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11.4 LOGISTIC REGRESSION 571
Standard
Wald
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 1.8744 0.9809 3.6518 0.0560
age 1 0.0379 0.0146 6.7083 0.0096
FIGURE 11.4.2
Table 11.4.3.
Partial SAS ® printout of the logistic regression analysis of the data in
documentation for further details.) A partial printout of the analysis is shown
in Figure 11.4.2.
The slope of our regression is -.0379, and the intercept is 1.8744. The
regression equation, then, is
yN i = 1.8744 - .0379x i
where yN i = ln3pN i >11 - pN i 24 and pN i is the predicted probability of attending
cardiac rehabilitation for a woman aged x i .
Test of H 0 that B 1 0
We reach a conclusion about the adequacy of the logistic model by testing the null
hypothesis that the slope of the regression line is zero. The test statistic is z = bN 1>sb
N 1
where z is the standard normal statistic, bN
1 is the sample slope 1-.03792, and s b N 1
is its
standard error (.0146) as shown in Figure 11.4.2. From these numbers we compute
z = -.0379>.0146 = -2.5959, which has an associated two-sided p value of .0094. We
conclude, therefore, that the logistic model is adequate. The square of z is chi-square with
1 degree of freedom, a statistic that is shown in Figure 11.4.2.
Using the Logistic Regression to Estimate p
We may use Equation 11.4.5 and the results of our analysis to estimate p, the probability
that a woman of a given age (within the range of ages represented by the data) will
attend a cardiac rehabilitation program. Suppose, for example, that we wish to estimate
the probability that a woman who is 50 years of age will participate in a rehabilitation
program. Substituting 50 and the results shown in Figure 11.4.2 into Equation 11.4.5
gives
pN =
exp31.8744 - 1.0379215024
1 + exp31.8744 - 1.0379215024 = .49485
SAS ® calculates the estimated probabilities for the given values of X. We can see the
estimated probabilities of attending cardiac rehabilitation programs for the age range
of the subjects enrolled in the study in Figure 11.4.3. Since the slope was negative,
we see a decreasing probability of attending a cardiac rehabilitation program for older
women.