Bartlett's Test (original) (raw)
Related papers
A New Alternative in Testing for Homogeneity of Variances
Journal of Statistical Research, 2006
A new alternative procedure is proposed for testing the hypothesis that the variances of k independent groups are equal under non-normality. Extensive simulations indicated that the new procedure always gave the experimenter more control over the probability of a Type I error than do the Bartlett χ 2 and Levene's tests. The same statement was true for unequal sample sizes. Overall, the new procedure was more powerful than the other procedures tested, especially when the variance ratios were high.
In this note we propose a general testing procedure for parametric models based on Bartlett Identities. A well-known example is the Information Matrix test, which is based on the Bartlett Identity of order 1. The Identities are shown to induce a sequence of testable restrictions on the data generating process. When all the restrictions are considered jointly, they are often complete, in the sense that they are satisfied if and only if the model is correctly specified. We show that this is the case for normal, exponential and Poisson models. A test of the joint validity of an arbitrarily chosen subset of restrictions is proposed, and its first order asymptotic properties are presented. An interpretation of the test as a score test for neglected parameter heterogeneity is also given.
A Comparison of Five Bootstrap and Non-Bootstrap Levene-Type Tests of Homogeneity of Variances
Iranian Journal of Science and Technology, Transactions A: Science, 2018
The test of homogeneity of variances arises in various statistical problems. In this paper, we compared five Levene-type tests including Levene's test, Welch's test, James's test, Brown and Forsythe's test and Alexander-Govern's test in terms of power and size. The simulation results show that the bootstrapping after applying Noguchi and Gel (J Nonparametr Stat 22:897-913, 2010) correction and removing structural zeros significantly improve the performance of Type I error rates. We identify the best in terms of controlling the Type I error rate and power for each number of groups.
A Monte Carlo Study of Seven Homogeneity of Variance Tests
2010
Problem statement: The decision by SPSS (now PASW) to use the unmodified Levene test to test homogeneity of variance was questioned. It was compared to six other tests. In total, seven homogeneity of variance tests used in Analysis Of Variance (ANOVA) were compared on robustness and power using Monte Carlo studies. The homogeneity of variance tests were (1) Levene, (2) modified Levene, (3) Z-variance, (4) Overall-Woodward Modified Z-variance, (5) O'Brien, (6) Samiuddin Cube Root and (7) F-Max. Approach: Each test was subjected to Monte Carlo analysis through different shaped distributions: (1) normal, (2) platykurtic, (3) leptokurtic, (4) moderate skewed and (5) highly skewed. The Levene Test is the one used in all of the latest versions of SPSS. Results: The results from these studies showed that the Levene Test is neither the best nor worst in terms of robustness and power. However, the modified Levene Test showed very good robustness when compared to the other tests but lower power than other tests. The Samiuddin test is at its best in terms of robustness and power when the distribution is normal. The results of this study showed the strengths and weaknesses of the seven tests. Conclusion/Recommendations: No single test outperformed the others in terms of robustness and power. The authors recommend that kurtosis and skewness indices be presented in statistical computer program packages such as SPSS to guide the data analyst in choosing which test would provide the highest robustness and power.
Statistics in Medicine, 2004
A common problem that arises in the meta-analysis of several studies, each with independent treatment and control groups, is to test for the homogeneity of e ect sizes without the assumptions of equal 11 variances of the treatment and the control groups and of equal variances among the separate studies. A commonly used test statistic, frequently denoted as Q, is the weighted sum of squares of the di erences 13 of the individual e ect sizes from the mean e ect size, with weights inversely proportional to the variances of the e ect sizes. The primary contributions of this article are the presentation of improved 15 and very accurate approximations to the distributions of the Q statistic when the e ect size is a linear contrast such as the di erence between the treatment and control means. Our improved approximation to 17 the distribution of Q under the null hypothesis is based on a multiple of an F-distribution; its use yields a substantial reduction in the type I error rate of the homogeneity test. Our improved approximation to 19 the distribution of Q under an alternative hypothesis is based on a shift of a chi-square distribution; its use allows for much greater accuracy in the computation of the power of the homogeneity test. These 21 two improved approximate distributions are developed using the Welch methodology of approximating the moments of Q by the use of multivariate Taylor expansions. The quality of these approximations 23 is studied by simulation.
Power comparisons of two-sided tests of equality of two covariance matrices based on six criteria
Annals of the Institute of Statistical Mathematics, 1979
Power studies of tests of equality of covariance matrices of two p-variate normal populations ~v~=Z2 against two-sided alternatives have been made based on the following six criteria: 1) Roy's largest root, 2) Hotelling's trace, 3) Pillai's trace, 4) Wilks' criterion, 5) Roy's largestsmallest roots and 6) modified likelihood ratio. A general theorem has been proved establishing the local unbiasedness conditions connecting the two critical values for tests 1) to 5). Extensive unbiased power tabulations have been made for 29=2, for various values of n,, n2, 21 and 22 where n~ is the df of the SP matrix from the ith sample and 2~ is the ith latent root of I~/:; ~ (i=1, 2). Further, comparisons of powers of tests 1) to 5) have been made with those of the modified likelihood ratio after obtaining the exact distribution of the latter for n2=2nl and p=2. Equal tail areas approach has also been used further to compute powers of tests 1) to 4) for p = 2 for studying the bias. Again, a separate study has been made to compare the powers of the largest-smallest roots test with its three biased approximate approaches as well as the largest root. Since the largest root test was observed to have some advantage over the others, critical values were also obtained for this test in the unbiased as well as equal tail areas case for p-3. 1. Introduction Let X1 (pxnl) and X2 (p• p<__n~, i = 1 , 2, be independent matrix variates, columns of X1 being independently distributed as N(O, I~) and AMS 1970 subject classifications: 62H10, 62H15.
Preliminary Tests of Homogeneity -Type I Error Rates under Non-Normality
Biostatistics and Biometrics Open Access Journal, 2018
Many statistical procedures utilize preliminary tests to enhance the accuracy of the final inferences. Preliminary tests like Goldfeld-Quandt (GQ) and Levene-type tests are used to assess the assumption of equality of population variances with normality as the underlying distributional assumption. Such tests must be used with care as the final inferences are conditional on the performance of these tests at first stage. This study explores the size distortions of GQ and Levene-type tests under non-normality. The results do not warrant the use of GQ & Levene test under non-normality as the size distortions are as high as 88 & 48% for the respective statistics. However, the modified form of Levene test (BF-test) retains its size properties except for the multi-model alternatives with relatively big outliers.