Introduction to Categorical Data Analysis

3.2 GENERALIZED LINEAR MODELS FOR BINARY DATA 69
If we ignored the binary nature of Y and used ordinary regression, the estimates
of the parameters would be the least squares estimates. They are the ML estimates
under the assumption of a normal response. These estimates exist, because for a
normal response an estimated mean of Y can be any real number and is not restricted
to the 0–1 range. Of course, an assumption of normality for a binary response is not
sensible; when ML fitting with the binomial assumption fails, the least squares method
is also likely to give estimated probabilities outside the 0–1 range for some x values.
3.2.2 Example: Snoring and Heart Disease
Table 3.1 is based on an epidemiological survey of 2484 subjects to investigate snoring
as a possible risk factor for heart disease. The subjects were classified according to
their snoring level, as reported by their spouses. The linear probability model states
that the probability of heart disease π(x) is a linear function of the snoring level x.
We treat the rows of the table as independent binomial samples with probabilities
π(x). We use scores (0, 2, 4, 5) for x = snoring level, treating the last two snoring
categories as closer than the other adjacent pairs.
Software for GLMs reports the ML model fit, ˆπ = 0.0172 + 0.0198x. For
example, for nonsnorers (x = 0), the estimated probability of heart disease is
ˆπ = 0.0172 + 0.0198(0) = 0.0172. The estimated values of E(Y) for a GLM are
called fitted values. Table 3.1 shows the sample proportions and the fitted values for
the linear probability model. Figure 3.1 graphs the sample proportions and fitted values.
The table and graph suggest that the model fits these data well. (Section 5.2.2
discusses goodness-of-fit analyses for binary-response GLMs.)
The model interpretation is simple. The estimated probability of heart disease
is about 0.02 (namely, 0.0172) for nonsnorers; it increases 2(0.0198) = 0.04 for
occasional snorers, another 0.04 for those who snore nearly every night, and another
0.02 for those who always snore. This rather surprising effect is significant, as the
standard error of ˆβ = 0.0198 equals 0.0028.
Suppose we had chosen snoring-level scores with different relative spacings than
the scores {0, 2, 4, 5}. Examples are {0, 2, 4, 4.5} or {0, 1, 2, 3}. Then the fitted values
would change somewhat. They would not change if the relative spacings between
Table 3.1. Relationship Between Snoring and Heart Disease
Heart Disease
Proportion Linear Logit Probit
Snoring Yes No Yes Fit Fit Fit
Never 24 1355 0.017 0.017 0.021 0.020
Occasional 35 603 0.055 0.057 0.044 0.046
Nearly every night 21 192 0.099 0.096 0.093 0.095
Every night 30 224 0.118 0.116 0.132 0.131
Note: Model fits refer to proportion of “yes” responses.
Source: P. G. Norton and E. V. Dunn, Br. Med. J., 291: 630–632, 1985, published by BMJ
Publishing Group.