Introduction to Categorical Data Analysis

9.4 TRANSITIONAL MODELING, GIVEN THE PAST 289
Table 9.8. Child’s Respiratory Illness by Age and Maternal
Smoking
No Maternal Maternal
Smoking Smoking
Child’s Respiratory Illness Age 10 Age 10
Age 7 Age 8 Age 9 No Yes No Yes
No No No 237 10 118 6
Yes 15 4 8 2
Yes No 16 2 11 1
Yes 7 3 6 4
Yes No No 24 3 7 3
Yes 3 2 3 1
Yes No 6 2 4 2
Yes 5 11 4 7
Source: Thanks to Dr. James Ware for these data.
Let yt denote the response on respiratory illness at age t. For the regressive logistic
model
logit[P(Yt = 1)] =α + βyt−1 + β1s + β2t, t = 8, 9, 10
each subject contributes three observations to the model fitting. The data set consists
of 12 binomials, for the 2 × 3 × 2 combinations of (s, t, yt−1). For instance, for the
combination (0, 8, 0), from Table 9.8 we see that y8 = 0 for 237 + 10 + 15 + 4 = 266
subjects and y8 = 1 for 16 + 2 + 7 + 3 = 28 subjects.
The ML fit of this regressive logistic model is
logit[ ˆ
P(Yt = 1)] =−0.293 + 2.210yt−1 + 0.296s − 0.243t
The SE values are 0.158 for the yt−1 effect, 0.156 for the s effect, and 0.095 for
the t effect. Not surprisingly, yt−1 has a strong effect – a multiplicative impact of
e 2.21 = 9.1 on the odds. Given that and the child’s age, there is slight evidence of a
positive effect of maternal smoking: The likelihood-ratio statistic for H0: β1 = 0is
3.55 (df = 1, P = 0.06). The maternal smoking effect weakens further if we add
yt−2 to the model (Problem 9.13).
9.4.3 Comparisons that Control for Initial Response
The transitional type of model can be especially useful for matched-pairs data. The
marginal models that are the main focus of this chapter would evaluate how the marginal
distributions of Y1 and Y2 depend on explanatory variables. It is often more relevant
to treat Y2 as a univariate response, evaluating effects of explanatory variables while
controlling for the initial response y1. That is the focus of a transitional model.