TESTS FOR HOMOGENEITY OF VARIANCES 355 introduced by Siegel and Tukey(1960)at about the resulting chi squared test is adjusted from (k-1)to same time.The only advantage of the S-T test is that vi,where v is available in the same reference.We do tables for the Mann-Whitney test may be used;no not examine this test because in general the range is special exact tables are required.We do not examine less efficient than the sample variance. the B-D and S-T tests here because the results would Mill.The innovative jackknife procedure was ap- be essentially the same as those found for F-A-B plied to variance testing by Miller(1968).The jack- Sch/.The test statistic of this parametric procedure, knife procedure relies on partitioning the samples into attributed by Layard (1973)to Scheffe (1959),resem- subsamples of some predetermined size m.We take bles in some respects the numerator of an F statistic m =1,to remove the chance variation involved with computed on si,weighted by the degrees of freedom m 1.We do not examine Mill:med. n-1.The denominator is a function of the(assumed) Bar3.Dixon and Massey (1969)reported a vari- common kurtosis,which in practice must be esti- ation of Bar that uses the F distribution.We also mated.We use the sample kurtosis for y,and also examine Bar3:med. examine Schl:med.The variations Sch2 and Sch2:med Sam.The cube root of s2 is more nearly normal arise when Layard's estimator for y is used. than s,which leads to this test by Samuiddin (1976) Leh/.Lehmann's(1959)suggested procedure is the We also examined Sam:med. same as Sch1,but with y=0 as in normal dis- F-K.Fligner and Killeen (1976)suggest ranking tributions.Ghosh(1972)shows that multiplication by Xi and assigning increasing scores aN.i=i, (N-k)/(N-2k)gives a distribution closer to the chi aw,i=i2,and aw.t=Φ-(1/2+(/2(W+1)》based on square.We call this variation Leh2 and examine those ranks.We suggest using the ranks ofXj Leh1:med and Leh2:med also and call the first test T-G after Talwar and Gentle Levl.Levene(1960)suggested using the one-way (1977),who used a trimmed mean instead of X:.The analysis of variance on the variables Zi=Xy-Xi second test,called the squared ranks test S-R,was as a method of incorporating the robustness of that discussed by Conover and Iman(1978),but has roots test into a test for variance.Further variations sugges- in earlier papers by Shorack (1965),Duran and ted by Levene involve Z2(Lev2),In Zi;(Lev3),and Mielke(1968),and others.We denote the third test by Z(Lev4).We also consider Lev1:med,recommended F-K,even though we have taken liberties with their by Brown and Forsythe(1974),and Lev4:med,but do suggestion.We also examine,as with Mood,the four not examine Lev3:med because In 0=-oo occurs variations associated with each test.We do not exam- with odd sample sizes.We also do not consider use of ine Fligner and Killeen's suggestion of using the grand the trimmed mean as Brown and Forsythe did,largely median in place of because their results indicated no advantages in using This list of tests does not include others such as one this variation by Moses (1963)that relies on a random pairing Capon.Instead of using scores that are a quadratic within samples or one by Sukhatme (1958)that is function of the ranks as Mood had done,Capon closely related to some of the linear rank tests already (1961)suggested choosing scores that give optimum included.Also,the Box-Anderson(1955)permutation power in some sense.The result is this normal scores test for two samples,which Shorack(1965)highly test,which is locally most powerful among rank tests recommends,was found by Hall(1972)to have Type I against the normal-type alternatives,and asymptoti- error rates as high as 27 percent in the multisample cally locally most powerful among all tests for this case with normal populations at =.05,so it is not alternative. included in our study.However,the list is extensive Klotz.Shortly thereafter,Klotz (1962)introduced enough for our purposes,namely,to obtain a listing of another normal scores test that used the more con- tests for variances that appear to have well-controlled venient normal quantiles.The result has possibly less Type I error rates,and to compare the power of the power locally for small sample sizes,but has the same tests.This is accomplished in the next section. asymptotic properties as Capon.Because of its con- venience,we examine the Klotz test,but not the very 3.THE RESULTS OF A SIMULATION STUDY similar Capon.As in Mood,four variations of Klotz In the search for one or more tests that are robust are considered. as well as powerful,it became necessary to obtain Bar:range.Implicit in the literature since Patnaik's (1950)paper on the use of the range instead of the pseudorandom samples from several distributions, using several sample sizes and various combinations variance,but not explicitly mentioned until Gartside of variances.The simulation study is described in this (1972),is this variation of Bar that uses the standard- section.The results in terms of percent of times the ized range instead of the variance.The standardizing null hypothesis was rejected are summarized in Tables constants di are available from Pearson and Hartley 5 and 6. (1970,p.201).The number of degrees of freedom of the For symmetric distributions we chose the uniform, TECHNOMETRICS©，VOL.23,NO.4,NOVEMBER1981 This content downloaded from 61.190.7.73 on Mon,30 Sep 2013 22:38:50 PM All use subject to JSTOR Terms and ConditionsTESTS FOR HOMOGENEITY OF VARIANCES introduced by Siegel and Tukey (1960) at about the same time. The only advantage of the S-T test is that tables for the Mann-Whitney test may be used; no special exact tables are required. We do not examine the B-D and S-T tests here because the results would be essentially the same as those found for F-A-B. Schl. The test statistic of this parametric procedure, attributed by Layard (1973) to Scheffe (1959), resem￾bles in some respects the numerator of an F statistic computed on si, weighted by the degrees of freedom ni - 1. The denominator is a function of the (assumed) common kurtosis, which in practice must be esti￾mated. We use the sample kurtosis for y, and also examine Schl :med. The variations Sch2 and Sch2 :med arise when Layard's estimator for y is used. Lehl. Lehmann's (1959) suggested procedure is the same as Schl, but with y = 0 as in normal dis￾tributions. Ghosh (1972) shows that multiplication by (N - k)/(N - 2k) gives a distribution closer to the chi square. We call this variation Leh2 and examine Lehl :med and Leh2:med also. Levi. Levene (1960) suggested using the one-way analysis of variance on the variables Zij = I Xij - xi as a method of incorporating the robustness of that test into a test for variance. Further variations sugges￾ted by Levene involve Zh/2 (Lev2), In Zij (Lev3), and Zj (Lev4). We also consider Levl :med, recommended by Brown and Forsythe (1974), and Lev4:med, but do not examine Lev3 :med because In 0 = - oo occurs with odd sample sizes. We also do not consider use of the trimmed mean as Brown and Forsythe did, largely because their results indicated no advantages in using this variation. Capon. Instead of using scores that are a quadratic function of the ranks as Mood had done, Capon (1961) suggested choosing scores that give optimum power in some sense. The result is this normal scores test, which is locally most powerful among rank tests against the normal-type alternatives, and asymptoti￾cally locally most powerful among all tests for this alternative. Klotz. Shortly thereafter, Klotz (1962) introduced another normal scores test that used the more con￾venient normal quantiles. The result has possibly less power locally for small sample sizes, but has the same asymptotic properties as Capon. Because of its con￾venience, we examine the Klotz test, but not the very similar Capon. As in Mood, four variations of Klotz are considered. Bar :range. Implicit in the literature since Patnaik's (1950) paper on the use of the range instead of the variance, but not explicitly mentioned until Gartside (1972), is this variation of Bar that uses the standard￾ized range instead of the variance. The standardizing constants di are available from Pearson and Hartley (1970, p. 201). The number of degrees of freedom of the resulting chi squared test is adjusted from (k - 1) to vi, where vi is available in the same reference. We do not examine this test because in general the range is less efficient than the sample variance. Mill. The innovative jackknife procedure was ap￾plied to variance testing by Miller (1968). The jack￾knife procedure relies on partitioning the samples into subsamples of some predetermined size m. We take m = 1, to remove the chance variation involved with m > 1. We do not examine Mill :med. Bar3. Dixon and Massey (1969) reported a vari￾ation of Bar that uses the F distribution. We also examine Bar3 :med. Sam. The cube root of s2 is more nearly normal than s2, which leads to this test by Samuiddin (1976). We also examined Sam :med. F-K. Fligner and Killeen (1976) suggest ranking |Xij| and assigning increasing scores aN, = i, aN, i = i2, and aN. i = - 1(1/2 + (i/2(N + 1))) based on those ranks. We suggest using the ranks of| Xi - Xi | and call the first test T-G after Talwar and Gentle (1977), who used a trimmed mean instead of Xi. The second test, called the squared ranks test S-R, was discussed by Conover and Iman (1978), but has roots in earlier papers by Shorack (1965), Duran and Mielke (1968), and others. We denote the third test by F-K, even though we have taken liberties with their suggestion. We also examine, as with Mood, the four variations associated with each test. We do not exam￾ine Fligner and Killeen's suggestion of using the grand median in place of Xi. This list of tests does not include others such as one by Moses (1963) that relies on a random pairing within samples or one by Sukhatme (1958) that is closely related to some of the linear rank tests already included. Also, the Box-Anderson (1955) permutation test for two samples, which Shorack (1965) highly recommends, was found by Hall (1972) to have Type I error rates as high as 27 percent in the multisample case with normal populations at a = .05, so it is not included in our study. However, the list is extensive enough for our purposes, namely, to obtain a listing of tests for variances that appear to have well-controlled Type I error rates, and to compare the power of the tests. This is accomplished in the next section. 3. THE RESULTS OF A SIMULATION STUDY In the search for one or more tests that are robust as well as powerful, it became necessary to obtain pseudorandom samples from several distributions, using several sample sizes and various combinations of variances. The simulation study is described in this section. The results in terms of percent of times the null hypothesis was rejected are summarized in Tables 5 and 6. For symmetric distributions we chose the uniform, TECHNOMETRICS ?, VOL. 23, NO. 4, NOVEMBER 1981 355 This content downloaded from 61.190.7.73 on Mon, 30 Sep 2013 22:38:50 PM All use subject to JSTOR Terms and Conditions