Statistical Methods in Medical Research 4ed

492 Modelling categorical data
DF. Adding the main effect of A to the model reduces the deviance to 13 59 with 11 DF, so
that the deviance test for the effect of A, after allowing for B, C and D, is 45 47 13 59 ˆ
31 88 as an approximate x2 p
…1† . This test is numerically similar to Wald's test, since
31 88 ˆ 5 65,
but in general such close agreement would not be expected. Although the
deviance tests of main effects are not necessary here, in general they are needed. For
example, if a factor with more than two levels were fitted, using dummy variables (§11.7),
a deviance test with the appropriate degrees of freedom would be required.
The deviance associated with the model including all the main effects is 13 59 with 11
DF, and this represents the 11 interactions not included in the model. Taking the deviance
as a x2 …11† , there is no evidence that the interactions are significant and the model with just
main effects is a good fit. However, there is still scope for one of the two-factor interactions
to be significant and it is prudent to try including each of the six two-factor
interactions in turn to the model. As an example, when the interaction of the two kinds
of reading, AC, is included, the deviance reduces to 10 72 with 10 DF. Thus, this interaction
has an approximate x2 …1† of 2 87, which is not significant (P ˆ 0 091). Similarly,
none of the other interactions is significant.
The adequacy of the fit can be visualized by comparing the observed and fitted
proportions over the 16 cells. The fitted proportions are shown in column (4) of Table
14.1 and seem in reasonable agreement with the observed values in column (3). A formal
test may be constructed by calculating the expected frequencies, E(r) and E…n r†, for
each factor combination and calculating the Pearson's x2 statistic (8.28). This has the
value 13 61 with 11 DF (16 5, since five parameters have been fitted). This test statistic is
very similar to the deviance in this example, and the model with just the main effects is
evidently a good fit.
The data of Example 13.2 could be analysed using logistic regression. In this
case the observed proportions are each based on one observation only. As a
model we could suppose that the logit of the population probability of survival,
Y, was related to haemoglobin, x1, and bilirubin, x2 by the linear logistic
regression formula (14.6)
Y ˆ b 0 ‡ b 1x 1 ‡ b 2x 2:
Application of the maximum likelihood method gave the following estimates of
b 0, b 1, and b 2 with their standard errors:
^b 0 ˆ 2 354 2 416
^b 1 ˆ 0 5324 0 1487
^b 2 ˆ 0 4892 0 3448:
…14:7†
The picture is similar to that presented by the discriminant analysis of Example
13.2. Haemoglobin is an important predictor; bilirubin is not. An interesting
point is that, if the distributions of the xs are multivariate normal, with the same
variances and covariances for both successes and failures (the basic model for
discriminant analysis), the discriminant function (13.4) can also be used to
predict Y. The formula is: