmetricay trimmed at tai computed with W Because there are four symmetric trimming percentages (5% 0% 5% 0%) there are four variants with this designation (E) In variants Eαα et Z t α The transformed vaues were trimmed symmetricay α % used with F tα Therefore variant E 50 signifies transformation Z where t 5 t 5 are group 5% symmetric trimmed means these vaues were subected to 0% symmetric trimming before being used with F Because there are four different t α ta s four symmetric trimming percentages there are 6 variants with this designation (E3) In variants E3αβ H et Z t α The transformed vaues were asymmetricay trimmed at β % invoving seven hinge estimators (HQ HQ HH 3 HQ HH HSK HSK 5 ) used with F Hence E 3 05HH 3 tβ 59

9 ROBUST MODIFICATIONS OF THE LEVENE AND O BRIEN TESTS FOR SPREAD signifies transformation where Z t 0 t 0 are group 0% symmetric trimmed means; these vaues were subected to 5% asymmetric trimming cacuated using the HH 3 hinge estimator before appying F tβ Because there are four different t α s four asymmetric trimming percentages with seven different hinge estimators there are variants for this designation (E4) Variants E4 α H use t α in pace of (A 3 ) or M (B 3 ) in getting the Z vaues The Zs were then used with the Wech test F W Hence E 4 0 signifies transformed vaues Z t 0 where t 0 are group 0% symmetric trimmed means used with F W Since there are four symmetric trimming percentages there are four variants with this designation A tota of 65 O Brien-type procedures were created (see Tabe for a summary of the O Brien methods examined): () O Brien transformation based upon group means variances used with the usua F-test W (that is r in W) (Q) Variants are designated Qα The O Brien transformation based upon symmetric trimmed means Winsorized variances of These trimmed means were cacuated at α 5% 0% 5% 0% The transformed vaues r were used with the usua F-test (that is r in W) Because there are four symmetrica trimming percentages there are four variants with this designation Therefore variants Q0 signifies transformation of with 0% symmetric trimmed mean before used with W (Q) Variants are designated Q αα The O Brien transformation based upon group symmetric trimmed means Winsorized variances of These trimmed means Winsorized variances were cacuated at symmetric trimming percentages (5% 0% 5% 0%) The resutant transformed vaues r were symmetricay trimmed based on the same percentages used for used with the robust ANOVA F t test (that is symmetricay trimmed r in F t ) Because there are four symmetric trimming percentages used twice there are 6 variants with this designation Hence variant Q 55 signifies transformation of with 5% symmetric trimmed mean 5% symmetric trimmed mean for the transform vaue r before used with F t (Q3) Variants are designated Q 3 αβ H The O Brien transformation based upon symmetric trimmed means Winsorized variances of These trimmed means were cacuated with the four symmetric trimming percentages (5% 0% 5% 0%) The resutant transformed vaues r were then asymmetricay trimmed at β 0% 5% 0% 5% invoving seven hinge estimators used with the robust ANOVA F t -test (that is asymmetricay trimmed r in F t ) Because there are four symmetric trimming percentages on four asymmetric trimming percentages on r with seven hinge estimators there are variants with this designation 60

10 OTHMAN YIN KESELMAN WILCO & ALGINA (Q4) Because variants are designated Q 4 α The O Brien transformation based upon symmetric trimmed means Winsorized variances of These trimmed means were cacuated at the four symmetric trimming percentages (5% 0% 5% 0%) The transformed vaues r were then used with the Wech test F W (that is r in F W ) Because there are four symmetrica trimming percentages there are four variants with this designation Hence variants Q 4 05 signifies transformation of with 5% symmetric trimmed mean before used with F W Variants are designated Kβ H The O Brien transformation based upon asymmetric trimmed means Winsorized variances of These trimmed means were cacuated at β 0% 5% 0% 5% invoving seven hinge estimators (HQ HQ HH 3 HQ HH HSK HSK 5 ) The transformed vaues r were used with the usua F-test (that is r in W) Because there are four asymmetrica trimming percentages seven hinge estimators there are 8 variants with this designation Hence variants K5HSK signifies transformation of with 5% asymmetric trimmed mean cacuated using the HSK hinge estimator before used with W Study Conditions Four variabes were empoyed in the 3 study: (a) tota sampe size; (b) degree of sampe size inequaity; (c) shape of the popuation distribution; (d) type amount of tota trimming Tota Sampe Size The effect of sampe size on the performance of the various procedures was evauated by varying the tota sampe size (N) The tota sampe size was manipuated setting the average group size to n 0 40 The average group-sizes correspond to tota sampe sizes of N 60 N 0 Degree of Sampe Size Inequaity Three conditions of sampe size equaity/inequaity were investigated which are referred to as: equa n moderatey unequa n extremey unequa n (see beow for vaues) These conditions were evauated because Keseman et a (998) found that unbaanced designs were more common than baanced designs Sampe Size Vaues ns Shape of the Popuation Distribution This study investigated distributions ranging from symmetric to skewed patykurtic to norma-taied to eptokurtic distributions In tota seven distributions were empoyed to compare the procedures The distributions used were: (i) the Feishman (978) transformation of the stard norma distribution into a skewed patykurtic distribution with skewness γ 05 kurtosis γ 05; (ii) a second Feishman transformation of the stard norma distribution into a skewed norma-taied distribution with γ 075 γ 0; (iii) the Beta (05 05) distribution representing symmetric patykurtic distributions; (iv) a g h distribution (Hoagin 985) where g h 0 which is the stard norma distribution with γ γ 0 ; (v) a g 0 h 05 ong-taied distribution with γ 0 γ 5484 representing symmetric eptokurtic distributions; (vi) a g 076 h 0098 distribution with skew kurtosis equa to that of an exponentia 6

11 ROBUST MODIFICATIONS OF THE LEVENE AND O BRIEN TESTS FOR SPREAD Tabe : Description of the Levene (960) Transformations Used In the Simuations Designation Description Cases A Z : use group means from Z : symmetricay trimmed at tai proportions: appy robust ANOVA F-test 4 A Z : use group means from Z : asymmetricay trimmed at tota proportions: hinge estimators: Q Q H3 Q H SK SK5 appy robust ANOVA F-test 8 A3 Z : use group means from Z : appy Wech F-test B3 Z : use group medians from Z : appy Wech F-test E Z : use group symmetric trimmed means from symmetricay trimmed at tai proportions: Z : appy usua ANOVA F-test 4 E E3 Z : use group symmetric trimmed means from symmetricay trimmed at tai proportions: Z : symmetricay trimmed at tai proportions: appy robust ANOVA F-test Z : use group symmetric trimmed means from symmetricay trimmed at tai proportions: Z : asymmetricay trimmed at tota proportions: keeping hinge estimator constant appy robust ANOVA F- test 6 E4 Z : use group symmetric trimmed means from symmetricay trimmed at tai proportions: Z : appy Wech F-test 4 Tota 70 6

12 OTHMAN YIN KESELMAN WILCO & ALGINA Tabe : Description of the O Brien (979) Designations Used In the Simuations Designation Description Cases R : use group means variances from R : appy usua ANOVA F-test Q R : use group symmetric trimmed means Winsorized variances from symmetricay trimmed at tai proportions: R : appy usua F-test 4 Q R : use group symmetric trimmed means Winsorized variances from symmetricay trimmed at tai proportions: R : symmetricay trimmed at tai proportions: appy robust ANOVA F-test 6 Q3 R : use group symmetric trimmed means Winsorized variances from symmetricay trimmed at tai proportions: R : asymmetricay trimmed at tota proportions: hinge estimators: Q Q H3 Q H SK SK5 Appy robust ANOVA F-test Q4 R : use group symmetric trimmed means Winsorized variances from symmetricay trimmed at tai proportions: R : appy Wech F-test 4 K R : use group asymmetric trimmed means Winsorized variances from asymmetricay trimmed at tota proportions: hinge estimators: Q Q H3 Q H SK SK5 R : appy usua F-test 8 Tota: 65 63

13 ROBUST MODIFICATIONS OF THE LEVENE AND O BRIEN TESTS FOR SPREAD distribution ( γ γ 6 ); (vii) a g 05 h 05 distribution which is aso a ong-taied skewed distribution ( γ 490 γ ) The ast two distributions represent skewed eptokurtic distributions with (vii) more severe than (vi) These distribution conditions were seected in order to evauate the operating characteristics of the procedures across a variety of distributions because they have been examined in other studies (eg Agina Keseman & Penfied 007) The Feishman (978) power transformation is of the form Y a + bz + cz + dz 3 where Z are stard norma variates Feishman provided a tabe of vaues for the coefficients b c d that enabes the stard norma distribution to be transformed into a nonnorma distribution aso having mean zero variance one but with different degrees of skewness kurtosis The extra coefficient a is obtained through the reation a c as a direct resut of constraining Ε (Y) 0 Two sets of coefficients (b c d) were seected from Feishman (978) used in the preceding equation to generate Zs from the RANDGEN function (SAS 006) with the norma distribution option to produce distributions (i) (ii) This RANDGEN SAS subroutine aows a user to generate 0 known distributions both discrete continuous Data from the third distribution was aso generated using the RANDGEN function but with the beta distribution option Beta (05 05) is a symmetric u-shaped distribution hence the negative kurtosis To generate data from a g h distribution stard unit norma variabes ( Z ) were converted to g h distributed rom variabes via ( ) exp g Z hz Y exp g (0) where both g h are non-zero When g is zero hz Y Z exp () The Z scores were generated using the generator RANDGEN with the norma distribution option Observations generated for distributions (iii) (v) (vi) (vii) where the variances were not equa to one were stardized so that they were one to refect the nu hypothesis H 0 : σ σ σ3 ; that is in the simuations σ σ σ 3 Percentages of Tota Trimming Four vaues of tota trimming namey 0% 5% 0% 5% were examined when data were asymmetricay trimmed whether to obtain the vaues used in the transformation of the data or when trimming was carried out on the Levene transformed vaues Z or O Brien s transformed vaues r r t Symmetric trimming vaues of 5% 0% 5% 0% were aso investigated As noted the iterature varies on the amount of recommended (symmetric) trimming thus these vaues were chosen to cover the range of vaues recommended For each condition 5000 repications were conducted the nomina eves of significance for a tests were Resuts Tabes 4 5 summarize the ten best resuts for the modified Levene (960) tests for spread Tabe 4 shows the average rates of Type I error the absoute vaues of the difference between the average rates 005 the percent of cases faing in three intervas ( ) ( ) ( ) The ast coumn indicates tota percentage of cases faing in ( ); using simpe set theory agebra this is ust percent of cases in ( ) in ( ) minus percent of cases in ( ) Based on these findings the foowing are noted: A ten methods examined provided very good Type I error contro Indeed the 64

14 OTHMAN YIN KESELMAN WILCO & ALGINA empirica rates ranged from 0046 to 00579; In order to identify the best method(s) the percentages reported in the ast coumn were reied upon From this information B 3 was identified as the best of the Levene (960) modifications defined examined Tabe 5 presents Type I error rates for each characteristic of the distributions investigated as we as the overa rate indicates that the method that seected as best contains average Type I errors of 0048 The same information is presented in Tabes 6 7 for the ten best modified O Brien (98) tests for spread Based on these findings the ten best O Brien variants provided tight Type I error contro ranging from 0490 to The ast coumn of Tabe 6 identifies two of the modified procedures Q 3 05HQ K0HH 3 as the best of the O Brien (98) modifications Tabe 7 presents Type I error rates for each characteristic of the distributions investigated as we as the overa rate indicates that the both methods that seected as best contain Type I errors averaging 0050 Concusion This study examined the Type I error rate (for α 005) of various modifications of Levene s (960) O Brien s (98) procedures that coud be used to compare variabiity across groups in independent groups designs specificay variations not examined by Keseman et a (008) The procedures examined used Levene (960) or O Brien (98) type transformations of the origina scores or transformed scores except as opposed to using the measures of centra tendency variabiity suggested by Levene O Brien robust measures of centra tendency /or variabiity were adopted The robust vaues of centra tendency variabiity (ie the trimmed means Winsorized variances) were based on symmetric or asymmetric trimming rues that is rues that either set a priori the amount of tota trimming or determined empiricay the amount to be trimmed from the tais (if at a) based on varied recommendations for tota trimming These approaches were aso appied to various test statistics: the ANOVA F-test a robust F-test (Lee & Fung 985) the Wech (95) test bootstrapped versions of these statistics The procedures were compared under seven distributions when group sizes were equa moderatey or very unequa The skewness kurtosis of the distributions examined varied from the norma distribution ( γ 0 γ 0 respectivey) to distributions that were nonnorma γ 49 γ respectivey) The procedures were compared on four measures: the average rate of Type I error across the 4 conditions examined the percentage of empirica Type I errors that fe within the intervas ( ) ( ) ( ) the absoute vaue of the difference between the mean Type I error rate 005 Finay it shoud be noted that though it was intended to examine bootstrapped versions of these procedures this was not pursued because very good Type I error contro was achieved without resorting to bootstrapping Resuts indicated that the resuts reported by Keseman et a (008) coud not be improved upon with respect to the Levene (960) test That is though the new Levene modifications a worked very we in controing Type I error rates they did not resut in as many cases faing into the three intervas defined for good Type I error contro as reported by Keseman et a (008) Conversey two of the O Brien (960) modifications did perform we at east as we as the variants examined by Keseman et a (008) their recommended Levene variant These were Q 3 05HQ K0HH 3 with tighter Type I error contro a decent number of cases faing into the three intervas defined for good Type I error contro by Keseman et a (008) Acknowedgements This research was supported by grants provided by the Socia Sciences Humanities Research Counci of Canada the Fundamenta Research Grant Scheme of Maaysia 65

15 ROBUST MODIFICATIONS OF THE LEVENE AND O BRIEN TESTS FOR SPREAD Tabe 4: Type I Error Rates for the 0 Best Performing New Levene s Variants Percentages of Type I Error Rates within Various Intervas No Variant Average p-vaues Mean-005 Percent Within 550 Percent Within 4555 Percent Within 4550 Tota % B E E 3 055HH E 3 05HH E 3 055HH E 3 055HHQ E 3 055HHQ E 3 05HHQ E E 3 05HHQ Tabe 5: Type I Error Rates for the 0 Best Performing Variants of Levene s Procedure No Variant Overa Skewed Symmetric Leptokurtic Norma Taied Patykurtic B E E 3 055HH E 3 05HH E 3 055HH E 3 055HHQ E 3 055HHQ E 3 05HHQ E E 3 05HHQ Tabe 6: Type I Error Rates for the 0 Best Performing O Brien (979) Variants Percentages of Type I Error Rates within Various Intervas No Variant Average p-vaues Mean-005 Percent Within 5 50 Percent Within Percent Within Tota % Q 3 05HH Q 3 55HH K0HQ Q 3 05HQ Q K0HH K0HH Q 3 00HQ Q 3 05HQ K5HQ

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