Rao\u27s Quadratic Entropy and Some New Applications

Many problems in statistical inference are formulated as testing the diversity of populations. The entropy functions measure the similarity of a distribution function to the uniform distribution and hence can be used as a measure of diversity. Rao (1982a) proposed the concept of quadratic entropy. Its concavity property makes the decomposition similar to ANOVA for categorical data feasible. In this thesis, after reviewing the properties and providing a modification to quadratic entropy, various applications of quadratic entropy are explored. First, analysis of quadratic entropy with the suggested modification to analyze the contingency table data is explored. Then its application to ecological biodiversity is established by constructing practically equivalent confidence intervals. The methods are applied on a real dinosaur diversity data set and simulation experiments are performed to study the validity of the intervals. Quadratic entropy is also used for clustering multinomial data. Another application of quadratic entropy that is provided here is to test the association of two categorical variables with multiple responses. Finally, the gene expression data inspires another application of quadratic entropy in analyzing large scale data, where a hill-climbing type iterative algorithm is developed based on a new minimum quadratic entropy criterion. The algorithm is illustrated on both simulated and real data

Models for Paired Comparison Data: A Review with Emphasis on Dependent Data

Thurstonian and Bradley-Terry models are the most commonly applied models in the analysis of paired comparison data. Since their introduction, numerous developments have been proposed in different areas. This paper provides an updated overview of these extensions, including how to account for object- and subject-specific covariates and how to deal with ordinal paired comparison data. Special emphasis is given to models for dependent comparisons. Although these models are more realistic, their use is complicated by numerical difficulties. We therefore concentrate on implementation issues. In particular, a pairwise likelihood approach is explored for models for dependent paired comparison data, and a simulation study is carried out to compare the performance of maximum pairwise likelihood with other limited information estimation methods. The methodology is illustrated throughout using a real data set about university paired comparisons performed by students.Comment: Published in at http://dx.doi.org/10.1214/12-STS396 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

An Overview of Methods in the Analysis of Dependent ordered catagorical Data: Assumptions and Implications

Subjective assessments of pain, quality of life, ability etc. measured by rating scales and questionnaires are common in clinical research. The resulting responses are categorical with an ordered structure and the statistical methods must take account of this type of data structure. In this paper we give an overview of methods for analysis of dependent ordered categorical data and a comparison of standard models and measures with nonparametric augmented rank measures proposed by Svensson. We focus on assumptions and issues behind model specifications and data as well as implications of the methods. First we summarise some fundamental models for categorical data and two main approaches for repeated ordinal data; marginal and cluster-specific models. We then describe models and measures for application in agreement studies and finally give a summary of the approach of Svensson. The paper concludes with a summary of important aspects.Dependent ordinal data; GEE; GLMM; Logit; modelling

Power Analysis of Longitudinal Data with Time-Dependent Covariates Using Generalized Method of Moments

Longitudinal data occur in different fields such as biomedical and health studies, education, engineering, and social studies. Planning advantageous research projects with both high power and minimum sample size is an important step in any study. The extensive use of longitudinal data in different fields and the importance of their power estimation, yet the limited resources about their respective power estimation tools, made it worthwhile to study their power estimation techniques. The presence of time-dependent covariates triggers the need to use more efficient models such as generalized method of moments than the existing models which are based on generalized estimating equations. Not taking into consideration the correlation among observations and the covariates that change over time while calculating power and minimum sample size will cause expensive research being conducted without using data that are capable of answering the research questions (Williams, 1995). Two different power estimation and minimum sample size calculation techniques for longitudinal data in the presence of time-dependent covariate using generalized method of moments approaches are constructed in this study and their performances are evaluated

An introduction to mixed models for experimental psychology

This chapter describes a class of statistical model that is able to account for most of the cases of nonindependence that are typically encountered in psychological experiments, linear mixed-effects models, or mixed models for short. It introduces the concepts underlying mixed models and how they allow accounting for different types of nonindependence that can occur in psychological data. The chapter discusses how to set up a mixed model and how to perform statistical inference with a mixed model. The most important concept for understanding how to estimate and how to interpret mixed models is the distinction between fixed and random effects. One important characteristic of mixed models is that they allow random effects for multiple, possibly independent, random effects grouping factors. Mixed models are a modern class of statistical models that extend regular regression models by including random-effects parameters to account for dependencies among related data points