The Impact of Levene’s Test of Equality of Variances on Statistical Theory and Practice

Abstract

In many applications, the underlying scientific question concerns whether the variances of k samples are equal. There are a substantial number of tests for this problem. Many of them rely on the assumption of normality and are not robust to its violation. In 1960 Professor Howard Levene proposed a new approach to this problem by applying the F -test to the absolute deviations of the observations from their group means. Levene's approach is powerful and robust to nonnormality and became a very popular tool for checking the homogeneity of variances.This paper reviews the original method proposed by Levene and subsequent robust modifications. A modification of Levene-type tests to increase their power to detect monotonic trends in variances is discussed. This procedure is useful when one is concerned with an alternative of increasing or decreasing variability, for example, increasing volatility of stocks prices or "open or closed gramophones" in regression residual analysis. A major section of the paper is devoted to discussion of various scientific problems where Levene-type tests have been used, for example, economic anthropology, accuracy of medical measurements, volatility of the price of oil, studies of the consistency of jury awards in legal cases and the effect of hurricanes on ecological systems.

Microfinance and Performance of Micro and Small Enterprises: Does Training have an Impact

“…In order to further strengthen the hypothesis and to be sure that the picture which is depicted by the descriptive analysis is correct, Levene's test for equality of variances has been applied (Gastwirth, Gel, & Miao, 2009 Table 3: t-test for equality of means At this stage table 3 has been used to find the significant differences among the two groups.…”

A Monte Carlo Study of Seven Homogeneity of Variance Tests

“…A question arose as to whether the designers at SPSS chose the "best" test for homogeneity of variance, since there are many others available. A subsequent literature search produced some research on the Levene test (Gastwirth et al, 2009;Carroll and Schneider, 1985;Tomarken and Serlin, 1986). A further search found other tests of homogeneity of variance in studies by Overall and Woodward (1974;; O'Brien (1981) and Levy (1975) that may have been a better choice than the Levene test.…”

“…The Bartlett test does not fare well for data that follow a leptokurtic or skewed distribution (Overall and Woodward, 1974). According to Levene (Gastwirth et al, 2009), the test he proposed was less sensitive to departures from normality. This says that the Levene Test had fewer Type 1 errors than the Bartlett Test for distributions that were aberrant from normality.…”

A Directional Bayesian Significance Test for Equality of Variances

“…This is an issue for example in toxicological and genetic applications (Gastwirth et al 2009). As shown in Box (1954), the robustness of the one-way ANOVA overall F-test to non-normality is dependent on the degree of inequality among the group variances.…”

“…The work in Gastwirth et al (2009) reviews several such approaches where typically bootstrap sampling methods and absolute value based (L 1 norm) measures of departure are used to extend standard tests. Earlier work in this regard can be found for example in Box and Tiao (1973) where Bayesian inference is developed for variance components, tests of homogeneity, and the modeling of variation in linear models with random effects.…”