The PROBIT Procedure
- Overview
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Getting Started
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SyntaxPROC PROBIT StatementBY StatementCDFPLOT StatementCLASS StatementEFFECTPLOT StatementESTIMATE StatementINSET StatementIPPPLOT StatementLPREDPLOT StatementLSMEANS StatementLSMESTIMATE StatementMODEL StatementOUTPUT StatementPREDPPLOT StatementSLICE StatementSTORE StatementTEST StatementWEIGHT Statement
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DetailsMissing ValuesResponse Level OrderingComputational MethodDistributionsINEST= SAS-data-setModel SpecificationLack-of-Fit TestsRescaling the Covariance MatrixTolerance DistributionInverse Confidence LimitsOUTEST= SAS-data-setXDATA= SAS-data-setTraditional High-Resolution GraphicsDisplayed OutputODS Table NamesODS Graphics
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Examples
- References
Two goodness-of-fit tests can be requested from the PROBIT procedure: a Pearson’s chi-square test and a log-likelihood ratio chi-square 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 goodness-of-fit test statistics only for the events/trials syntax.
If the Pearson’s goodness-of-fit chi-square test is requested and the p-value for the test is too small, variances and covariances are adjusted by a heterogeneity factor (the goodness-of-fit chi-square divided by its degrees of freedom) and a critical value from the t distribution is used to compute the fiducial limits. The Pearson’s chi-square test statistic is computed as
![\[ \chi _ P^2 = \sum _{i=1}^ m \sum _{j=1}^{k} \frac{(r_{ij} - n_ i {\hat p}_{ij})^2}{n_ i {\hat p}_{ij}} \]](images/statug_probit0072.png){kind=small}
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 chi-square test statistic is computed as
![\[ \chi _ D^2 = 2 \sum _{i=1}^ m \sum _{j=1}^{k} r_{ij} \ln \left( \frac{r_{ij}}{n_ i {\hat p}_{ij}} \right) \]](images/statug_probit0075.png){kind=small}
This quantity is sometimes called the deviance. If the modeled probabilities fit the data, these statistics should be approximately
distributed as chi-square 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 chi-square, there must be sufficient replication within the groupings. When this is not true, the data are sparse, and the p-values 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.