  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.
