Introduction to Categorical Data Analysis

5.3 EFFECTS OF SPARSE DATA 155
Table 5.7. Clinical Trial Relating Treatment (X) to Response
(Y ) for Five Centers (Z), with XY and YZ Marginal Tables
Response (Y ) YZ Marginal
Center (Z) Treatment (X) Success Failure Success Failure
1 Active drug 0 5
Placebo 0 9
2 Active drug 1 12
Placebo 0 10
3 Active drug 0 7
Placebo 0 5
4 Active drug 6 3
Placebo 2 6
5 Active drug 5 9
Placebo 2 12
XY Active drug 12 36
Marginal Placebo 4 42
Source: Diane Connell, Sandoz Pharmaceuticals Corp.
0 14
1 22
0 12
8 9
7 21
(1 = success, 0 = failure). For these data, let Y = Response, X = Treatment (Active
drug or Placebo), and Z = Center. Centers 1 and 3 had no successes. Thus, the 5 × 2
marginal table relating center to response, collapsed over treatment, contains zero
counts. This marginal table is shown in the last two columns of Table 5.7.
For these data, consider the model
logit[P(Y = 1) = α + βx + β Z k
Because centers 1 and 3 had no successes, the ML estimates of the terms βZ 1 and βZ 3
pertaining to their effects equal −∞. The fitted logits for those centers equal −∞,
for which the fitted probability of success is 0.
In practice, software notices that the likelihood function continually increases as
βZ 1 and βZ 3 decrease toward −∞, but the fitting algorithm may “converge” at large
negative values. For example, SAS (PROC GENMOD) reports ˆβ Z 1 and ˆβ Z 3 to both
be about −26 with standard errors of about 200,000. Since the software uses default
coding that sets βZ 5 = 0, these estimates refer to contrasts of each center with the last
one. If we remove α from the model, then one of {βZ k } is no longer redundant. Each
center parameter then refers to that center alone rather than a contrast with another
center. Most software permits fitting a model parameterized in this way by using a