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《实用非参数统计》课程教学资源（阅读材料）A Comparative Study of Tests for Homogeneity of Variances, with Applications to the Outer Continental Shelf Bidding Data

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TECHNOMETRICS©，VOL.23,NO.4,NOVEMBER1981 This paper was presented at the TECHNOMETRICS Session of the 25th Annual Fall Technical Conference of the Chemical Division of the American Society for Quality Control and the Section on Physical and Engineering Sciences of the American Statistical Associ- ation in Gatlinburg,Tennessee,October 29-30,1981. A Comparative Study of Tests for Homogeneity of Variances,with Applications to the Outer Continental Shelf Bidding Data W.J.Conover Mark E.Johnson and Myrle M.Johnson College of Business Statistics Group,S-1 Administration Los Alamos National Texas Tech University Laboratory Lubbock,TX 79409 Los Alamos,NM 87545 Many of the existing parametric and nonparametric tests for homogeneity of variances,and some variations of these tests,are examined in this paper.Comparisons are made under the null hypothesis(for robustness)and under the alternative(for power).Monte Carlo simulations of various symmetric and asymmetric distributions,for various sample sizes,reveal a few tests that are robust and have good power.These tests are further compared using data from outer continental shelf bidding on oil and gas leases. KEY WORDS:Test for homogeneity of variances;Bartlett's test;Robustness;Power;Non- parametric tests;Monte Carlo 1.INTRODUCTION to test variances rather than means.Many are based Tests for homogeneity of variances are often of on nonparametric methods,although their modifi- interest as a preliminary to other analyses such as cation for the case in which the means are unknown analysis of variance or a pooling of data from different often makes these tests distributionally dependent. sources to yield an improved estimated variance.For Among the many possible tests for equality of vari- example,in the data base described in Section 4,if the ances,one would hope that at least one is robust to variance of the logs of the bids on each offshore lease variations in the underlying distribution and yet sensi- is homogeneous within a sale,then the scale pa- tive to departures from the equal variance hypothesis. rameter of the lognormal distribution can be esti- However,recent comparative studies are not reassur- mated using all the bids in the sale.In quality control ing in this regard.For example,Gartside(1972)stud- work,tests for homogeneity of variances are often a ied eight tests and concluded that the only robust useful endpoint in an analysis procedure was a log-anova test that not only has poor The classical approach to hypothesis testing usually power,but also depends on the unpleasant process of begins with the likelihood ratio test under the assump- dividing each sample at random into smaller subsam- tion of normal distributions.However,the dis- ples.Layard(1973)reached a similar conclusion re- tribution of the statistic in the likelihood ratio test for garding the log-anova test,but indicated that two equality of variances in normal populations depends other tests in his study of four tests,Miller's jackknife on the kurtosis of the distribution(Box 1953),which procedure and Scheffe's chi squared test,did not suffer helps to explain why that test is so sensitive to depar- greatly from lack of robustness and had considerably tures from normality.This nonrobust (sometimes more power,at least when sample sizes were equal. called"puny")property of the likelihood ratio test has These tests are included in our study as Mill and Sch2. prompted the invention of many alternative tests for Layard indicated a reluctance to use these tests when variances.Some of these are modifications of the like- sample sizes are less than 10,and yet this is the case of ihood ratio test.Others are adaptations of the F test interest to us,as we explain later.The jackknife pro- 351 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

TECHNOMETRICS ?, VOL. 23, NO. 4, NOVEMBER 1981 This paper was presented at the TECHNOMETRICS Session of the 25th Annual Fall Technical Conference of the Chemical Division of the American Society for Quality Control and the Section on Physical and Engineering Sciences of the American Statistical Association in Gatlinburg, Tennessee, October 29-30, 1981. A Comparative Study of Tests for Homogeneity of Variances, with Applications to the Outer Continental Shelf Bidding Data W. J. Conover College of Business Administration Texas Tech University Lubbock, TX 79409 Mark E. Johnson and Myrle M. Johnson Statistics Group, S-1 Los Alamos National Laboratory Los Alamos, NM 87545 Many of the existing parametric and nonparametric tests for homogeneity of variances, and some variations of these tests, are examined in this paper. Comparisons are made under the null hypothesis (for robustness) and under the alternative (for power). Monte Carlo simulations of various symmetric and asymmetric distributions, for various sample sizes, reveal afew tests that are robust and have good power. These tests are further compared using data from outer continental shelf bidding on oil and gas leases. KEY WORDS: Test for homogeneity of variances; Bartlett's test; Robustness; Power; Nonparametric tests; Monte Carlo. 1. INTRODUCTION Tests for homogeneity of variances are often of interest as a preliminary to other analyses such as analysis of variance or a pooling of data from different sources to yield an improved estimated variance. For example, in the data base described in Section 4, if the variance of the logs of the bids on each offshore lease is homogeneous within a sale, then the scale parameter of the lognormal distribution can be estimated using all the bids in the sale. In quality control work, tests for homogeneity of variances are often a useful endpoint in an analysis. The classical approach to hypothesis testing usually begins with the likelihood ratio test under the assumption of normal distributions. However, the distribution of the statistic in the likelihood ratio test for equality of variances in normal populations depends on the kurtosis of the distribution (Box 1953), which helps to explain why that test is so sensitive to departures from normality. This nonrobust (sometimes called "puny") property of the likelihood ratio test has prompted the invention of many alternative tests for variances. Some of these are modifications of the likelihood ratio test. Others are adaptations of the F test to test variances rather than means. Many are based on nonparametric methods, although their modification for the case in which the means are unknown often makes these tests distributionally dependent. Among the many possible tests for equality of variances, one would hope that at least one is robust to variations in the underlying distribution and yet sensitive to departures from the equal variance hypothesis. However, recent comparative studies are not reassuring in this regard. For example, Gartside (1972) studied eight tests and concluded that the only robust procedure was a log-anova test that not only has poor power, but also depends on the unpleasant process of dividing each sample at random into smaller subsamples. Layard (1973) reached a similar conclusion regarding the log-anova test, but indicated that two other tests in his study of four tests, Miller's jackknife procedure and Scheff6's chi squared test, did not suffer greatly from lack of robustness and had considerably more power, at least when sample sizes were equal. These tests are included in our study as Mill and Sch2. Layard indicated a reluctance to use these tests when sample sizes are less than 10, and yet this is the case of interesto us, as we explain later. The jackknife pro- 351 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

352 W.J.CONOVER,MARK E.JOHNSON,AND MYRLE M.JOHNSON cedure appeared to be the best of the six procedures described.A final section presents the summary and investigated by Hall(1972)in an extensive simulation conclusions of this study. study,while Keselman,Games,and Clinch(1979)con- clude that the jackknife procedure(Mill)has unstable 2.A SURVEY OF k-SAMPLE TESTS FOR error rates (Type I error)when the sample sizes are EQUALITY OF VARIANCES unequal.They conclude from their study of 10 tests For i=1,...,k,let {Xu be random samples of size that "the current tests for variance heterogeneity are ni from populations with means u;and variances of. either sensitive to nonnormality or,if robust,lacking To test the hypothesis of equal variances,one ad- in power.Therefore these tests cannot be rec- ditional assumption is necessary (Moses 1963).One ommended for the purpose of testing the validity of possible assumption is that the Xi's are normally the ANOVA homogeneity assumption."The four tests distributed.This leads to a large number of tests,some studied by Levy (1978)all "were grossly affected by with exact tables available and some with only violations of the underlying assumption of normality." asymptotic approximations available,for the dis- The potential user of a test for equality of variances tributions of the test statistics.Another possible as- is thus presented with a confusing array of infor- sumption is that the Xif's are identically distributed mation concerning which test to use.As a result,many when the null hypothesis is true.This assumption users default to Bartlett's (1937)modification of the enables various nonparametric tests to be formulated likelihood ratio test,a modification that is well known In practice,neither assumption is entirely true,so that to be nonrobust and that none of the comparative all of these tests for variances are only approximate.It studies recommends except when the populations are is appropriate to examine all of the available tests for known to be normal.The purpose of our study is to their robustness to violations of the assumptions.In provide a list of tests that have a stable Type I error this section we present a(nearly)chronological listing rate when the normality assumption may not be true, of tests for equal variances and a summary of these when sample sizes may be small and/or unequal,and tests in Tables 1 through 4.Most of the tests in Tables when distributions may be skewed and heavy-tailed. 1 through 3 are based on some modification of the The tests that show the desired robustness are com- likelihood ratio test statistic derived under the as- pared on the basis of power.Further,we hope that sumption of normality.Tests that are essentially our method of comparing tests may be useful in future modifications of the likelihood ratio test or that other- studies for evaluating additional tests of variance. wise rely on the assumption of normality are given in The tests examined in this study are described Table 1.Modifications to those tests,employing an briefly in Section 2.Fifty-six tests for equality of vari- estimate of the kurtosis,appear in Table 2.They are ances are compared,most of which are variations of asymptotically distribution free for all parent popu- the most popular and most useful parametric and lations,with only minor restrictions.Tests based on a nonparametric tests available for testing the equality modification of the F test for means are given in Table of k variances (k 2)in the presence of unknown 3,along with the jackknife test,which does not seem means.Some tests not studied in detail are also men- to fit anywhere else.Finally,Table 4 presents modifi- tioned in Section 2,along with the reason for their cations of nonparametric tests.The modification con- exclusion.This coverage is by far the most extensive sists of using the sample mean or sample median that we are aware of and should provide valuable instead of the population mean when computing the comparative information regarding tests for variances. test statistic.Only nonparametric tests in the class of The simulation study is described in Section 3.Each linear rank tests are included here,because this class test statistic is computed 1,000 times in each of 91 of tests includes all locally most powerful rank tests situations,representing various distributions,sample (Hajek and Sidak 1967).Therefore,in Table 4,only sizes,means,and variances.Nineteen of these sample the scores,a.i,for these tests are presented.From situations have equal variances and are therefore these scores,chi squared tests may be formulated studies of the Type I error rate,while the remaining 72 based on the statistic situations represent studies of the power The basic motivation for this study is described in Section 4.The lease production,and revenue(LPR) X2=∑n:(a-a2/W2, (2.1) data base includes,among other data,the actual amount of each sealed bid submitted by oil and gas where A;=mean score in the ith sample,a overall companies on individual tracts offered by the federal government in all of the sales of offshore oil and gas mean score 1/N >aN.i,and V2 =(1/N -1) leases in the United States since 1954.The results of 1(aw.-a)2,which is compared with quantiles from a chi squared distribution with k-1 degrees of several tests for variances applied to those sales are freedom.Alternatively,the statistic 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 Conditions

W. J. CONOVER, MARK E. JOHNSON, AND MYRLE M. JOHNSON cedure appeared to be the best of the six procedures investigated by Hall (1972) in an extensive simulation study, while Keselman, Games, and Clinch (1979) conclude that the jackknife procedure (Mill) has unstable error rates (Type I error) when the sample sizes are unequal. They conclude from their study of 10 tests that "the current tests for variance heterogeneity are either sensitive to nonnormality or, if robust, lacking in power. Therefore these tests cannot be recommended for the purpose of testing the validity of the ANOVA homogeneity assumption." The four tests studied by Levy (1978) all "were grossly affected by violations of the underlying assumption of normality." The potential user of a test for equality of variances is thus presented with a confusing array of information concerning which test to use. As a result, many users default to Bartlett's (1937) modification of the likelihood ratio test, a modification that is well known to be nonrobust and that none of the comparative studies recommends except when the populations are known to be normal. The purpose of our study is to provide a list of tests that have a stable Type I error rate when the normality assumption may not be true, when sample sizes may be small and/or unequal, and when distributions may be skewed and heavy-tailed. The tests that show the desired robustness are compared on the basis of power. Further, we hope that our method of comparing tests may be useful in future studies for evaluating additional tests of variance. The tests examined in this study are described briefly in Section 2. Fifty-six tests for equality of variances are compared, most of which are variations of the most popular and most useful parametric and nonparametric tests available for testing the equality of k variances (k > 2) in the presence of unknown means. Some tests not studied in detail are also mentioned in Section 2, along with the reason for their exclusion. This coverage is by far the most extensive that we are aware of and should provide valuable comparative information regarding tests for variances. The simulation study is described in Section 3. Each test statistic is computed 1,000 times in each of 91 situations, representing various distributions, sample sizes, means, and variances. Nineteen of these sample situations have equal variances and are therefore studies of the Type I error rate, while the remaining 72 situations represent studies of the power. The basic motivation for this study is described in Section 4. The lease production, and revenue (LPR) data base includes, among other data, the actual amount of each sealed bid submitted by oil and gas companies on individual tracts offered by the federal government in all of the sales of offshore oil and gas leases in the United States since 1954. The results of several tests for variances applied to those sales are described. A final section presents the summary and conclusions of this study. 2. A SURVEY OF k-SAMPLE TESTS FOR EQUALITY OF VARIANCES For i = 1, ..., k, let {Xij} be random samples of size ni from populations with means pi and variances of. To test the hypothesis of equal variances, one additional assumption is necessary (Moses 1963). One possible assumption is that the Xij's are normally distributed. This leads to a large number of tests, some with exact tables available and some with only asymptotic approximations available, for the distributions of the test statistics. Another possible assumption is that the Xij's are identically distributed when the null hypothesis is true. This assumption enables various nonparametric tests to be formulated. In practice, neither assumption is entirely true, so that all of these tests for variances are only approximate. It is appropriate to examine all of the available tests for their robustness to violations of the assumptions. In this section we present a (nearly) chronological listing of tests for equal variances and a summary of these tests in Tables 1 through 4. Most of the tests in Tables 1 through 3 are based on some modification of the likelihood ratio test statistic derived under the assumption of normality. Tests that are essentially modifications of the likelihood ratio test or that otherwise rely on the assumption of normality are given in Table 1. Modifications to those tests, employing an estimate of the kurtosis, appear in Table 2. They are asymptotically distribution free for all parent populations, with only minor restrictions. Tests based on a modification of the F test for means are given in Table 3, along with the jackknife test, which does not seem to fit anywhere else. Finally, Table 4 presents modifications of nonparametric tests. The modification consists of using the sample mean or sample median instead of the population mean when computing the test statistic. Only nonparametric tests in the class of linear rank tests are included here, because this class of tests includes all locally most powerful rank tests (Hajek and Sidak 1967). Therefore, in Table 4, only the scores, a, i, for these tests are presented. From these scores, chi squared tests may be formulated based on the statistic k X2 = E ni(Ai-a)2/V2, i= 1 (2.1) where Ai = mean score in the ith sample, a = overall mean score = 1/N EiN= aN.i, and V2 = (1/N - 1) 1= (aN.- a)2, which is compared with quantiles from a chi squared distribution with k - 1 degrees of freedom. Alternatively, the statistic TECHNOMETRICS ?, VOL. 23, NO. 4, NOVEMBER 1981 352 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

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 Conditions

TESTS FOR HOMOGENEITY OF VARIANCES Table 1. Tests That Are Classically Based on an Estimate of Sampling Fluctuation Assuming Normality 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 distribution 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 resulted 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

354 W.J.CONOVER,MARK E.JOHNSON,AND MYRLE M.JOHNSON Table 3.Tests Based on a Modification of the F tractive to the practitioner.For this reason we do not Test for Means see equation (2.3)for F()) include these tests in our study.A Monte Carlo com- w FK-1.N-k-F(1X13-X11) parison of these methods with the jackknife methods (see Mill)is presented by Martin and Games(1977). Lev2 -,M-k·F%-X)) Mood.The first nonparametric test for the variance -1.krn%-风) problem was presented by Mood(1954).It,like all of the nonparametric tests,assumes identical dis- Levs 男-1.k”F%11:2 tributions under the null hypothesis.In particular,this requires equal means,or a known transformation to 5-1,Nek”ru}tere"时"ng in af-(n-i)in achieve equal means,which is often not met in appli- ad划2nn)21n1 cations.Therefore,we adapt the Mood test and all of the nonparametric tests as follows.Instead of letting Rij be the rank of Xu when the means are equal or of Cad.A desire for simplification led to replacing the (Xy-u)when the means are unequal but known,we variance in Hart with the sample range in a paper by let Rij be the rank of (Xij).Each Xij is then Cadwell (1953).Exact tables for equal sample sizes are replaced by the score aN,Ri;based on this rank.The given by Harter(1963)for k=2 and Leslie and Brown result is a test that is not nonparametric but may be as (1966)for k s 12.We do not examine this test because robust and powerful as some of its parametric com- we feel that the computational advantages are no petitors.The use ofX instead ofi results in longer real with present-day software Mood:med,which we also examine.The chi squared Barl.Box (1953)showed that the asymptotic dis- approximation and the F approximation for each test tribution of Bar was dependent on the common kur- lead to four variations,which are studied. tosis of the sampled distributions and that by dividing F-A-B.Although the Mood test is a quadratic func- Bar by (1 +y/2),where y=E(X-u)/of-3,the tion of Ri,this test introduced by Freund and Ansari test would be asymptotically distribution free,pro- (1957)and further developed by Ansari and Bradley vided the assumption of common kurtosis was met. (1960)is a linear function of Rij.Again,we let Rij be Our form for this modification of Bar involves esti- the rank of (Xij-X).We examine four variations of mating y with the sample moments,a suggestion that F-A-B(see Mood).The B-D test was introduced by Layard (1973)attributes to Scheffe (1959).We also Barton and David (1958)shortly after the F-A-B test examine Barl:med.Bar2 and Bar2:med result from a and is similar to the F-A-B test in principle.Whereas different estimator for y as given by Layard the F-A-B scores are triangular in shape,the B-D Box.An interesting approach to obtaining a more scores follow a V shape with the large scores at the robust test for variance involves using the one-way extremes and the small scores at the grand median. layout F statistic,which is known to be quite robust. The result is a test with the same robustness and A concept suggested by Bartlett and Kendall(1946) power as F-A-B.The same can be said for the S-T test, was developed by Box(1953)into a test known as the log-anova test.For a preselected,arbitrary integer m>2,each sample is divided into subsamples of size Table 4.Linear Rank Tests scores may be used in m in some random manner.(See Martin and Games equations(2.1),(2.2),or(2.3)) 1975,1977 and Martin 1976 for suggestions on the size of m.)Remaining observations either are not used Score Function whereR the rank of: or are included in the final subsample.The sample 4-42 % variance sij is computed for each subsample, i=1,...,k,j=1,...,[n/m]=Ji.A log trans- F-A-B ---1,2,313.2,1 % formation Yy=In sy then makes the variables more P 3,2,1,12,… x nearly normal,and F(Y)is used as a test statistic. Subsequent studies by Gartside(1972),Layard(1973), 1,4,5…6,3,2 」 and Levy (1975)confirmed the robustness of this 9200 【E,1here,11the1边 method,but also revealed a lack of power as com- oradele pared with other tests that have the same robustness. ()2 where (x)ts the A modification that leads to a more nearly normal faea5doa1aatrtbnton sample is attributed to Bargmann by Gartside(1972). It uses Wij=wi(ln sij c),where wi and ci are nor- x malizing constants.However,the random method of 2 x subdividing samples and the possibility of not using all of the observations make these procedures unat- ◆小空+20* (SeeK1 otr for)】 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 Conditions

W. J. CONOVER, MARK E. JOHNSON, AND MYRLE M. JOHNSON Table 3. Tests Based on a Modification of the F Test for Means ( see equation ( 2. 3 ) for F ( ) ) Levl Fk-, N-k F(IX -Xil) Lev2 Fk-1, N-k = F((Xij-Xi)2) Lev3 Fk-1, N-k = F(ln(Xij -i)2) Lev4 Fk-l, N-k = F ( X I i l) Mill F 1 Nk = F(Uij) where Uj = ni in si -(ni -)ln sij 2 1 2 2 and sij2 = n-2 [(ni-1)si-ni(Xij-Xi) /(ni-O)] Cad. A desire for simplification led to replacing the variance in Hart with the sample range in a paper by Cadwell (1953). Exact tables for equal sample sizes are given by Harter (1963) for k = 2 and Leslie and Brown (1966) for k < 12. We do not examine this test because we feel that the computational advantages are no longer real with present-day software. Barl. Box (1953) showed that the asymptotic distribution of Bar was dependent on the common kurtosis of the sampled distributions and that by dividing Bar by (1 + y/2), where y = E(Xij - )4/a - 3, the test would be asymptotically distribution free, provided the assumption of common kurtosis was met. Our form for this modification of Bar involves estimating y with the sample moments, asuggestion that Layard (1973) attributes to Scheffe (1959). We also examine Barl:med. Bar2 and Bar2:med result from a different estimator for y as given by Layard. Box. An interesting approach to obtaining a more robust test for variance involves using the one-way layout F statistic, which is known to be quite robust. A concept suggested by Bartlett and Kendall (1946) was developed by Box (1953) into a test known as the log-anova test. For a preselected, arbitrary integer m > 2, each sample is divided into subsamples of size m in some random manner. (See Martin and Games 1975, 1977 and Martin 1976 for suggestions on the size of m.) Remaining observations either are not used or are included in the final subsample. The sample variance si is computed for each subsample, i = 1, ..., k, j= 1, ..., [ni/m] = Ji. A log transformation Yij = In sij then makes the variables more nearly normal, and F(Y1j) is used as a test statistic. Subsequent studies by Gartside (1972), Layard (1973), and Levy (1975) confirmed the robustness of this method, but also revealed a lack of power as compared with other tests that have the same robustness. A modification that leads to a more nearly normal sample is attributed to Bargmann by Gartside (1972). It uses Wij = wi(ln sij + ci), where wi and ci are normalizing constants. However, the random method of subdividing samples and the possibility of not using all of the observations make these procedures unattractive to the practitioner. For this reason we do not include these tests in our study. A Monte Carlo comparison of these methods with the jackknife methods (see Mill) is presented by Martin and Games (1977). Mood. The first nonparametric test for the variance problem was presented by Mood (1954). It, like all of the nonparametric tests, assumes identical distributions under the null hypothesis. In particular, this requires equal means, or a known transformation to achieve equal means, which is often not met in applications. Therefore, we adapt the Mood test and all of the nonparametric tests as follows. Instead of letting Rij be the rank of Xij when the means are equal or of (Xij - p) when the means are unequal but known, we let Rij be the rank of (Xij - X). Each Xij is then replaced by the score aN, Rij based on this rank. The result is a test that is not nonparametric but may be as robust and powerful as some of its parametric competitors. The use of Xi instead of Xi results in Mood:med, which we also examine. The chi squared approximation and the F approximation for each test lead to four variations, which are studied. F-A-B. Although the Mood test is a quadratic function of Rij, this test introduced by Freund and Ansari (1957) and further developed by Ansari and Bradley (1960) is a linear function of Rj. Again, we let Rij be the rank of (Xij - X). We examine four variations of F-A-B (see Mood). The B-D test was introduced by Barton and David (1958) shortly after the F-A-B test and is similar to the F-A-B test in principle. Whereas the F-A-B scores are triangular in shape, the B-D scores follow a V shape with the large scores at the extremes and the small scores at the grand median. The result is a test with the same robustness and power as F-A-B. The same can be said for the S-T test, Table 4. Linear Rank Tests (scores may be used in equations (2. 1), (2.2), or (2.3) ) Abbreviation of Test Score aNR is a function of Rij, Score Function aN,i where Rij is the rank of: Mood (i- N1)2 F-A-B 2 - ji- 2 1-1, 2, 3,...3, 2, 1 B-D ...,3, 2, 1, 1, 2, 3, ... S-T 1, 4, 5,..., 6, 3, 2 Capon [E(ZN,i)2 where ZN, is the i th order statistic from a standard normal random sample of size N Klotz -i1 2 [~ (N~+) where n(x) is the standard normal distribution function T-G i S-R i2 (e-1 1 + i (See Klotz for 0) (Xi-Xi) (X -X ) (X ij-X) (X ij-Xi ) (X ij-X i) I xij-x i I Xij -Xi I Xl - il TECHNOMETRICS ?, VOL. 23, NO. 4, NOVEMBER 1981 354 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

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