Introduction to Categorical Data Analysis

102 LOGISTIC REGRESSION
of observations at each point. It appears that y = 1 occurs relatively more often at
higher x values. Since y takes only values 0 and 1, however, it is difficult to determine
whether a logistic regression model is reasonable by plotting y against x.
Better information results from grouping the width values into categories and
calculating a sample proportion of crabs having satellites for each category. This
reveals whether the true proportions follow approximately the trend required by this
model. Consider the grouping shown in Table 4.1. In each of the eight width categories,
we computed the sample proportion of crabs having satellites and the mean width
for the crabs in that category. Figure 4.2 contains eight dots representing the sample
proportions of female crabs having satellites plotted against the mean widths for the
eight categories.
Section 3.3.2 that introduced the horseshoe crab data mentioned that software can
smooth the data without grouping observations. Figure 4.2 also shows a curve based
on smoothing the data using generalized additive models, which allow the effect of x
to be much more complex than linear. The eight plotted sample proportions and this
smoothing curve both show a roughly increasing trend, so we proceed with fitting
models that imply such trends.
4.1.3 Horseshoe Crabs: Interpreting the Logistic Regression Fit
For the ungrouped data in Table 3.2, let π(x) denote the probability that a female
horseshoe crab of width x has a satellite. The simplest model to interpret is the
linear probability model, π(x) = α + βx. During the ML fitting process, some predicted
values for this GLM fall outside the legitimate 0–1 range for a binomial
parameter, so ML fitting fails. Ordinary least squares fitting (such as GLM software
reports when you assume a normal response and use the identity link function) yields
ˆπ(x) =−1.766 + 0.092x. The estimated probability of a satellite increases by 0.092
Table 4.1. Relation Between Width of Female Crab and Existence of Satellites, and
Predicted Values for Logistic Regression Model
Number Predicted
Number of Having Sample Estimated Number of Crabs
Width Cases Satellites Proportion Probability with Satellites
29.25 14 14 1.00 0.93 13.1
Note: The estimated probability is the predicted number (in the final column) divided by the number of
cases.