Two goodnessoffit tests can be requested from the PROBIT procedure: a Pearson’s chisquare test and a loglikelihood ratio chisquare test.
To compute the test statistics, you can use the AGGREGATE or AGGREGATE= option grouping the observations into subpopulations. If neither AGGREGATE nor AGGREGATE= is specified, PROC PROBIT assumes that each observation is from a separate subpopulation and computes the goodnessoffit test statistics only for the events/trials syntax.
If the Pearson’s goodnessoffit chisquare test is requested and the pvalue for the test is too small, variances and covariances are adjusted by a heterogeneity factor (the goodnessoffit chisquare divided by its degrees of freedom) and a critical value from the t distribution is used to compute the fiducial limits. The Pearson’s chisquare test statistic is computed as

where the sum on i is over grouping, the sum on j is over levels of response, is the frequency of response level j for the ith grouping, is the total frequency for the ith grouping, and is the fitted probability for the jth level at the ith grouping.
The likelihood ratio chisquare test statistic is computed as

This quantity is sometimes called the deviance. If the modeled probabilities fit the data, these statistics should be approximately distributed as chisquare with degrees of freedom equal to , where k is the number of levels of the multinomial or binomial response, m is the number of sets of independent variable values (covariate patterns), and q is the number of parameters fit in the model.
In order for the Pearson’s statistic and the deviance to be distributed as chisquare, there must be sufficient replication within the groupings. When this is not true, the data are sparse, and the pvalues for these statistics are not valid and should be ignored. Similarly, these statistics, divided by their degrees of freedom, cannot serve as indicators of overdispersion. A large difference between the Pearson’s statistic and the deviance provides some evidence that the data are too sparse to use either statistic.