Introduction to Categorical Data Analysis

286 MODELING CORRELATED, CLUSTERED RESPONSES
The response is the patient’s reported time (in minutes) to fall asleep after going to
bed. Patients responded before and following a 2 week treatment period. The two
treatments, active drug and placebo, form a binary explanatory variable. The subjects
were randomly allocated to the treatment groups. Here, each subject forms a cluster,
with the observations in a cluster being the ordinal response at the two occasions of
observation.
Table 9.7 displays sample marginal distributions for the four treatment – occasion
combinations. From the initial to follow-up occasion, time to falling asleep seems
to shift downwards for both treatments. The degree of shift seems greater for the
active drug, indicating possible interaction. Let t denote the occasion (0 = initial,
1 = follow-up) and let x denote the treatment (0 = placebo, 1 = active drug). The
cumulative logit model
logit[P(Yt ≤ j)]=αj + β1t + β2x + β3(t × x) (9.1)
permits interaction between occasion and treatment. Like the cumulative logit models
of Section 6.2, it makes the proportional odds assumption of the same effects for each
response cutpoint.
Table 9.7. Sample Marginal Distributions of Table 9.6
Response
Treatment Occasion 60
Active Initial 0.101 0.168 0.336 0.395
Follow-up 0.336 0.412 0.160 0.092
Placebo Initial 0.117 0.167 0.292 0.425
Follow-up 0.258 0.242 0.292 0.208
For independence working correlation, the GEE estimates (with SE values in
parentheses) are:
ˆβ1 = 1.038(0.168), ˆβ2 = 0.034(0.238), ˆβ3 = 0.708(0.244)
The SE values are not the naive ones assuming independence, but the ones adjusted
for the actual empirical dependence. At the initial observation, the estimated odds
that time to falling asleep for the active treatment is below any fixed level equal
exp(0.034) = 1.03 times the estimated odds for the placebo treatment. In other words,
initially the two groups had similar distributions, as expected by the randomization
of subjects to treatments. At the follow-up observation, the effect is exp(0.034 +
0.708) = 2.1. Those taking the active drug tended to fall asleep more quickly.
The ˆβ3 and SE values indicate considerable evidence of interaction. The test
statistic z = 0.708/0.244 = 2.9 provides strong evidence that the distribution of time
to fall asleep decreased more for the treatment group than for the placebo group
(two-sided P -value = 0.004).