TESTS FOR HOMOGENEITY OF VARIANCES 353 Table 1.Tests That Are Classically Based on an In tests for equal variances,F is computed on some Estimate of Sampling Fluctuation Assuming Nor- transformation of the Xi's rather than on the Xi's mality themselves. Comments on the various tests are now presented. Areyation Test Statistic and Distribution The notation med refers to the replacement of with -卫 5学名-点n安为 in the test statistic in an attempt to improve the robustness of the test Aar 21~是here52·00a2-上 N-P.The test proposed by Neyman and Pearson ac1+)l:安 (1931)is the likelihood ratio test under normality.We also examine the modification N-P:med. Cach 6e0e381a18e:t1oy（a970.p.209 Bar.Bartlett (1937)modified N-P to "correct for bias."The resulting test is probably the most common used for equality of variances.It is well known to be B-K 1n(e等-la(ta a/25 6toam1rt1e1970.p.177 sensitive to departures from normality.Recent papers average sample site) by Glaser(1976),Chao and Glaser(1978),and Dyer 8eg818a18art1ey1970.p202 and Keating(1980)give methods for finding the exact distribution of the test statistic.We also examine Bar:med. End 58218rt1eyi97o.p.2 Coch.The test introduced by Cochran(1941)was considerably easier to compute than the tests up to T Bar? 1w“-37aev1c-w2 that time.With today's computers the difference in andb·c+z computation time is slight,however.We also look at (See Bat for c and T2) Coch:med. 名齿学 he1”《 92 2 B-K.Another attempt to simplify calculations re- sulted in this test by Bartlett and Kendall (1946), 星-29-1 which relies on the fact that In s2 is approximately normal and uses tables for the normalized range in w.3 normal samples.We do not examine this test because 03 of its equivalence to the following test. Hart.Four years after B-K this test by Hartley (1950)was presented.Well known as the"F-max"test, Bartrange 0-n点o-3-1.号21c it is merely an exponential transformation of B-K.An advantage of this test is the exact tables available for Lthl 名0à京 equal sample sizes (David 1952).We also examine Hart:med. nd男·ns Table 2.Tests That Attempt To Estimate Kurtosis 爱-1·信-0，/2-6) (See Lehl for T) Test Statistie and Distribution X2k-1) F= (N-1-X2)/N-k) (2.2) Barl g may be compared with quantiles from the F dis- 名a咖i好 tribution with k-1,N-k degrees of freedom. (See Bar for T,and c) In the following descriptions of the tests,we let, and r denote the ith sample mean,median,and range,respectively,while X denotes the overall mean. Bar2 名点7.野 c(1+y/2) The ith sample variance,with divisor n-1,is s.In 【2 addition, N=∑n,s2=∑m-1)s:/N-k, (ee Lathl.for Narl for) and 2+(1- ∑n以X:-X)2/k-) FX)=2x-XPw-内 (2.3) is the usual one-way analysis of variance test statistic. Sch2 (See Lehl for Ta.Bar2 for Y) 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 Table 1. Tests That Are Classically Based on an Estimate of Sampling Fluctuation Assuming Nor￾mality Abbreviation of Test Test Statistic and Distribution 2 H-k 2nk 1 2 N-P 1b - T T1 = N ln(-- s ) - n (i n n- s) Bar. x_1 -2 T where T2 - (N-k)ln s2 - n s and C - 1 + 3(- - i I - and C" 1 + 3(23s-1 -_) - ]N-k Coch max s. i i B-K ln(max s )-ln(min si) (n/2)1 Hart 2 max si min si max ri min ri See Pearson and Hartley (1970), p. 203 for special tables. See Pearson and Hartley (1970), p. 177 for special tables. (n-average sample size) See Pearson and Hartley (1970), p. 202 for special tables. See Pearson and Hartley (1970), p. 264 for special tables. B--ar3, F~w T2 where w - (k+l)/(C-1)2 Far k-l,w (k-)(b-T2) and b - Cw (See Bar for C and T2) 2 k (mi_m)2 2- Sam Xk-1 ' E where m - (1- 2 )s-2/3 a 9(ni-l) a2i 2/[9(n-l)s4/3] (m/a2) and m 2 i E(1/a ) Bar:range [(N-k)ln(H E (ni-l)(~ )2) - Z(ni-)ln( )2 1/C H-k i i d5 i S) (d I (See Bar for C) See Pearson and Hartley (1970), p. 201 for special tables. Lehl X l " T3/2 where T3 - E(ni-l)(Pi- k 2 Xk- 1 (nj- )2 3 3 i N-k E (n3-S)P) and P - ln s2 Leh2 k_1 - (N-k)T3/(2N-4k) (See Lehl for T3) F= X2/(k- 1) (2.2) (N - 1 - X2)/(N - k) (2.2) may be compared with quantiles from the F dis￾tribution with k - 1, N - k degrees of freedom. In the following descriptions of the tests, we let Xi,, Xi, and ri denote the ith sample mean, median, and range, respectively, while X denotes the overall mean. The ith sample variance, with divisor ni - 1, is si. In addition, N= ni,, s2 = (n,i - l)s,/(N- k), and F(Xj) - ,i nX - X)2/(k - 1) (2.3) i, tZ u a (X - o ,)vi(N - k) ( .3) is the usual one-way analysis of variance test statistic. In tests for equal variances, F is computed on some transformation of the Xij's rather than on the X1j's themselves. Comments on the various tests are now presented. The notation med refers to the replacement of Xi with Xi in the test statistic in an attempt to improve the robustness of the test. N-P. The test proposed by Neyman and Pearson (1931) is the likelihood ratio test under normality. We also examine the modification N-P :med. Bar. Bartlett (1937) modified N-P to "correct for bias." The resulting test is probably the most common used for equality of variances. It is well known to be sensitive to departures from normality. Recent papers by Glaser (1976), Chao and Glaser (1978), and Dyer and Keating (1980) give methods for finding the exact distribution of the test statistic. We also examine Bar:med. Coch. The test introduced by Cochran (1941) was considerably easier to compute than the tests up to that time. With today's computers the difference in computation time is slight, however. We also look at Coch :med. B-K. Another attempt to simplify calculations re￾sulted in this test by Bartlett and Kendall (1946), which relies on the fact that In s2 is approximately normal and uses tables for the normalized range in normal samples. We do not examine this test because of its equivalence to the following test. Hart. Four years after B-K this test by Hartley (1950) was presented. Well known as the "F-max" test, it is merely an exponential transformation of B-K. An advantage of this test is the exact tables available for equal sample sizes (David 1952). We also examine Hart :med. Table 2. Tests That Attempt To Estimate Kurtosis Abbreviation of Test Bar 1 Bar2 Schl Sch2 Test Statistic and Distribution (See Bar for T2 and C) T NEE(X ij-Xi4 2 2 - i1 i Xk-_ 1 where y - 3 C(l+y/2) [E(ni-l)sI2 T2 2 3 (See Lehl for T3, Barl for y) 2+(l- )Y 2 T3 Xk-1 + k - (See Lehl for T3, Bar2 for y) TECHNOMETRICS ?, VOL. 23, NO. 4, NOVEMBER 1981 353 Cad 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