Introduction to Categorical Data Analysis

4.2 INFERENCE FOR LOGISTIC REGRESSION 109
The term ˆα + ˆβx in the exponents of the prediction equation (4.4) is the estimated
linear predictor in the logit transform of π(x). This estimated logit has large-sample
SE given by the estimated square root of
Var( ˆα + ˆβx) = Var( ˆα) + x 2 Var( ˆβ) + 2x Cov( ˆα, ˆβ)
A 95% confidence interval for the true logit is ( ˆα + ˆβx) ± 1.96(SE). Substituting the
endpoints of this interval for α + βx in the two exponents in equation (4.4) gives a
corresponding interval for the probability.
For example, at x = 26.5 for the horseshoe crab data, the estimated logit is
−12.351 + 0.497(26.5) = 0.825. Software reports estimated covariance matrix for
( ˆα, ˆβ) of
Estimated Covariance Matrix
Parameter Intercept width
Intercept 6.9102 −0.2668
width −0.2668 0.0103
A covariance matrix has variances of estimates on the main diagonal and covariances
off that diagonal. Here, �Var( ˆα) = 6.9102, �Var( ˆβ) = 0.0103, �Cov( ˆα, ˆβ) =−0.2668.
Therefore, the estimated variance of this estimated logit equals
�Var( ˆα) + x 2 �Var( ˆβ) + 2x �Cov( ˆα, ˆβ) = 6.9102 + (26.5) 2 (0.0103)
+ 2(26.5)(−0.2668)
or 0.038. The 95% confidence interval for the true logit equals 0.825 ± (1.96) √ 0.038,
or (0.44, 1.21). From equation (4.4), this translates to the confidence interval
{exp(0.44)/[1 + exp(0.44)], exp(1.21)/[1 + exp(1.21)]} = (0.61, 0.77)
for the probability of satellites at width 26.5 cm.
4.2.7 Standard Errors of Model Parameter Estimates ∗
We have used only a single explanatory variable so far, but the rest of the chapter
allows additional predictors. The remarks of this subsection apply regardless of the
number of predictors.
Software fits models and provides the ML parameter estimates. The standard errors
of the estimates are the square roots of the variances from the main diagonal of the
covariance matrix. For example, from the estimated covariance matrix reported above
in Section 4.2.6, the estimated width effect of 0.497 in the logistic regression model
has SE = √ 0.0103 = 0.102.