Introduction to Categorical Data Analysis

282 MODELING CORRELATED, CLUSTERED RESPONSES
Table 9.3 reports the GEE estimates based on the independence working
correlations. For that case, the GEE estimates equal those obtained from ordinary
logistic regression, that is, using ML with 3 × 340 = 1020 independent observations
rather than treating the data as three dependent observations for each of 340
subjects. The empirical standard errors incorporate the sample dependence to adjust
the independence-based standard errors.
Table 9.3. Output from Using GEE to Fit Logistic Model to Table 9.1
Initial Parameter Estimates GEE Parameter Estimates
Empirical Std Error Estimates
Std Std
Parameter Estimate error Parameter Estimate Error
Intercept −0.0280 0.1639 Intercept -0.0280 0.1742
severity −1.3139 0.1464 severity -1.3139 0.1460
drug −0.0596 0.2222 drug -0.0596 0.2285
time 0.4824 0.1148 time 0.4824 0.1199
drug*time 1.0174 0.1888 drug*time 1.0174 0.1877
Working Correlation Matrix
Col1 Col2 Col3
Row1 1.0000 0.0000 0.0000
Row2 0.0000 1.0000 0.0000
Row3 0.0000 0.0000 1.0000
The estimated time effect is ˆβ3 = 0.482 for the standard drug (d = 0) and ˆβ3 +
ˆβ4 = 1.500 for the new one (d = 1). For the new drug, the slope is ˆβ4 = 1.017
(SE = 0.188) higher than for the standard drug. The Wald test of no interaction,
H0: β4 = 0, tests a common time effect for each drug. It has z test statistic equal to
1.017/0.188 = 5.4(P -value < 0.0001). Therefore, there is strong evidence of faster
improvement for the new drug. It would be inadequate to use the simpler model
lacking the drug-by-time interaction term.
The severity of depression estimate is ˆβ1 =−1.314 (SE = 0.146). For each drug–
time combination, the estimated odds of a normal response when the initial diagnosis
was severe equal exp(−1.314) = 0.27 times the estimated odds when the initial
diagnosis was mild. The estimate ˆβ2 =−0.060 (SE = 0.228) for the drug effect
applies only when t = 0 (i.e., after one week), for which the interaction term does
not contribute to the drug effect. It indicates an insignificant difference between the
drugs after 1 week. At time t, the estimated odds of normal response with the new
drug are exp(−0.060 + 1.017t) times the estimated odds for the standard drug, for
each initial diagnosis level. By the final week (t = 2), this estimated odds ratio has
increased to 7.2.
In summary, severity, drug treatment, and time all have substantial effects on the
probability of a normal response. The chance of a normal response is similar for the
two drugs initially and increases with time, but it increases more quickly for those
taking the new drug than the standard drug.