Levene type tests for the ratio of two scales

Tests for the equality of variances are of interest in many areas such as quality control, agricultural production systems, experimental education, pharmacology, biology as well as a preliminary to analysis of variance, dose-response modelling or discriminant analysis. The literature is vast. Traditional nonparametric tests are due to Mood, Miller and Ansari-Bradley. A test which usually stands out in terms of power and robustness against non normality is the W50 Brown-Forsythe (1974) modification of the Levene (1960) test. This paper deals with the two-sample scale problem and in particular with Levene type tests. We consider ten Levene type tests: the W50 test, the M50 and L50 tests (Pan, 1999), the R test (O’Brien, 1979), as well as the bootstrap and permutation versions of the W50, L50 and R tests. We consider also the F test, the modified Fligner-Killeen (1976) test, an adaptive test due to Hall and Padmanabhan (1997) and two tests due to Shoemaker (1995 and 1999). The aim is to identify effective methods for detecting scale differences. Our study is different with respect to the other ones since is focused on resampling versions of the Levene type tests, and many tests considered here have not ever been proposed and/or compared. The computationally simplest test found robust is W50. Higher power, while preserving robustness, is achieved by considering resampling Levene type tests like the permutation R test (recommended for normal- and light-tailed distributions) and the bootstrap L50 test (recommended for heavy-tailed and skewed distributions). Among non Levene type tests, the best one is the adaptive test due to Hall and Padmanabhan.