Introduction to Categorical Data Analysis

10.2 EXAMPLES OF RANDOM EFFECTS MODELS FOR BINARY DATA 303
the logits of the probabilities vary according to a normal distribution, the fitting process
“borrows from the whole,” using data from all the areas to estimate the probability in
any given one. The estimate for a given area is then a weighted average of the sample
proportion for that area alone and the overall proportion for all the areas.
Software provides ML estimates ˆα and ˆσ and predicted values {ûi} for the random
effects. The predicted value ûi depends on all the data, not only the data for area i.
The estimate of the probability πi in area i is then
ˆπi = exp(ûi +ˆα)/[1 + exp(ûi +ˆα)]
A benefit of using data from all the areas instead of only area i to estimate πi is that
the estimator ˆπi tends to be closer than the sample proportion pi to πi. The {ˆπi}
result from shrinking the sample proportions toward the overall sample proportion.
The amount of shrinkage increases as ˆσ decreases. If ˆσ = 0, then {ˆπi} are identical.
In fact, they then equal the overall sample proportion after pooling all n samples.
When truly all πi are equal, ˆπi is a much better estimator of that common value than
the sample proportion from sample i alone.
Foragiven ˆσ >0, the {ˆπi} give more weight to the sample proportions as {Ti}
grows. As each sample has more data, we put more trust in the separate sample
proportions.
The simple random effects model (10.4), which is natural for small-area estimation,
can be useful for any application that estimates a large number of binomial parameters
when the sample sizes are small. The following example illustrates this.
10.2.2 Example: Estimating Basketball Free Throw Success
In basketball, the person who plays center is usually the tallest player on the team.
Often, centers shoot well from near the basket but not so well from greater distances.
Table 10.2 shows results of free throws (a standardized shot taken from a distance
of 15 feet from the basket) for the 15 top-scoring centers in the National Basketball
Association after one week of the 2005–2006 season.
Let πi denote the probability that player i makes a free throw (i = 1,...,15). For
Ti observations of player i, we treat the number of successes yi as binomial with
index Ti and parameter πi. Table 10.2 shows {Ti} and {pi = yi/Ti}.
For the ML fit of model (10.4), ˆα = 0.908 and ˆσ = 0.422. For a player
with random effect ui = 0, the estimated probability of making a free throw is
exp(0.908)/[1 + exp(0.908)] =0.71. We predict that 95% of the logits fall within
0.908 ± 1.96(0.422), which is (0.08, 1.73). This interval corresponds to probabilities
in the range (0.52, 0.85).
The predicted random effect values (obtained using PROC NLMIXED in SAS)
yield probability estimates {ˆπi}, also shown in Table 10.2. Since {Ti} are small and
since ˆσ is relatively small, these estimates shrink the sample proportions substantially
toward the overall sample proportion of free throws made, which was 101/143 =
0.706. The {ˆπi} vary only between 0.61 and 0.76, whereas the sample proportions