Statistical Methods in Medical Research 4ed

494 Modelling categorical data
in a smaller number of distinct covariate patterns; this is particularly likely to be
the case if the covariates are categorical variables with just a few levels. The value
of the deviance is unaltered by grouping into covariate patterns, but the degrees
of freedom are equal to the number of covariate patterns less the number of
parameters fitted.
For individual data that do not reduce to a smaller number of covariate
patterns, tests based on grouping the data may be constructed. For a logistic
regression, grouping could be by the estimated probabilities and a x 2 test
produced by comparing observed and expected frequencies (Lemeshow & Hosmer,
1982; Hosmer & Lemeshow, 1989, §5.2.2). In this test the individuals are
ranked in terms of the size of the estimated probability, P, obtained from the
fitted logistic regression model. The individuals are then divided into g groups;
often g ˆ 10. One way of doing this is to have the groups of equal sizeÐthat is,
the first 10% of subjects are in the first group, etc. Another way is to define the
groups in terms of the estimated probabilities so that the first group contains
those with estimated probabilities less than 0 1, the second 0 1to02, etc. A g 2
table is then formed, in which the columns represent the two categories of the
dichotomous outcome variable, containing the observed and expected numbers
in each cell. The expected numbers for each group are the sum of the estimated
probabilities, P, and the sum of 1 P, for all the individuals in that group. A x 2
goodness-of-fit statistic is then calculated (11.73). Based on simulations, Hosmer
and Lemeshow (1980) showed that this test statistic is distributed approximately
as a x 2 with g 2 degrees of freedom. This test can be modified when some
individuals have the same covariate pattern (Hosmer & Lemeshow, 1989, §5.2.2),
provided that the total number of covariate patterns is not too different from the
total number of individuals.
Diagnostic methods based on residuals similar to those used in classical
regression (§11.9) can be applied. If the data are already grouped, as in Example
14.1, then standardized residuals can be produced and assessed, where each
residual is standardized by its estimated standard error. In logistic regression
the standardized residual is
r n^m
p
‰n^m…1 ^m† Š ,
where there are r events out of n. For individual data the residual may be defined
using the above expression, with r either 0 or 1, but the individual residuals are of
little use since they are not distributed normally and cannot be assessed individually.
For example, if ^m ˆ 0 01, the only possible values of the standardized
residual are 9 9 and 0 1; the occurrence of the larger residual does not
necessarily indicate an outlying point, and if accompanied by 99 of the smaller
residuals the fit would be perfect. It is, therefore, necessary to group the residuals,
defining groups as individuals with similar values of the xi.