Introduction to Categorical Data Analysis

4.2 INFERENCE FOR LOGISTIC REGRESSION 107
Exponentiating the endpoints yields an interval for e β , the multiplicative effect on the
odds of a 1-unit increase in x.
When n is small or fitted probabilities are mainly near 0 or 1, it is preferable
to construct a confidence interval based on the likelihood-ratio test. This interval
contains all the β0 values for which the likelihood-ratio test of H0: β = β0 has
P -value >α. Some software can report this (such as PROC GENMOD in SAS with
its LRCI option).
For the logistic regression analysis of the horseshoe crab data, the estimated effect
of width on the probability of a satellite is ˆβ = 0.497, with SE = 0.102. A 95% Wald
confidence interval for β is 0.497 ± 1.96(0.102), or(0.298, 0.697). The likelihoodratio-based
confidence interval is (0.308, 0.709). The likelihood-ratio interval for the
effect on the odds per cm increase in width equals (e 0.308 , e 0.709 ) = (1.36, 2.03).We
infer that a 1 cm increase in width has at least a 36 percent increase and at most a
doubling in the odds that a female crab has a satellite.
From Section 4.1.1, a simpler interpretation uses a straight-line approximation to
the logistic regression curve. The term βπ(x)[1 − π(x)] approximates the change
in the probability per 1-unit increase in x. For instance, at π(x) = 0.50, the estimated
rate of change is 0.25 ˆβ = 0.124. A 95% confidence interval for 0.25β equals
0.25 times the endpoints of the interval for β. For the likelihood-ratio interval, this
is [0.25(0.308), 0.25(0.709)] =(0.077, 0.177). So, if the logistic regression model
holds, then for values of x near the width value at which π(x) = 0.50, we infer that
the rate of increase in the probability of a satellite per centimeter increase in width
falls between about 0.08 and 0.18.
4.2.3 Significance Testing
For the logistic regression model, H0: β = 0 states that the probability of success is
independent of X. Wald test statistics (Section 1.4.1) are simple. For large samples,
z = ˆβ/SE
has a standard normal distribution when β = 0. Refer z to the standard normal table
to get a one-sided or two-sided P -value. Equivalently, for the two-sided Ha: β �= 0,
z 2 = ( ˆβ/SE) 2 has a large-sample chi-squared null distribution with df = 1.
Although the Wald test is adequate for large samples, the likelihood-ratio test is
more powerful and more reliable for sample sizes often used in practice. The test
statistic compares the maximum L0 of the log-likelihood function when β = 0to
the maximum L1 of the log-likelihood function for unrestricted β. The test statistic,
−2(L0 − L1), also has a large-sample chi-squared null distribution with df = 1.
For the horseshoe crab data, the Wald statistic z = ˆβ/SE = 0.497/0.102 =
4.9. This shows strong evidence of a positive effect of width on the presence of
satellites (P