ABSTRACTS

A Clarification of Type III Hypotheses in Multi-Factor Models
L. R. LaMotte
Biostatistics Program
Louisiana State University Health Sciences Center
New Orleans

Type III hypotheses on effects in multi-factor models are routinely tested in statistical computing packages. They are controversial in unbalanced settings. Milliken and Johnson (1984, p. 185) say, "we think that the Type III hypotheses are the worst hypotheses to consider in this situation because there seems to be no reasonable way to interpret them." A precise, mathematically explicit, general description of Type III hypotheses does not seem to be available. Instead there are murky verbal descriptions and worked-out examples. They are defined, by default, by the programs employed in statistical computing packages. Much of their appeal is their connection with definitions of effects in terms of contrasts on marginal means.
 
In this paper, Type III hypotheses are described in general in terms of explicitly-defined linear subspaces. A general formulation is presented for multi-factor models in terms of sets of contrasts on marginal means. The connection between hypotheses on effects in terms of marginal means, on the one hand, and Type III hypotheses, on the other, is described. In particular, it is shown that the Type III hypothesis on an effect includes all estimable marginal means hypotheses on that effect.

Collapsibility of Regression Coefficients and Its Extensions
P. Vellaisamy
Department of Statistics
Michigan State University
(On leave from Dept. of Mathematics, Indian Institute of Technology, Bombay)

Collapsibility with respect to a measure of association implies that the measure of association can be obtained from the marginal model. We first discuss model collapsibility and collapsibility with respect to regression coefficients for linear regression models. For parallel regression models, we give simple and different proofs of some of the known results and obtain also certain new results. For random coefficient regression models, we define (average) A-collapsibility and discuss conditions under which it holds. We consider Poisson regression and logistic regression models also, and derive conditions for collapsibility and A- collapsibility respectively. These results generalize some of the results available in the literature. Some suitable examples will also discussed.

Classification of Functional Data: A Segmentation Approach
Bin Li
Department of Experimental Statistics
Louisiana State University
Baton Rouge

We suggest a classification and feature extraction method on functional data where the predictor variables are curves. The method, called functional segment discriminant analysis (FSDA), combines the classical linear discriminant analysis and support vector machine. FSDA is particularly useful for irregular functional data, characterized by spatial heterogeneity and local patterns like spikes. FSDA not only reduces the computation and storage burden by using a fraction of the spectrum, but also identifies important predictors and extracts features. FSDA is highly flexible, easy to incorporate information from other data sources and/or prior knowledge from the investigators. We apply FSDA to public domain data sets and discuss the understanding developed from the study.

Exact Properties of A New Test and Other Tests for Differences Between Several Binomial Proportions
Jie Peng
Department of Mathematics
University of Louisiana at Lafayette
Lafayette

The problem of testing equality of several binomial proportions is considered. An approximate unconditional test (AU-test) is proposed by extending a result for the two-sample case. Exact binomial distributions are used to evaluate Type I error rates of the usual chi-square test, an exact conditional test, a conditional test based on mid p-values and the AU-test are evaluated numerically. The AU test and the conditional test based on mid p-values control the Type I error rates very satisfactorily even for small samples whereas the exact conditional test is too conservative. The powers of the chi-square test, conditional test based on mid-p values, and the AU test are evaluated and compared. Power comparison shows that all three tests exhibit similar power properties when their sizes are within the nominal level. The AU-test practically behaves like an exact test even for small samples, and can be safely used for applications. The results are illustrated using an example where small proportions are to be compared.

A Nonparametric Approach Using Dirichlet Process for Hierarchical Generalized Linear Mixed Models
Jing Wang
Department of Experimental Statistics
Louisiana State University
Baton Rouge

We propose a nonparametric approach using the Dirichlet processes (DP) as a class of prior distributions for the distribution G of the random effects in the hierarchical generalized linear mixed model (GLMM). The support of the prior distribution (and the posterior distribution) is large, allowing for a wide range of shapes for G. This provides great flexibility in estimating G and therefore produces a more flexible estimator than does the parametric analysis. We present some computation strategies for posterior computations involved in DP modeling. The proposed method is illustrated with real examples as well as simulations.

Heteroscedastic ANOVA - old solutions, new views
Julia Volaufova
Biostatistics Program
Louisiana State University Health Sciences Center
New Orleans

The generalization of the Behrens-Fisher problem to comparing k > 2 means from nonhomogenous populations has attracted the attention of statisticians for many decades. Several approaches offer different approximations to the distribution of the test statistic. The question of statistical properties of these approximations is still alive. Here we present a brief overview of several approaches suggested in the literature. We illustrate by simulation the behavior of p-values under the null hypothesis, particularly the accuracy of the p-value. In addition to Welch test, the Kenward-Roger test, the simple ANOVA F-test, the parametric bootstrap test, and the generalized F-test will be briefly discussed.

Bilinear Varying-Coefficient Models with an Application to Poisson Death Counts
Zhe Liu
Department of Experimental Statistics
Louisiana State University
Baton Rouge

Strong seasonal patterns can be often found in monthly death counts data and their strengths vary from year to year. Varying-coefficient models with cosine and sine regressors are good choice for such data, but in many cases, these models are limited by the shape of (co)sine functions: they may not be able to fit sharp peaks in winter and at troughs in summer very good. In this study, we propose a bilinear model. In the bilinear model, we use a general carrier wave (an unknown twelve vector) modulated over year to model the seasonal pattern. We also propose another bilinear model that combine the (co)sine regressors (for major seasonal pattern) and the carrier wave (for shocks). We estimate the bilinear models in two steps: (i) Estimate the varying coefficient for a known carrier wave; (ii) Estimate the carrier wave for known varying coefficients. The P-spline method is used to estimate the varying-coefficients. Difference penalty terms are added to the likelihood function to enforce smoothness of the estimation. Akaike Information Criterion (AIC) is used to measure model performance and find optimal penalty tuning parameters. For an example, we ap- plied our models to the death counts of respiratory diseases of women aged 50 in the United States during the years 1960 to 1998. The character and strength of the seasonal patterns is quantified. The combined bilinear model shows better fit than the pure VCM model.

Unified Approach for Weibull Analysis
Yin Lin
Department of Mathematics
University of Louisiana at Lafayette
Lafayette

In this talk, we prose generalized variable (GV) approach for making inference for Weibull distributions. We consider one-sample as well as two-sample problems. We show that the GV approach produces exact inferential procedures for one-sample problems such as setting confidence limits on quantiles and survival probabilities. GV procedures are also given for comparing two Weibull means, comparing scale parameters when the shape parameters are unknown and arbitrary, setting confidence bounds on stress-strength reliability involving two independent Weibull random variables, and estimating the ratio of two survival probabilities. The procedures can be easily extended to the type II censored samples, and they can be used to find approximate inferential procedures for type I censored samples. The proposed GV methods are conceptually simple and easy to use. The GV procedures are illustrated using some practical examples.