Introduction to Categorical Data Analysis

152 BUILDING AND APPLYING LOGISTIC REGRESSION MODELS
Figure 5.2. Observed proportion (x) and estimated probability of heart disease (curve) for linear logit
model.
patterns to what might be chance variation from a model. Also, these deletion diagnostics
all relate to removing an entire binomial sample at a blood pressure level instead
of removing a single subject’s binary observation. Such subject-level deletions have
little effect for this model.
Another useful graphical display for showing lack of fit compares observed and
fitted proportions by plotting them against each other, or by plotting both of them
against explanatory variables. For the linear logit model, Figure 5.2 plots both the
observed proportions and the estimated probabilities of heart disease against blood
pressure. The fit seems decent.
5.3 EFFECTS OF SPARSE DATA
The log likelihood function for logistic regression models has a concave (bowl) shape.
Because of this, the algorithm for finding the ML estimates (Section 3.5.1) usually
converges quickly to the correct values. However, certain data patterns present difficulties,
with the ML estimates being infinite or not existing. For quantitative or
categorical predictors, this relates to observing only successes or only failures over
certain ranges of predictor values.
5.3.1 Infinite Effect Estimate: Quantitative Predictor
Consider first the case of a single quantitative predictor. The ML estimate for its effect
is infinite when the predictor values having y = 0 are completely below or completely
above those having y = 1, as Figure 5.3 illustrates. In it, y = 0atx = 10, 20, 30,
40, and y = 1atx = 60, 70, 80, 90. An ideal (perfect) fit has ˆπ = 0 for x ≤ 40 and
ˆπ = 1 for x ≥ 60. One can get a sequence of logistic curves that gets closer and closer
to this ideal by letting ˆβ increase without limit, with ˆα =−50 ˆβ. (This ˆα value yields