Introduction to Categorical Data Analysis

304 RANDOM EFFECTS: GENERALIZED LINEAR MIXED MODELS
Table 10.2. Estimates of Probability of Centers Making
a Free Throw, Based on Data from First Week of 2005–2006
NBA Season
Player ni pi ˆπi Player ni pi ˆπi
Yao 13 0.769 0.730 Curry 11 0.545 0.663
Frye 10 0.900 0.761 Miller 10 0.900 0.761
Camby 15 0.667 0.696 Haywood 8 0.500 0.663
Okur 14 0.643 0.689 Olowokandi 9 0.889 0.754
Blount 6 0.667 0.704 Mourning 9 0.778 0.728
Mihm 10 0.900 0.761 Wallace 8 0.625 0.692
Ilgauskas 10 0.600 0.682 Ostertag 6 0.167 0.608
Brown 4 1.000 0.748
Note: pi = sample, ˆπi = estimate using random effects model.
Source: nba.com.
vary between 0.17 and 1.0. Relatively extreme sample proportions based on few
observations, such as the sample proportion of 0.17 for Ostertag, shrink more. If you
are a basketball fan, which estimate would you think is more sensible for Ostertag’s
free throw shooting prowess, 0.17 or 0.61?
Are the data consistent with the simpler model, logit(πi) = α, in which πi is
identical for each player? To answer this, we could test H0: σ = 0 for model (10.4).
The usual tests do not apply to this hypothesis, however, because ˆσ cannot be negative
and so is not approximately normally distributed about σ under H0. We will learn
how to conduct the analysis in Section 10.5.2.
10.2.3 Example: Teratology Overdispersion Revisited
Section 9.2.4 showed results of a teratology experiment in which female rats on irondeficient
diets were assigned to four groups. Group 1 received only placebo injections.
The other groups received injections of an iron supplement at 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. Because of unmeasured covariates
that vary among rats in a given treatment, it is natural to permit the probability of
death to vary from litter to litter within each treatment group.
Let yi denote the number dead out of the Ti fetuses in litter i. Let πit denote the
probability of death for fetus t in litter i. Section 9.2.4 used the model
logit(πit) = α + β2zi2 + β3zi3 + β4zi4
where zig = 1 if litter i is in group g and 0 otherwise. The estimates and standard
errors treated the {yi} as binomial. This approach regards the outcomes for fetuses in a
litter as independent and identical, with the same probability of death for each fetus in
each litter within a given treatment group. This is probably unrealistic. Section 9.2.4
used the GEE approach to allow observations within a litter to be correlated.