Statistical Methods in Medical Research 4ed

where
and
b 0 0
Y ˆ b 0 0 ‡ b0 1 x 1 ‡ b0 2 x 2 ‡ ...‡ b0 pxp,
b 0 j ˆ bj
ˆ 1
2 ‰b0 1 …xA1 ‡ xB1†‡...‡ b 0 p …xAp ‡ xBp†Š ‡ ln…nA=nB†: …14:8†
In Example 13.2, using the discriminant function coefficients b1 and b2 given
there, we find
b 0 0
b 0 1
b 0 2
ˆ 4 135,
ˆ 0 6541,
ˆ 0 3978,
14.2 Logistic regression 493
which lead to values of Y not differing greatly from those obtained from (14.7),
except for extreme values of x1 and x2.
An example of the use of the linear discriminant function to predict the
probability of coronary heart disease is given by Truett et al. (1967). The point
should be emphasized that, in situations in which the distributions of xs are far
from multivariate normal, this method may be unreliable, and the maximum
likelihood solution will be preferable.
To test the adequacy of the logistic regression model (14.6), after fitting by
maximum likelihood, an approximate x 2 test statistic is given by the deviance.
This was the approach in Example 14.1, where the deviance after fitting the four
main effects was 13 59 with 11 DF (since four main effects and a constant term
had been estimated from 16 groups). The fit is clearly adequate, suggesting that
there is no need to postulate interactions, although, as was done in the example,
a further refinement to testing the goodness of fit is to try interactions, since a
single effect with 1 DF could be undetected when tested with other effects
contributing 10 DF.
In general terms, the adequacy of the model can be assessed by including
terms such as x 2 i , to test for linearity in xi, and xixj, to test for an interaction
between xi and xj.
The approximation to the distribution of the deviance by x 2 is unreliable for
sparse dataÐthat is, if a high proportion of the observed counts are small. The
extreme case of sparse data is where all values of n are 1. Differences between
deviances can still be used to test for the inclusion of extra terms in the model.
For sparse data, tests based on the differences in deviances are superior to the
corresponding Wald test (Hauck & Donner, 1977). Goodness-of-fit tests should
be carried out after forming groups of individuals with the same covariate
patterns. Even for a case of individual data, it may be that the final model results