Gerald L. Sievers

A weighted dispersion function for estimation in linear models

Robust estimates for the parameters in the general linear model are proposed which are based on weighted rank statistics. The method is based on the minimization of a dispersion function defined by a weighted Ginis mean difference. The asymptotic distribution of the estimate is derived with an asymptotic linearity result. An influence function is determined to measure how the weights can reduce the influence of high-leverage points. The weights can also be used to base the ranking on a restricted set of comparisons. This is illustrated in several examples with stratified samples, treatment vs control groups and ordered alternatives.

Weighted Rank Statistics for Simple Linear Regression

Abstract This article is concerned with statistical inferences for the slope parameter β in the simple linear regression model. Rank procedures are proposed which extend the procedures of Theil and Sen by using weights for the pairwise slopes. Estimation, confidence interval, and hypothesis testing problems are considered.

Coefficients of determination for least absolute deviation analysis

The least-absolute deviation or l1 analysis of a linear model is an important alternative to the classical least squares analysis from the point of view of efficiency for longer-tailed error distributions and robustness to the presence of outliers. In this paper two coefficients of determination are proposed for the least-absolute deviation analysis. It is shown that they have desirable properties as measures of multiple association. Both fixed and random predictor variable cases are considered. In the case of random predictor variables, the sample coefficients of determination are shown to be consistent estimators of appropriate population parameters.

Predictors of response to alprazolam and placebo in patients with panic disorder

Date from a panic disorder treatment study with 506 patients, comparing alprazolam and placebo in a double-blind manner for 8 weeks, were analyzed to identify demographic and clinical characteristics of the patients that might predict response to treatment. The strongest predictors of response to alprazolam were age over 40, and lower baseline levels of anxiety and phobic symptoms. Predictors of response to placebo were weaker and, in addition to lower levels anxiety and panic attacks, included a lack of previous psychiatric treatment.

A Robust Two-Stage Multiple Comparison Procedure with Application to a Random Drug Screen

SUMMARY A high-volume random drug screen is used to detect drugs that are effective in reducing atherogenic low density lipoproteins. A two-stage analysis, consisting of an overall F-test along with a multiple comparison procedure to detect differences between treatments and a placebo, seemed appropriate. Due to observations with heavy right tails and more than occasional outliers in the left tail, this analysis based on least squares estimates had insufficient power. As a solution to this problem we present a similar analysis based on the class of robust R-estimates. In general, it was more powerful than the least squares analysis. Furthermore, by a prudent choice of rank score functions, this robust analysis can take advantage of the underlying skewed error structure. Its validity and power are verified in a Monte Carlo study. Atherosclerosis is a leading cause of death in the United States. The search continues for an effective agent to treat this condition. Rather than searching for drugs that reduce serum total cholesterol, the random drug screen discussed in this paper attempts to find agents effective in reducing atherogenic low density lipoproteins. The screen uses the normocholesterolemic male SEA (Susceptible to Experimental Atherosclerosis) Japanese quail as the animal model; see Chapman, Stafford, and Day (1976) for further details on the use of this animal for such studies. Prior to drug testing the birds were randomly allocated to 10-15 groups of 10 quails each. They were housed individually in ten-cage units and fed a commercial diet for 5 days. The drugs to be tested were dissolved or dispersed in ethanol and mixed with the diet. Control groups received a diet mixed with ethanol alone. After 1 week on the diets each bird was bled from the right jugular vein and serum samples were obtained. The response of interest was the amount of beta cholesterol obtained from these birds. From a statistical point of view each application of the screen results in a one-way design. A two-stage procedure seems to be the appropriate analysis. This would include an overall

Rank scores suitable for analyses of linear models under asymmetric error distributions

Linear models having asymmetric error distributions often occur in practice. In this article, rank scores suitable for skewed error distributions are proposed. Using these scores, a rank analysis of a linear model based on R estimates will tend to be a more powerful analysis than that based on scores suitable for symmetrically distributed errors or least squares. Score functions are presented for the generalized F family of error distributions. These scores are bounded, and hence the corresponding rank analysis is robust. This family contains distributions used to model lifetime data. Practical procedures for score selection are discussed, including a method to estimate the scores. Two examples are discussed in detail, along with the results of a small Monte Carlo study.

Rank estimation of regression coefficients using iterated reweighted least squares

This paper is concerned with the rank estimator for the parameter vector β in a linear model which is obtained by the minimization of a rank dispersion function. The rank estimator has many advantages over the regular least squares estimator, but the inaccessibility of software to carry out its computation has limited its use. An iterated reweighted least squares algorithm is presented for the computation of the rank estimator. The method is simple in concept and can be carried out readily with a wide variety of statistical software. Details of the method are discussed along with some results on its asymptotic distribution and numerical stability. Some examples are presented to show advantages of the rank method.

On the robust rank analysis of linear models with nonsymmetric error distributions

Abstract The robust analysis of linear models based on R-estimates involves an estimate of a scale parameter which is used in the analysis as a standardizing constant. The consistency of previous estimates of this scale parameter required that the underlying errors be symmetrically distributed. This assumption is not always warranted, for instance in survival models. A new estimate is proposed for the scale parameter and it is shown to be consistent for nonsymmetric and symmetric error distributions. With this new scale estimate, a complete robust analysis of a linear model can be accomplished without assuming symmetry. The small sample properties of the analysis are examined in a Monte Carlo study of several different situations.

Smoothed Mann–Whitney–Wilcoxon Procedure for Two-Sample Location Problem

This study is mainly concerned with estimating a shift parameter in the two-sample location problem. The proposed Smoothed Mann–Whitney–Wilcoxon method smooths the empirical distribution functions of each sample by using convolution technique, and it replaces unknown distribution functions F(x) and G(x − Δ0) with the new smoothed distribution functions F s (x) and G s (x − Δ0), respectively. The unknown shift parameter Δ0 is estimated by solving the gradient function S n (Δ) with respect to an arbitrary variable Δ. The asymptotic properties of the new estimator are established under some conditions that are similar to the Generalized Wilcoxon procedure proposed by Anderson and Hettmansperger (1996). Some of these properties are asymptotic normality, asymptotic level confidence interval, and hypothesis testing for Δ0. Asymptotic relative efficiency of the proposed method with respect to the least squares, Generalized Wilcoxon and Hodges and Lehmann (1963) procedures are also calculated under the contaminated normal model.