Kinley Larntz

OPTIMAL BAYESIAN DESIGN APPLIED TO LOGISTIC REGRESSION EXPERIMENTS

Abstract A traditional way to design a binary response experiment is to design the experiment to be most efficient for a best guess of the parameter values. A design which is optimal for a best guess however may not be efficient for parameter values close to that best guess. We propose designs which formally account for the prior uncertainty in the parameter values. A design for a situation where the best guess has substantial uncertainty attached to itis very different from a design for a situation where approximate values of the parameters are known. We derive a general theory for concave design critria for non-linear models and then apply the theory to logistic regression. Designs found by numerical optimization are examined for a range of prior distributions and a range of criteria. The theoretical results are used to verify that the designs are indeed optimal.

Small-Sample Comparisons of Exact Levels for Chi-Squared Goodness-of-Fit Statistics

Abstract The small-sample properties of three goodness-of-fit statistics for the analysis of categorical data are examined with respect to the adequacy of the asymptotic chi-squared approximation. The approximate tests based on the likelihood ratio and Freeman-Tukey statistics yield exact levels that are typically in excess of the nominal levels for moderate expected values. In contrast, the Pearson statistic attains exact levels that are quite close to the nominal values. The reason for the large number of rejections for the likelihood ratio and Freeman-Tukey statistics is related to their handling of small observed counts.

An Empirical Investigation of Goodness-of-Fit Statistics for Sparse Multinomials

Abstract Traditional discussions of goodness-of-fit tests for multinomial data consider asymptotic chi-squared properties under the assumption that all expected cell frequencies become large. This condition is not always satisfied, however, and another asymptotic theory must be considered. For testing a specified simple hypothesis, Morris (1975) and Hoist (1972) gave conditions for the asymptotic normality of the Pearson and likelihood ratio statistics when both the sample size and number of cells become large (even if the expected cell frequencies remain small). Monte Carlo techniques are used to examine the applicability of the normal approximations for moderate sample sizes with moderate numbers of cells.

Bayesian design for accelerated life testing

Abstract In this paper we study experimental design for accelerated life tests where the lifetimes have either lognormal distributions or Weibull distributions. We assume that the quantities of interest are quantiles of the lifetime distribution at a specified stress and that the log lifetime decreases as a function of stress. There is an upper limit on the stress that can be used in the experiment and a fixed period of time in which to perform the experiment so that there is type I censoring when the experiment is terminated. Examples are given and it is shown that locally optimal designs, which correspond to putting a point mass prior distribution on the parameters, are very similar to Bayesian designs, which allow for prior uncertainty in the parameter values.

Jackknifing in Nonlinear Regression

The standard jackknife and two linear jackknife methods based on a single fit are compared in the context of nonlinear regression fitting. Emphasis is on determination of confidence regions for parameters, where we find that the standard jackknife may be inferior.

Measuring Visual Response to Clothing

Two groups of females differing in age, students and nonstudents, were presented with five photographs of clothed bodies. Each subject was asked to respond to each photograph using an instrument made up of 56 semantic scales. The scales were chosen to represent a range of responses encountered in the visual perception of the clothed body. Multivariate analysis methods were used to test for differences among the costumes, the two observer groups, and the observers within the groups. Consistency in response to the costumes was found between the two groups. The residual vectors were analyzed by principal components and word pairs which were rated similarly were grouped and interpreted. The evaluative component which explained the largest variance included words such as like-dislike and fashionable-unfash ionable. The next four components included word pairs relating to uncertainty, complexity, and potency.

Statistical issues for HIV surrogate endpoints: point/counterpoint

This paper summarizes the proceedings of an NIAID-sponsored workshop on statistical issues for HIV surrogate endpoints. The workshop brought together statisticians and clinicians in an attempt to shed light on some unresolved issues in the use of HIV laboratory markers (such as HIV RNA and CD4+ cell counts) in the design and analysis of clinical studies and in patient management. Utilizing a debate format, the workshop explored a series of specific questions dealing with the relationship between markers and clinical endpoints, and the choice of endpoints and methods of analysis in clinical studies. This paper provides the position statements from the two debaters on each issue. Consensus conclusions, based on the presentations and discussion, are outlined. While not providing final answers, we hope that these discussions have helped clarify a number of issues, and will stimulate further consideration of some of the highlighted problems. These issues will be critical in the proper assessment and use of future therapies for HIV disease.

Use of Perceptions of Female Dress as an Indicator of Role Definition

The purpose of this study was to compare 60 male and 59 female responses to social and business female dress. Suits and dresses were selected as stimuli to represent a social-to- business continuum. A semantic differential instrument was adapted for use from a previous study. Use of word pairs was similar in nature for male and female responses, although there was a difference in relative influence. Separation of responses was mostly evaluative in nature. The social-to-business continuum was a key parameter in male-female differences in response. Commonality in female response led to the conclusion that females accepted the business image represented by the stimuli. The male response was more diverse for business and social images.

An empirical Bayes approach to estimating the relation of mortality to exposure to particulate matter.

As part of its assessment of the health risks associated with exposure to particulate matter (PM), the U.S. Environmental Protection Agency analyzed the risks associated with current levels, and the risk reductions that might be achieved by attainment of alternative PM standards, in two locations in the United States, Philadelphia, and Los Angeles. The concentration-response function describing the relation between a health endpoint and ambient PM concentrations is an important component, and a source of substantial uncertainty, in such risk analyses. In the absence of location-specific estimates, the concentration-response functions necessary for risk assessments in Philadelphia and Los Angeles must be inferred from the available information in other locations. Although the functional form of the concentration-response relations is assumed to be the same everywhere, the value of the PM coefficient in that function may vary from one location to another. Under this model, a distribution describes the probability that the PM coefficient in a randomly selected location will lie in any range of interest. An empirical Bayes estimation technique was used to improve the estimation of location-specific concentration-response functions relating mortality to short-term exposure to particles of aerodynamic diameter less than or equal to 2.5 microm (PM-2.5), for which functions have previously been estimated in several locations. The empirical Bayes-adjusted parameter values and their SEs were used to derive an estimate of the distribution of PM-2.5 coefficients for mortality associated with short-term exposures. From this distribution, distributions of relative risks corresponding to different specified changes in PM-2.5 concentrations could be derived.

Multiplicative Models for Dyad Formation

Abstract The analysis of k × k upper triangular contingency tables in which both margins have the same polytomy is considered. Such a table will arise when k subjects form pairs (dyads) in which the pair (i, j) cannot be distinguished from (j, i), and individuals cannot form dyads with themselves. A quasi-independence model for such data is developed and extended to account for additional multiplicative factors. The resulting estimation procedures and goodness-of-fit tests are illustrated in two examples.