Introduction to Categorical Data Analysis

PROBLEMS 319
and yi = number of those for which mother’s age was under 18. Let πi be the
probability that a pregnancy in county i is to a mother of age under 18. The
logistic-normal model, logit(πi) = ui + α, has ˆα =−3.24 and ˆσ = 0.33.
a. Find ˆπi for a county estimated to be (i) at the mean, (ii) two standard
deviations below the mean, (iii) two standard deviations above the mean
on the random effects distribution.
b. For estimating {πi}, what advantage does this model have over the fixed
effects model, logit(πi) = βi?
10.4 Table 10.9 shows the free-throw shooting, by game, of Shaq O’Neal of the
Los Angeles Lakers during the 2000 NBA (basketball) playoffs. In game i,
let yi = number made out of Ti attempts.
a. Fit the model, logit(πi) = ui + α, where {ui} are independent N(0,σ),
and given {ui}, {yi} are independent binomial variates for {Ti} trials with
success probabilities {πi}. Report ˆα and ˆσ .
b. Use ˆα to summarize O’Neal’s free-throw shooting in an average game
(for which ui = 0).
c. Use ˆα and ˆσ to estimate how O’Neal’s free-throw shooting varies among
games.
Table 10.9. Shaq O’Neal Basketball Data for Problem 10.4
No. No. No. No. No. No.
Game Made Attempts Game Made Attempts Game Made Attempts
1 4 5 9 4 12 17 8 12
2 5 11 10 1 4 18 1 6
3 5 14 11 13 27 19 18 39
4 5 12 12 5 17 20 3 13
5 2 7 13 6 12 21 10 17
6 7 10 14 9 9 22 1 6
7 6 14 15 7 12 23 3 12
8 9 15 16 3 10
Source: www.nba.com.
10.5 For 10 coins, let πi denote the probability of a head for coin i. You
flip each coin five times. The sample numbers of heads are {2, 4, 1, 3, 3,
5, 4, 2, 3, 1}.
a. Report the sample proportion estimates of πi. Formulate a model for which
these are the ML estimates.
b. Formulate a random effects model for the data. Using software, find the
ML estimates of the parameters. Interpret.
c. Using software, for the model in (b) obtain predicted values {ˆπi}.