Some skew models for quantal response analysis

Data involving quantal responses occurs across scientific disciplines such as biology, pharmacology, psychology, sociology, and economics. Traditional approaches to analyzing quantal response data require the relevant distribution function or the quantile function associated with the dose-response relationship in convenient closed form. The most common and conventional methods of analyzing binary response data are probit and logit analyses, which assume that the underlying dose-response relationship is normal and logistic, respectively. However, it is well understood that these methods, which impose the assumption of symmetry on the dose-response relationship, provide inadequate fits in many cases.;This research effort is motivated by numerous factors, the principal being development of models with less restrictive assumptions and greater applicability than the major methodologies in current use, constructing methods for assessing appropriateness of the popular methods, and, obviously, to satisfy some of our curiosities.;The investigation, which began with a study of the skew-normal distribution, eventually led to the consideration of several skew models for quantal response analysis. The skew-normal distribution, an elegant skew alternative to the normal distribution, has been extensively studied in recent literature for its structural properties, inference, and theory. However, its distribution function and quantile function are not available in closed form, which makes it inconvenient for applications such as quantal response analysis. Hence, using the basic idea underlying the skew-normal distribution, the development of alternative skew families containing the normal distribution, which are more convenient for analyzing quantal response data, is considered. One alternative, called the uniform-skewnormal distribution, is proposed, and its properties and usefulness for general model-fitting as well as for quantal response analysis are explored.;In a similar manner, skew extensions of other traditional quantal response models are considered. First, three skew extensions of the logistic distribution are explored in the context of quantal response analysis. The motivation for studying these particular distributions in the quantal response context is that they accommodate both a symmetric dose-response relationship as well as a spectrum of skew dose-response relationships. Perhaps more importantly, one of these logistic extensions, the exponentiated logistic model, contains logistic as a particular case, which provides a simple procedure for assessing the goodness-of-fit of the established logistic model.;The two-parameter Weibull quantal response model has been applied in the literature because of its relationship to the (medium-tail) extreme value distribution, which is used to model the dose-response relationship in the often used complementary log-log quantal response model, and the realization that a response occurs when the dose exceeds a tolerance threshold. The generalized extreme value distribution, a unified expression of the short-tail, medium-tail and long-tail distributions, includes the Weibull distribution as a particular case and the extreme value distribution as a limiting case. In this dissertation, two extensions of the two-parameter Weibull distribution, which provide more flexibility through the inclusion of an additional shape parameter, are considered. These extensions are well-established in the literature for various other applications. The generalized models provide not only flexibility, but, as in the case of the exponentiated logistic model, one of the models, the exponentiated Weibull, contains Weibull as a particular case and offers a means of testing the goodness-of-fit of the two-parameter Weibull model.;In conclusion, while the study of the uniform-skewnormal model in this research provides an informative theoretical contribution to the literature, it is not likely to be a practical model for analyzing quantal response data due to its complicated two-part form which contains a threshold parameter. The exponentiated logistic model seems to be a strong candidate for a practical, easily applied skew quantal response model. It is relatively simple in form and provides a continuous transition between the symmetric logistic model and the skew alternatives. The two Weibull extensions also provide flexible fits, but it is likely that the two-parameter Weibull distribution would be sufficient in practice. However, as in the case of the exponentiated logistic model, the exponentiated Weibull is useful in that it provides a parametric goodness-of-fit test for the two-parameter Weibull model.