Introduction to Categorical Data Analysis

288 MODELING CORRELATED, CLUSTERED RESPONSES
Analyses when many data are missing should be made with caution. At a minimum,
one should compare results of the analysis using all available cases for all clusters
to the analysis using only clusters having no missing observations. If results differ
substantially, conclusions should be tentative until the reasons for missingness can
be studied.
9.4 TRANSITIONAL MODELING, GIVEN THE PAST
Let Yt denote the response at time t, t = 1, 2,...,in a longitudinal study. Some studies
focus on the dependence of Yt on the previously observed responses {y1,y2,...,yt−1}
as well as any explanatory variables. Models that include past observations as
predictors are called transitional models.
A Markov chain is a transitional model for which, for all t, the conditional distribution
of Yt,givenY1,...,Yt−1, is assumed identical to the conditional distribution of Yt
given Yt−1 alone. That is, given Yt−1, Yt is conditionally independent of Y1,...,Yt−2.
Knowing the most recent observation, information about previous observations before
that one does not help with predicting the next observation. A Markov chain model
is adequate for modeling Yt if the model with yt−1 as the only past observation used
as a predictor fits as well, for practical purposes, as a model with {y1,y2,...,yt−1}
as predictors.
9.4.1 Transitional Models with Explanatory Variables
Transitional models usually also include explanatory variables other than past observations.
With binary y and k such explanatory variables, one might specify a logistic
regression model for each t,
logit[P(Yt = 1)] =α + βyt−1 + β1x1 +···+βkxk
This Markov chain model is called a regressive logistic model. Given the predictor
values, the model treats repeated observations by a subject as independent. Thus, one
can fit the model with ordinary GLM software, treating each observation separately.
This model generalizes so a predictor xj can take a different value for each t.
For example, in a longitudinal medical study, a subject’s values for predictors such
as blood pressure could change over time. A higher-order Markov model could also
include in the predictor set yt−2 and possibly other previous observations.
9.4.2 Example: Respiratory Illness and Maternal Smoking
Table 9.8 is from the Harvard study of air pollution and health. At ages 7–10 children
were evaluated annually on whether they had a respiratory illness. Explanatory variables
are the age of the child t (t = 7, 8, 9, 10) and maternal smoking at the start of
the study (s = 1 for smoking regularly, s = 0 otherwise).