Introduction to Categorical Data Analysis

156 BUILDING AND APPLYING LOGISTIC REGRESSION MODELS
“no intercept” option. When SAS (PROC GENMOD) does this, ˆβ Z 1 and ˆβ Z 3 are both
about −28 with standard errors of about 200,000.
The counts in the 2 × 2 marginal table relating treatment to response, shown in
the bottom panel of Table 5.7, are all positive. The empty cells in Table 5.7 affect
the center estimates, but not the treatment estimates, for this model. The estimated
log odds ratio equals 1.55 for the treatment effect (SE = 0.70). The deviance (G2 )
goodness-of-fit statistic equals 0.50 (df = 4, P = 0.97).
The treatment log odds ratio estimate of 1.55 also results from deleting centers
1 and 3 from the analysis. In fact, when a center has outcomes of only one type, it
provides no information about the odds ratio between treatment and response. Such
tables also make no contribution to the Cochran–Mantel–Haenszel test (Section 4.3.4)
or to a small-sample, exact test of conditional independence between treatment and
response (Section 5.4.2).
An alternative strategy in multi-center analyses combines centers of a similar type.
Then, if each resulting partial table has responses with both outcomes, the inferences
use all data. For estimating odds ratios, however, this usually has little impact. For
Table 5.7, perhaps centers 1 and 3 are similar to center 2, since the success rate is
very low for that center. Combining these three centers and re-fitting the model to
this table and the tables for the other two centers yields an estimated treatment effect
of 1.56 (SE = 0.70). Centers with no successes or with no failures can be useful for
estimating some parameters, such as the difference of proportions, but they do not
help us estimate odds ratios for logistic regression models or give us information
about whether a treatment effect exists in the population.
5.3.4 Effect of Small Samples on X 2 and G 2 Tests
When a model for a binary response has only categorical predictors, the true sampling
distributions of goodness-of-fit statistics are approximately chi-squared, for large
sample size n. The adequacy of the chi-squared approximation depends both on n
and on the number of cells. It tends to improve as the average number of observations
per cell increases.
The quality of the approximation has been studied carefully for the Pearson X 2
test of independence for two-way tables (Section 2.4.3). Most guidelines refer to the
fitted values. When df > 1, a minimum fitted value of about 1 is permissible as long
as no more than about 20% of the cells have fitted values below 5. However, the
chi-squared approximation can be poor for sparse tables containing both very small
and very large fitted values. Unfortunately, a single rule cannot cover all cases. When
in doubt, it is safer to use a small-sample, exact test (Section 2.6.1).
The X 2 statistic tends to be valid with smaller samples and sparser tables than
G 2 . The distribution of G 2 is usually poorly approximated by chi-squared when
n/(number of cells) is less than 5. Depending on the sparseness, P -values based
on referring G 2 to a chi-squared distribution can be too large or too small. When
most fitted values are smaller than 0.50, treating G 2 as chi-squared gives a highly
conservative test; that is, when H0 is true, reported P -values tend to be much larger