Introduction to Categorical Data Analysis

PROBLEMS 171
b. Try to fit a main-effects logistic regression model containing all three predictors.
Explain why the ML estimate for the effect of lymphocytic infiltration
is infinite.
c. Using conditional logistic regression, conduct an exact test of the hypothesis
of no effect of lymphocytic infiltration, controlling for the other variables.
Interpret.
d. Using conditional logistic regression, find a 95% confidence interval for
the effect in (c). Interpret.
5.26 Table 5.15 describes results from a study in which subjects received a drug and
the outcome measures whether the subject became incontinent (y = 1, yes;
y = 0, no). The three explanatory variables are lower urinary tract variables
that represent drug-induced physiological changes.
a. Report the prediction equations when each predictor is used separately in
logistic regressions.
b. Try to fit a main-effects logistic regression model containing all three
predictors. What does your software report for the effects and their standard
errors? (The ML estimates are actually −∞ for x1 and x2 and ∞
for x3.)
c. Use conditional logistic regression to find an exact P -value for testing H0:
β3 = 0. [The exact distribution is degenerate, and neither ordinary ML or
exact conditional ML works with these data. For alternative approaches, see
articles by D. M. Potter (Statist. Med., 24: 693–708, 2005) and G. Heinze
and M. Schemper (Statist. Med., 22: 1409–1419, 2002).]
Table 5.15. Data from Incontinence Study of
Problem 5.26
y x1 x2 x3 y x1 x2 x3
0 −1.9 −5.3 −43 0 −1.5 3.9 −15
0 −0.1 −5.2 −32 0 0.5 27.5 8
0 0.8 −3.0 −12 0 0.8 −1.6 −2
0 0.9 3.4 1 0 2.3 23.4 14
1 −5.6 −13.1 −1 1 −5.3 −19.8 −33
1 −2.4 1.8 −9 1 −2.3 −7.4 4
1 −2.0 −5.7 −7 1 −1.7 −3.9 13
1 −0.6 −2.4 −7 1 −0.5 −14.5 −12
1 −0.1 −10.2 −5 1 −0.1 −9.9 −11
1 0.4 −17.2 −9 1 0.7 −10.7 −10
1 1.1 −4.5 −15
Source: D. M. Potter, Statist. Med., 24: 693–708, 2005.