Applied Statistics Using SPSS, STATISTICA, MATLAB and R

7.5 Logit and Probit Models 325
sections, are no longer applicable. For the logit and probit models, some sort of the
chi-square test described in Chapter 5 is usually applied in order to assess the
goodness of fit of the model. SPSS and STATISTICA afford another type of chisquare
test based on the log-likelihood of the model. Let L0 represent the loglikelihood
for the null model, i.e., where all slope parameters are zero, and L1 the
log-likelihood of the fitted model. In the test used by STATISTICA, the following
quantity is computed:
L = −2(L0 − L1),
which, under the null hypothesis that the null model perfectly fits the data, has a
chi-square distribution with p − 1 degrees of freedom. The test used by SPSS is
similar, using only the quantity –2 L1, which, under the null hypothesis, has a chisquare
distribution with n − p degrees of freedom.
In Example 7.21, the chi-square test is significant for both the logit and probit
models; therefore, we reject the null hypothesis that the null model fits the data
perfectly. In other words, the estimated parameters b1 (0.23 and 0.138 for the logit
and probit models, respectively) have a significant contribution for the fitted
models.
1.2
1.0
0.8
0.6
0.4
0.2
0.0
HG
-0.2
-5 0 5 10 15 20 25 30 35 40 45
Figure 7.24. Logistic response for the clay classification problem, using variable
HG (obtained with STATISTICA). The circles represent the observed data.
Table 7.18. Classification matrix for the clay dataset, using predictor HG in the
logit or probit models.
Predicted Age = 1 Predicted Age = 0 Error rate
Observed Age = 1 65 4 94.2
Observed Age = 0 10 15 60.0