Discrete Data Analysis with R

Visualization and Modeling Techniques for Categorical and Count Data

Michael Friendly, David Meyer

• Publisher: Chapman and Hall/CRC
• Year: 2015
• Print ISBN: 9781498725835
• eBook ISBN: 9781498725866
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

An Applied Treatment of Modern Graphical Methods for Analyzing Categorical DataDiscrete Data Analysis with R: Visualization and Modeling Techniques for Categorical and Count Data presents an applied treatment of modern methods for the analysis of categorical data, both discrete response data and frequency data. It explains how to use graphical meth

Part III
Model-Building Methods

7

Logistic Regression Models

This chapter introduces the modeling framework for categorical data in the simple situation where we have a categorical response variable, often binary, and one or more explanatory variables. A fitted model provides both statistical inference and prediction, accompanied by measures of uncertainty. Data visualization methods for discrete response data must often rely on smoothing techniques, including both direct, non-parametric smoothing and the implicit smoothing that results from a fitted parametric model. Diagnostic plots help us to detect influential observations that may distort our results.

7.1 Introduction

All models are wrong, but some are useful.

George E. P. Box, (Box and Draper, 1987, p. 424)

Chapter 4, Chapter 5, Chapter 6 have been concerned primarily with simple exploratory methods for studying the relations among categorical variables and with testing hypotheses about their associations through non-parametric tests and with overall goodness-of-fit statistics.

This chapter begins our study of model-based methods for the analysis of discrete data. These models differ from those we have examined earlier primarily in that they consider explicitly an assumed probability distribution for the observations, and make clear distinctions between the systematic component, which is explained by the model, and the random component, which is not. More importantly, the model-based approach allows a compact summary of categorical data in terms of a (hopefully) small number of parameters accompanied by measures of uncertainty (standard errors), and the ability to estimate predicted values over the range of explanatory variables.

This model-fitting approach has several advantages: (a) Inferences for the model parameters include both hypothesis tests and confidence intervals. (b) The former help us to assess which explanatory variables affect the outcome; the size of the estimated parameters and the widths of their confidence intervals help us to assess the strength and importance of these effects. (c) There are a variety of methods for model selection, designed to help determine a favorable trade-off between goodness-of-fit and parsimony. (d) Finally, the predicted values obtained from the model effectively smooth the discrete responses, allow predictions for unobserved values of the explanatory variables, and provide important means to interpret the fitted relationship graphically.

Figure 7.1 provides a visual overview of the steps for fitting and graphing with model-based methods in R. (a) A modeling function such as glm() is applied to an input data frame. The result is a model object containing all the information from the fitting process. (b) As is standard in R, print() and summary() methods give, respectively, basic and detailed printed output. (c) Many modeling functions have plot() methods that produce different types of summary and diagnostic plots. (d) For visualizing the fitted model, most model methods provide a predict() method that can be used to plot the fitted values from the model over the ranges of the predictors. Such plots can be customized by the addition of points (showing the observations), lines, confidence bands, and so forth.

Figure 7.1: Overview of fitting and graphing for model-based methods in R.

In this chapter we consider models for a binary response, such as “success” or “failure,” or the number of “successes” in a fixed number of “trials,” where we might reasonably assume a binomial distribution for the random component. As we will see in Chapter 8, these methods extend readily to a polytomous response with more than two outcome categories, such as improvement in therapy, with categories “none,” “some,” and “marked.”

These models can be seen as simple extensions of familiar ANOVA and regression models for quantitative data. They are also important special cases of a more gen...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Table of Contents
• Preface
• I Getting Started
• II Exploratory and Hypothesis-Testing Methods
• III Model-Building Methods
• References
• Author Index
• Example Index
• Subject Index