Testing homogeneity of variances with unequal sample sizes

Abstract When sample sizes are unequal, problems of heteroscedasticity of the variables given by the absolute deviation from the median arise. This paper studies how the best known heteroscedastic alternatives to the ANOVA F test perform when they are applied to these variables. This procedure leads to testing homoscedasticity in a similar manner to Levene’s (1960) test. The difference is that the ANOVA method used by Levene’s test is non-robust against unequal variances of the parent populations and Levene’s variables may be heteroscedastic. The adjustment proposed by O’Neil and Mathews (Aust Nz J Stat 42:81–100, 2000) is approximated by the Keyes and Levy (J Educ Behav Stat 22:227–236, 1997) adjustment and used to ensure the correct null hypothesis of homoscedasticity. Structural zeros, as defined by Hines and O’Hara Hines (Biometrics 56:451–454, 2000), are eliminated. To reduce the error introduced by the approximate distribution of test statistics, estimated critical values are used. Simulation results show that after applying the Keyes–Levy adjustment, including estimated critical values and removing structural zeros the heteroscedastic tests perform better than Levene’s test. In particular, Brown–Forsythe’s test controls the Type I error rate in all situations considered, although it is slightly less powerful than Welch’s, James’s, and Alexander and Govern’s tests, which perform well, except in highly asymmetric distributions where they are moderately liberal.

[1]  Yulia R. Gel,et al.  lawstat: An R Package for Law, Public Policy and Biostatistics , 2008 .

[2]  J. Tukey,et al.  A Nonparametric Sum of Ranks Procedure for Relative Spread in Unpaired Samples , 1960 .

[3]  Dexter O. Cahoy.  A bootstrap test for equality of variances , 2010, Comput. Stat. Data Anal..

[4]  M. Akritas,et al.  Heteroscedastic One-Way ANOVA and Lack-of-Fit Tests , 2004 .

[5]  A. D. Beuckelaer,et al.  A closer examination on some parametric alternatives to the ANOVA F-test , 1996 .

[6]  H. Levene.  Robust tests for equality of variances , 1961 .

[7]  R. Wilcox.  ANOVA: A Paradigm for Low Power and Misleading Measures of Effect Size? , 1995 .

[8]  Morton B. Brown,et al.  Robust Tests for the Equality of Variances , 1974 .

[9]  M. Bartlett.  Properties of Sufficiency and Statistical Tests , 1992 .

[10]  A comparative study of nonparametric two–sample tests after Levene's transformation , 2007 .

[11]  Rand R. Wilcox,et al.  New monte carlo results on the robustness of the anova f, w and f statistics , 1986 .

[12]  Helmut Schneider,et al.  A note on levene's tests for equality of variances , 1985 .

[13]  L. Hothorn,et al.  Parametric location-scale and scale trend tests based on Levene's transformation , 2000 .

[14]  G. S. James.  THE COMPARISON OF SEVERAL GROUPS OF OBSERVATIONS WHEN THE RATIOS OF THE POPULATION VARIANCES ARE UNKNOWN , 1951 .

[15]  R. Wilcox.  A Bootstrap Modification of the Alexander-Govern ANOVA Method, Plus Comments on Comparing Trimmed Means , 1997 .

[16]  W. Loh,et al.  A comparison of tests of equality of variances , 1996 .

[17]  Comment on a Wilcox Test Statistic for Comparing Means When Variances Are Unequal , 1994 .

[18]  W. R. Buckland,et al.  Contributions to Probability and Statistics , 1960 .

[19]  Dennis D. Boos,et al.  Bootstrap Methods for Testing Homogeneity of Variances , 1989 .

[20]  Alexander and Govern's approximation : Providing an alternative to ANOVA under variance heterogeneity , 1997 .

[21]  S. Johansen,et al.  The Welch - James approximation to the distribution of the residual sum of squares in a weighted lin , 1980 .

[22]  Ivan A. Iachine,et al.  Robust tests for the equality of variances for clustered data , 2010 .

[23]  Rand R. Wilcox,et al.  Comparing the means of two independent groups , 2007 .

[24]  Ralph A. Alexander,et al.  A New and Simpler Approximation for ANOVA Under Variance Heterogeneity , 1994 .

[25]  Michael R. Harwell,et al.  Summarizing Monte Carlo Results in Methodological Research: The One- and Two-Factor Fixed Effects ANOVA Cases , 1992 .

[26]  M. O'Neill,et al.  Theory & Methods: A Weighted Least Squares Approach to Levene's Test of Homogeneity of Variance , 2000 .

[27]  Adjusting for Unequal Variances When Comparing Means in One-Way and Two-Way Fixed Effects ANOVA Models , 1989 .

[28]  R. Wilcox.  A new alternative to the ANOVA F and new results on James's second-order method , 1988 .

[29]  A robust I-sample analysis of means type randomization test for variances for unbalanced designs , 2004 .

[30]  G. Box.  Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, I. Effect of Inequality of Variance in the One-Way Classification , 1954 .

[31]  T. C. Oshima,et al.  Type I error rates for James's second-order test and Wilcox's Hm test under heteroscedasticity and non-normality , 1992 .

[32]  W G Hines,et al.  Increased power with modified forms of the Levene (Med) test for heterogeneity of variance. , 2000, Biometrics.

[33]  H. Keselman,et al.  Consequences of Assumption Violations Revisited: A Quantitative Review of Alternatives to the One-Way Analysis of Variance F Test , 1996 .

[34]  Arne C. Bathke,et al.  The ANOVA F test can still be used in some balanced designs with unequal variances and nonnormal data , 2004 .

[35]  D. Mehrotra.  Improving the brown-forsythe solution to the generalized behrens-fisher problem , 1997 .

[36]  H. Keselman,et al.  An evaluation of some non-parametric and parametric tests for location equality , 1977 .

[37]  W. G. Cochran.  Some Methods for Strengthening the Common χ 2 Tests , 1954 .

[38]  Martin S. Levy,et al.  Analysis of Levene’s Test Under Design Imbalance , 1997 .

[39]  Y. Gel,et al.  Combination of Levene-type tests and a finite-intersection method for testing equality of variances against ordered alternatives , 2010 .

[40]  Isabel Parra-Frutos,et al.  The behaviour of the modified Levene’s test when data are not normally distributed , 2009, Comput. Stat..

[41]  H. Keselman,et al.  Parametric Alternatives to the Analysis of Variance , 1982 .

[42]  G. Glass,et al.  Consequences of Failure to Meet Assumptions Underlying the Fixed Effects Analyses of Variance and Covariance , 1972 .

[43]  H. Keselman,et al.  Is the ANOVA F-Test Robust to Variance Heterogeneity When Sample Sizes are Equal?: An Investigation via a Coefficient of Variation , 1977 .

[44]  Edward P. Markowski,et al.  Conditions for the Effectiveness of a Preliminary Test of Variance , 1990 .

[45]  H. Scheffé,et al.  The Analysis of Variance , 1960 .

[46]  Morton B. Brown,et al.  The Small Sample Behavior of Some Statistics Which Test the Equality of Several Means , 1974 .

[47]  J. J. Dijkstra,et al.  Testing the equality of several means when the population variances are unequal , 1981 .

[48]  M. Layard,et al.  Robust Large-Sample Tests for Homogeneity of Variances , 1973 .

[49]  Dennis D. Boos,et al.  Comparing variances and other measures of dispersion , 2004 .

[50]  A. R. Othman,et al.  A comparative study of robust tests for spread: asymmetric trimming strategies. , 2008, The British journal of mathematical and statistical psychology.

[51]  M. E. Johnson,et al.  A Comparative Study of Tests for Homogeneity of Variances, with Applications to the Outer Continental Shelf Bidding Data , 1981 .

[52]  B. L. Welch.  ON THE COMPARISON OF SEVERAL MEAN VALUES: AN ALTERNATIVE APPROACH , 1951 .

[53]  A. John Bailer,et al.  Testing multiple-group variance equality with randomization procedures , 2007 .

[54]  Wei-Yin Loh,et al.  Some modifications of levene's test of variance homogeneity , 1987 .

[55]  D. A. Kenny,et al.  Consequences of violating the independence assumption in analysis of variance. , 1986 .