Introduction to Categorical Data Analysis

120 LOGISTIC REGRESSION
1.2, based on df = 1. The evidence of interaction is not strong (P = 0.28). Although
the sample slopes for the width effect are quite different for the two colors, the sample
had only 24 crabs of dark color. So, effects involving it have relatively large standard
errors.
Fitting the interaction model is equivalent to fitting the logistic regression model
with width as the predictor separately for the crabs of each color. The reduced model
has the advantage of simpler interpretations.
4.5 SUMMARIZING EFFECTS IN LOGISTIC REGRESSION
We have interpreted effects in logistic regression using multiplicative effects on the
odds, which correspond to odds ratios. However, many find it difficult to understand
odds ratios.
4.5.1 Probability-Based Interpretations
For a relatively small change in a quantitative predictor, Section 4.1.1 used a straight
line to approximate the change in the probability. This simpler interpretation applies
also with multiple predictors.
Consider a setting of predictors at which P(Y ˆ = 1) =ˆπ. Then, controlling for the
other predictors, a 1-unit increase in xj corresponds approximately to a ˆβj ˆπ(1 −ˆπ)
change in ˆπ. For example, for the horseshoe crab data with predictors x = width
and an indicator c that is 0 for dark crabs and 1 otherwise, logit( ˆπ) =−12.98 +
1.300c + 0.478x. When ˆπ = 0.50, the approximate effect on ˆπ of a 1 cm increase in
x is (0.478)(0.50)(0.50) = 0.12. This is considerable, since a 1 cm change in width
is less than half its standard deviation (which is 2.1 cm).
This straight-line approximation deteriorates as the change in the predictor values
increases. More precise interpretations use the probability formula directly. One way
to describe the effect of a predictor xj sets the other predictors at their sample means
and finds ˆπ at the smallest and largest xj values. The effect is summarized by reporting
those ˆπ values or their difference. However, such summaries are sensitive to outliers
on xj . To obtain a more robust summary, it is more sensible to use the quartiles of the
xj values.
For the prediction equation logit( ˆπ) =−12.98 + 1.300c + 0.478x, the sample
means are 26.3 cm for x = width and 0.873 for c = color. The lower and upper quartiles
of x are LQ = 24.9 cm and UQ = 27.7 cm. At x = 24.9 and c =¯c, ˆπ = 0.51.
At x = 27.7 and c =¯c, ˆπ = 0.80. The change in ˆπ from 0.51 to 0.80 over the middle
50% of the range of width values reflects a strong width effect. Since c takes only
values 0 and 1, one could instead report this effect separately for each value of c rather
than just at its mean.
To summarize the effect of an indicator explanatory variable, it makes sense to
report the estimated probabilities at its two values rather than at quartiles, which
could be identical. For example, consider the color effect in the prediction equation