Introduction to Categorical Data Analysis

4.1 INTERPRETING THE LOGISTIC REGRESSION MODEL 103
for each 1 cm increase in width. This model provides a simple interpretation and
realistic predictions over most of the width range, but it is inadequate for extreme
values. For instance, at the maximum width in this sample of 33.5 cm, its estimated
probability equals −1.766 + 0.092(33.5) = 1.3.
Table 4.2 shows some software output for logistic regression. The estimated
probability of a satellite is the sample analog of formula (4.2),
ˆπ(x) =
exp(−12.351 + 0.497x)
1 + exp(−12.351 + 0.497x)
Since ˆβ >0, the estimated probability ˆπ is larger at larger width values. At the
minimum width in this sample of 21.0 cm, the estimated probability is
exp(−12.351 + 0.497(21.0))/[1 + exp(−12.351 + 0.497(21.0))] =0.129
At the maximum width of 33.5 cm, the estimated probability equals
exp(−12.351 + 0.497(33.5))/[1 + exp(−12.351 + 0.497(33.5))] =0.987
The median effective level is the width at which ˆπ(x) = 0.50. This is x = EL50 =
−ˆα/ ˆβ = 12.351/0.497 = 24.8. Figure 4.1 plots the estimated probabilities as a
function of width.
At the sample mean width of 26.3 cm, ˆπ(x) = 0.674. From Section 4.1.1, the
incremental rate of change in the fitted probability at that point is ˆβ ˆπ(x)[1 −ˆπ(x)]=
0.497(0.674)(0.326) = 0.11. For female crabs near the mean width, the estimated
probability of a satellite increases at the rate of 0.11 per 1 cm increase in width. The
estimated rate of change is greatest at the x value (24.8) at which ˆπ(x) = 0.50; there,
the estimated probability increases at the rate of (0.497)(0.50)(0.50) = 0.12 per 1 cm
increase in width. Unlike the linear probability model, the logistic regression model
permits the rate of change to vary as x varies.
To describe the fit further, for each category of width Table 4.1 reports the predicted
number of female crabs having satellites (i.e., the fitted values). Each of these sums the
ˆπ(x)values for all crabs in a category. For example, the estimated probabilities for the
14 crabs with widths below 23.25 cm sum to 3.6. The average estimated probability
Table 4.2. Computer Output for Logistic Regression Model with Horseshoe Crab Data
Log Likelihood −97.2263
Standard Likelihood Ratio Wald
Parameter Estimate Error 95% Conf. Limits Chi-Sq Pr > ChiSq
Intercept −12.3508 2.6287 −17.8097 −7.4573 22.07