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 ConditionsTECHNOMETRICS ?, 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 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 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; Non￾parametric 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 pa￾rameter of the lognormal distribution can be esti￾mated 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 assump￾tion of normal distributions. However, the dis￾tribution 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 depar￾tures 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 like￾lihood ratio test. Others are adaptations of the F test to test variances rather than means. Many are based on nonparametric methods, although their modifi￾cation for the case in which the means are unknown often makes these tests distributionally dependent. Among the many possible tests for equality of vari￾ances, one would hope that at least one is robust to variations in the underlying distribution and yet sensi￾tive to departures from the equal variance hypothesis. However, recent comparative studies are not reassur￾ing in this regard. For example, Gartside (1972) stud￾ied 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 subsam￾ples. Layard (1973) reached a similar conclusion re￾garding 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