Model (4.11) for the probability of a satellite for horseshoe crabs with color and width predictors

Model (4.11) for the probability π of a satellite for horseshoe crabs with color and width predictors has fit logit(πˆ ) = −12.715 + 1.330c1 + 1.402c2 + 1.106c3 + 0.468x Consider this fit for crabs of width x = 20 cm.

a. Estimate π for medium-dark crabs (c3 = 1) and for dark crabs (c1 = c2 =

c3 = 0). Then, estimate the ratio of probabilities.

b. Estimate the odds of a satellite for medium-dark crabs and the odds for dark crabs. Show that the odds ratio equals exp(1.106) = 3.02. When each probability is close to zero, the odds ratio is similar to the ratio of probabilities, providing another interpretation for logistic regression parameters.

For widths at which ˆπ is small, ˆπ for medium-dark crabs is about three times that for dark crabs.