Introduction to Categorical Data Analysis

9.2 MARGINAL MODELING: THE GEE APPROACH 283
9.2.4 Example: Teratology Overdispersion
Table 9.4 shows results of a teratology experiment. Female rats on iron-deficient diets
were assigned to four groups. Group 1 received only placebo injections. The other
groups received injections of an iron supplement according to various schedules. The
rats were made pregnant and then sacrificed after 3 weeks. For each fetus in each rat’s
litter, the response was whether the fetus was dead.
We treat the fetuses in a given litter as a cluster. Let yi denote the number of dead
fetuses for the Ti fetuses in litter i. Let πit denote the probability of death for fetus t
in litter i. Let zig = 1 if litter i is in group g and 0 otherwise.
First, we ignore the clustering and suppose that yi is a bin(Ti,πit) variate. The
model
logit(πit) = α + β2zi2 + β3zi3 + β4zi4
treats all litters in a group g as having the same probability of death, exp(α + βg)/[1 +
exp(α + βg)], where β1 = 0. Here, βi is a log odds ratio comparing group i with the
placebo group (group number 1). Table 9.5 shows ML estimates and standard errors.
There is strong evidence that the probability of death is substantially lower for each
treatment group than the placebo group.
Because of unmeasured covariates that affect the response, it is natural to expect
that the actual probability of death varies from litter to litter within a particular treatment
group. In fact, the data show evidence of overdispersion, with goodness-of-fit
statistics X 2 = 154.7 and G 2 = 173.5 (df = 54). For comparison, Table 9.5 also
shows results with the GEE approach to fitting the logit model, assuming an
exchangeable working correlation structure for observations within a litter. The
estimated within-litter correlation between the binary responses is 0.19.
Table 9.4. Response Counts of (Litter Size, Number Dead) for 58 Litters of
Rats in a Low-Iron Teratology Study
Group 1: untreated (low iron)
(10, 1) (11, 4) (12, 9) (4, 4) (10, 10) (11, 9) (9, 9) (11, 11) (10, 10) (10, 7) (12, 12)
(10, 9) (8, 8) (11, 9) (6, 4) (9, 7) (14, 14) (12, 7) (11, 9) (13, 8) (14, 5) (10, 10)
(12, 10) (13, 8) (10, 10) (14, 3) (13, 13) (4, 3) (8, 8) (13, 5) (12, 12)
Group 2: injections days 7 and 10
(10, 1) (3, 1) (13, 1) (12, 0) (14, 4) (9, 2) (13, 2) (16, 1) (11, 0) (4, 0) (1, 0) (12, 0)
Group 3: injections days 0 and 7
(8, 0) (11, 1) (14, 0) (14, 1) (11, 0)
Group 4: injections weekly
(3, 0) (13, 0) (9, 2) (17, 2) (15, 0) (2, 0) (14, 1) (8, 0) (6, 0) (17, 0)
Source: D. F. Moore and A. Tsiatis, Biometrics, 47: 383–401, 1991.