Weiwen Miao

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

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.

A New Test of Symmetry about an Unknown Median

Many robust estimators of location, e.g. trimmed means, implicitly assume that the data come from a symmetric distribution. Consequently, it is important to check this assumption with an appropriate statistical test that does not assume a known value of the median or location parameter. This article replaces the mean and standard deviation in the classic Hotelling-Solomons measure of asymmetry by corresponding robust estimators; the median and mean deviation from the median. The asymptotic distribution theory of the test statistic is developed and the new procedure is compared to tests recently proposed by Cabilio and Masaro (1996) and Mira (1999). Using their approach to approximating the variance of this class of statistics, it is shown that the new test has greater power than the existing tests to detect the asymmetry of skewed contaminated normal data as well as a majority of skewed distributions belonging to the lambda family. The increased power of the new test suggests that the use of robust estimators in goodness of fit type tests deserves further study.

The Effect of Dependence on Confidence Intervals for a Population Proportion

The binomial model is widely used in statistical applications. Usually, the success probability, p, and its associated confidence 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 observations are dependent, even slightly, the coverage probabilities of the usual confidence intervals can deviate noticeably from their nominal level. Several modified confidence intervals that incorporate the dependence structure are proposed and examined. Our results show that the modified Wilson, Agresti-Coull, and Jeffreys confidence intervals perform well and can be recommended for general use.

Confidence intervals for the difference between two means

This paper compares three confidence intervals for the difference between two means when the distributions are non-normal and their variances are unknown. The confidence intervals considered are Welch-Satterthwaite confidence interval, the adaptive interval that incorporates a preliminary test (pre-test) of symmetry for the underlying distributions, and the adaptive interval that incorporates the Shapiro-Wilk test for normality as a pre-test. The adaptive confidence intervals use the Welch-Satterthwaite interval if the pre-test fails to reject symmetry (or normality) for both distributions; otherwise, apply the Welch-Satterthwaite confidence interval to the log-transformed data, then transform the interval back. Our study shows that the adaptive interval with pre-test of symmetry has best coverage among the three intervals considered. Simulation studies show that the adaptive interval with pre-test of symmetry performs as well as the Welch-Satterthwaite interval for symmetric distributions. However, for skewed distributions, the adaptive interval with pre-test of symmetry performs better than the Welch-Satterthwaite interval.

Shrinkage estimation for the difference between exponential guarantee time parameters

This paper studies the shrinkage estimation for the difference between location parameters of exponential distributions when it is suspected but uncertain whether the two parameters are equal. A pre-test estimator and a shrinkage estimator after pre-test are proposed. Both the suboptimal levels of significance for the pre-test estimator in a special case and suboptimal values of shrinkage coefficients for the shrinkage estimator are obtained based on a regret function.

Shrinkage estimation for the difference between a control and treatment mean

Estimation for the difference between a control and treatment mean in the one-way model when it is suspected but uncertain that the two mean values are equal is studied. A preliminary-test (pre-test) estimator and a shrinkage estimator after pre-test are proposed. Both the optimal level of significance for the pre-test estimator and optimal value of shrinkage coefficient for the pre-test shrinkage estimator are obtained on the basis of a regret function. The pre-test estimator performs better than the pre-test shrinkage estimator over a small region around the origin; however, the pre-test shrinkage estimator performs better than the pre-test estimator outside the dominance region, and the dominance region of the pre-test shrinkage estimator is larger than that of the pre-test estimator. Therefore, if the performance outside the dominance region and the size of dominance region are taken into account for comparison, then the pre-test shrinkage estimator is preferable to the pre-test estimator.

The Shonubi Case as an Example of the Legal System’s Failure to Appreciate Statistical Evidence

This article illustrates shortcomings in the legal system’s weighting of statistical evidence by examining one of the data sets submitted into evidence in the U.S. v Shonubi. Using outlier methods to remove several unusually large observations and developing several models for the learning curve of drug runners, one can determine whether violations in reasonable assumptions can have a critical impact on sentencing. Courts should not insist on a “perfect” statistical study but need to assess the potential effect of missing data, omitted variables, etc. on the ultimate inference. Two other areas of law, sampling in false claims cases and possible discrimination in layoffs where strong statistical evidence may not be fully appreciated, are discussed. In the second type of case, courts have used an ad hoc measure of the impact of a layoff on protected group rather than proper measures such as odds ratios.

New Statistical Tests for Detecting Disparate Impact Arising From Two-Stage Selection Processes

Statistical evidence of a significant difference between the performance of a protected group and the majority on a preemployment exam is often critical when a court decides whether the exam has a disparate impact, that is, whether the exam has a disproportionate adverse impact on minority candidates. In many cases, the hiring or promotion process consists of two steps. Since disparate impact can occur at each step, parties submitting evidence may use statistical tests at each stage without accounting for a potential multiple comparisons problem. Because different courts have focused on data concerning either one or the other step or a composite of both, they have reached opposite conclusions when faced with similar data. After illustrating the issues, two two-step tests are recommended to alleviate the problem. The large sample properties of these tests are obtained. A simulation study shows that in most situations, the new tests have higher power than the ones in current use.

The Annual Maximum Wind Speed at Pisa Airport in Italy

Structural engineers use the extreme or fastest values of wind speed with return periods such as 25 years for structures having no human occupants or where there is a negligible risk to human life, 50 years for most permanent structures, and 100 years for structures with an unusually high degree of hazard of life and property in case of failure. The data of 41 annual maximum 10-minute average wind speeds at Pisa Airport in Italy from 1951 to 1991 were analyzed and modeled. Three extreme value models for the data were considered and compared. Subsequently, the required design value with a given return period of exceedance was obtained.