Statistical Methods in Medical Research 4ed

498 Modelling categorical data
When there are only two response categories, (14.9) and (14.10) are entirely
equivalent, and both the cumulative logits model and the adjacent categories
model reduce to ordinary logistic regression. In the more general case,
with more than two categories, computer programs are available for estimation
of the coefficients. For example, SAS CATMOD uses weighted least
squares.
The mean response model
Suppose that scores x are assigned to the categories, as in §15.2, and denote by
M…x† the mean score for individuals with explanatory variables x. The model
specifies the same linear relation as in multiple regression
M…x† ˆa ‡ b 0 x: …14:11†
The approach is thus a generalization of that underlying the comparison of
mean scores by (15.8) in the simple two-group case. In the general case the
regression coefficients cannot be estimated accurately by standard multiple
regression methods, because there may be large departures from normality
and disparities in variance. Nor can exact variances such as (15.5) be easily
exploited.
Choice of model
The choice between the models described briefly above, or any others, is largely
empirical: which is the most convenient to use, and which best describes the
data? There is no universally best choice. The two logistic models attempt
to describe the relative frequencies of observations in the various categories,
and their adequacy for any particular data set may be checked by comparing
observed and expected frequencies. The mean response model is less
searching, since it aims to describe only the mean values. It may, therefore, be a
little more flexible in fitting data, and particularly appropriate where there is a
natural underlying continuous response variate or scoring system, but
less appropriate when the fine structure of the categorical response is under
study.
Further descriptions of these models are given in Agresti (1990, Chapter 9),
and an application to repeated measures data is described in Agresti (1989). An
example relating alcohol consumption in eight ordered categories to biochemical
and haematological variables was discussed by Ashby et al. (1986). These
authors also discussed a test of goodness of fit, which is essentially an extension
of the Hosmer±Lemeshow test, and a method of allocating an individual to one of
the groups.