Statistical Methods in Medical Research 4ed

486 Modelling categorical data
In this chapter we consider some of the properties of generalized linear
models and illustrate their application with examples. Readers interested in a
more detailed exposition are referred to the books by McCullagh and Nelder
(1989) and Dobson (1990). Cox and Snell (1989), Hosmer and Lemeshow (1989)
and Kleinbaum (1994) give details of logistic regression.
General theory
Consider a random variable y distributed according to the probability density
f …y; m†, where m is the expected value of y; that is,
E…y† ˆm: …14:1†
Suppose that for each observation y there is a set of explanatory variables
x1, x2, ..., xp, and that m depends on the values of these variables. Suppose
further that, after some transformation of m, g…m†, the relationship is linear;
that is,
Z ˆ g…m† ˆb 0 ‡ b 1x 1 ‡ b 2x 2 ‡ ...‡ bpxp: …14:2†
Then the relationship between y and the explanatory variables is a generalized
linear model. The transformation, g…m†, is termed the link function, since it
provides the link between the linear part of the model, Z, and the random part
represented by m. The linear function, Z, is termed the linear predictor and the
distribution of y, f …y; m†, is the error distribution.
When the error distribution is normal, the classic linear regression model is
E…y† ˆb 0 ‡ b 1x 1 ‡ b 2x 2 ‡ ...‡ bpxp:
Thus from (14.1) and (14.2), Z ˆ m, and the link function is the identity,
g…m† ˆm. Thus, the familiar multiple linear regression is a member of the family
of generalized linear models.
When the basic observations are proportions, then the basic form of random
variation might be expected to be binomial. Such data give rise to many of the
difficulties described in §10.8. Regression curves are unlikely to be linear, because
the scale of the proportion is limited by the values 0 and 1 and changes in any
relevant explanatory variable at the extreme ends of its scale are unlikely to
produce much change in the proportion. A sigmoid regression curve as in
Fig. 14.1(a) is, in fact, likely to be found. An appropriate transformation may
convert this to a linear relationship. In the context of the reasons for transformations
discussed in §10.8, this is a linearizing transformation. The variance of an
observed proportion depends on the expected proportion, as well as on the
denominator of the fraction (see (3.16)). Finally, the binomial distribution of
random error is likely to be skew in opposite directions as the proportion
approaches 0 or 1.