This course is designed as a broad, applied introduction to the statistical analysis of categorical (or discrete) data, such as counts, proportions, nominal variables, ordinal variables, discrete variables with few values, continuous variables grouped into a small number of categories, etc.
The course begins with methods designed for cross-classified table of counts, (i.e., contingency tables), using simple chi square-based methods.
It progresses to generalized linear models, for which log-linear models provide a natural extension of simple chi square-based methods.
This framework is then extended to comprise logit and logistic regression models for binary responses and generalizations of these models for polytomous (multicategory) outcomes.
Throughout, there is a strong emphasis on associated graphical methods for visualizing categorical data, checking model assumptions, etc. Lab sessions will familiarize the student with software using R for carrying out these analyses.
Course and lecture topics are listed below, in a visual overview.
Bold face items are considered essential. When there are assignments, some supplementary readings are mentioned there.
Discrete distributions are the gateway drug for categorical data analysis. Meet some — binomial, Poisson, Negative Binomial, and others — who will become your friends as you learn to analyze discrete data.
More importantly, learn some nifty graphical methods for fitting these distributions and understanding why a given one might not fit well.
How can we test for independence and measure the strength of association in two way tables? Get acquainted with some standard tests and statistics: Pearson \(\chi^2\), Odds ratio, Cramer’s \(V\), Cohen’s \(\kappa\) and even Bangdiwal’s \(W\).
More importantly, how can we visualize association? We’ll meet fourfold plots, sieve diagrams, spine plots. Much of this is prep for understanding how to formulate, test and visualize models for categorical data.
Some people think nothing is prettier Than algebra of models log-linear. But I've got the hots For my mosaic plots With all those squares in the interior.
— by Michael Greenacre (see his Statistical Songs, https://www.youtube.com/StatisticalSongs)
Correspondence analysis (CA) is one of the first things I think of when I meet a new frequency table and want to get a quick look at the relations among the row and column categories. Very much like PCA for quantitative data, think of CA as a multivariate juicer that takes a high-dimensional data set and squeezes it into a 2D (or 3D) space that best accounts for the associations (Pearson \(\chi^2\)) between the row and column categories. The category scores on the dimensions are in fact the best numerical values that can be defined. They can be used to permute the categories in mosaic displays to make the pattern of associations as clear as possible.
Logistic regression provides an entry to model-based methods for categorical data analysis. These provide estimates and tests for predictor variables of a binary outcome, but more importantly, allow graphs of a fitted outcome together with confidence bands representing uncertainty,
The ideas behind logistic regression can be extended in a variety of ways. The effects of predictors on a binary response can incorporate non-linear terms and interactions. When the outcome is polytomous (more than two categories), and the response categories are ordered, the proportional odds model provides a simple framework. Other methods for polytomous outcomes include nested dichotomies and the general multinomial logistic regression model.
Here we return to loglinear models to consider extensions to the
glm() framework. Models for ordinal factors have greater
power when associations reflect their ordered nature. The
gnm package extends these to generalized
non-linear models. For square tables, we can
fit a variety of specialized models, including
quasi-independence, symmetry and
Here we consider generalized linear models more broadly, with emphasis on those for a count or frequency response variable in the Poisson family. Some extensions allow for overdispersion, including the quasi-poisson and negative binomial model. When the data exhibits a greater frequency of 0 counts, zero-inflated versions of these models come to the rescue.
Logit models for a binary response simplify the specification and interpretation of loglinear models. In the same way, a model for a polytomous response can be simplified by considering a set of log odds defined for the set of adjacent categories. Similarly, when there are two response variables, models for their log odds ratios provide a new way to look at their associations in a structured way.
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