Introduction to Categorical Data Analysis

3.4 STATISTICAL INFERENCE AND MODEL CHECKING 85
3.4.2 Example: Snoring and Heart Disease Revisited
Section 3.2.2 used a linear probability model to describe the probability of heart
disease π(x)as a linear function of snoring level x for data in Table 3.1. The ML model
fit is ˆπ = 0.0172 + 0.0198x, where the snoring effect ˆβ = 0.0198 has SE = 0.0028.
The Wald test of H0: β = 0 against Ha: β �= 0 treats
z = ˆβ/SE = 0.0198/0.0028 = 7.1
as standard normal, or z 2 = 50.0 as chi-squared with df = 1. This provides extremely
strong evidence of a positive snoring effect on the probability of heart disease (P <
0.0001). We obtain similar strong evidence from a likelihood-ratio test comparing this
model to the simpler one having β = 0. That chi-squared statistic equals −2(L0 −
L1) = 65.8 with df = 1 (P < 0.0001). The likelihood-ratio 95% confidence interval
for β is (0.0145, 0.0255).
3.4.3 The Deviance
Let LM denote the maximized log-likelihood value for a model M of interest. Let
LS denote the maximized log-likelihood value for the most complex model possible.
This model has a separate parameter for each observation, and it provides a perfect
fit to the data. The model is said to be saturated.
For example, suppose M is the linear probability model, π(x) = α + βx, applied
to the 4 × 2 Table 3.1 on snoring and heart disease. The model has two parameters
for describing how the probability of heart disease changes for the four levels of x =
snoring. The corresponding saturated model has a separate parameter for each of the
four binomial observations: π(x) = π1 for never snorers, π(x) = π2 for occasional
snorers, π(x) = π3 for snoring nearly every night, π(x) = π4 for snoring every night.
The ML estimate for πi is simply the sample proportion having heart disease at level
i of snoring.
Because the saturated model has additional parameters, its maximized log
likelihood LS is at least as large as the maximized log likelihood LM for a simpler
model M. The deviance of a GLM is defined as
Deviance =−2[LM − LS]
The deviance is the likelihood-ratio statistic for comparing model M to the saturated
model. It is a test statistic for the hypothesis that all parameters that are in the saturated
model but not in model M equal zero. GLM software provides the deviance, so it is
not necessary to calculate LM or LS.
For some GLMs, the deviance has approximately a chi-squared distribution. For
example, in Section 5.2.2 we will see this happens for binary GLMs with a fixed
number of explanatory levels in which each observation is a binomial variate having
relatively large counts of successes and failures. For such cases, the deviance