A Parameteric, Hierarchical Statistical Framework for Inference with Skewed Distributions

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

This project seeks to develop a class of statistical models for the analysis of data having skewed distributions, especially data arising from hierarchical or multi-level settings. Skewed distributions are ubiquitous in the social sciences. Often, the higher-order characteristics of the distribution, such as the scale (variability) and shape, can provide important insight into substantive issues and provide for significant theoretical development. In addition to having skew, these distributions typically have variability at several levels. For example, businesses may be clustered by economic sectors and completion time data may be clustered by participant. Data in these contexts often are analyzed with linear models such as regression or ANOVA. Although these methods can account for the hierarchical nature of the data and often are well-suited to analyzing differences in means, it is difficult, if not impossible, to perform inference on higher order characteristics. The researcher team will develop a Bayesian approach to analyzing a broad class of models in which statistical inference about location, scale, and shape is both possible and practical. Bayesian statistics is adopted because it is ideally suited to hierarchical models. Bayesian analysis depends on the researcher's informed knowledge of experimental conditions -- the "prior distribution." In some cases, Bayesian analysis is relatively insensitive to this prior; however, in other cases subtle errors in prior specification can lead to erroneous inference. For these reasons, the research team will develop appropriate "noninformative priors." The project will produce software tools so that other researchers can perform Bayesian analysis on these hierarchical models. In the social sciences, researchers have a well-developed set of statistical tools for analyzing the overall effects of manipulations on outcomes. For example, experimental psychologists study how practice (a manipulation) improves performance (an outcome). Current statistical tools are well-suited for assessing the overall (e.g., average) improvement with practice but are ill-suited for assessing whether practice affects the variability of performance or the skew in the pattern of performance (skew would occur if performance is good on many trials and poor on a few). The goal of the project is the development of statistical tools for assessing differences in variability and skew of outcome measures as well as overall effects due to manipulations. The results will lead not only to better understanding of the data but, more importantly, to better theoretical development. For example, learning theories which predict that practice affects the variability of performance can be rigorously tested. The developed statistical tools would be broad and applicable to many social science fields such as psychology, education, economics, and other social sciences.