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.

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Paired data arises in a wide variety of applications where often the underlying distribution of the paired differences is unknown. When the differences are normally distributed, the t-test is optimum. On the other hand, if the differences are not normal, the t-test can have substantially less power than the appropriate optimum test, which depends on the unknown distribution. In textbooks, when the normality of the differences is questionable, typically the non-parametric Wilcoxon signed rank test is suggested. An adaptive procedure that uses the Shapiro-Wilk test of normality to decide whether to use the t-test or the Wilcoxon signed rank test has been employed in several studies. Faced with data from heavy tails, the U.S. Environmental Protection Agency (EPA) introduced another approach: it applies both the sign and t-tests to the paired differences, the alternative hypothesis is accepted if either test is significant. This paper investigates the statistical properties of a currently used adaptive test, the EPA's method and suggests an alternative technique. The new procedure is easy to use and generally has higher empirical power, especially when the differences are heavy-tailed, than currently used methods.

The binomial model is widely used in statistical applications. Usually, the success probability, p, and its associated con dence interval are estimated from a random sample. Thus, the observations are independent and identically distributed. Motivated by a legal case where some grand jurors could serve a second year, this article shows that when the observationsare dependent,even slightly, the coverage probabilities of the usual con dence intervals can deviate noticeably from their nominal level. Several modi ed con dence intervals that incorporate the dependence structure are proposed and examined. Our results show that the modi ed Wilson, Agresti-Coull, and Jeffreys con dence intervals perform well and can be recommended for general use.