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 The LOGISTIC Procedure

## Score Statistics and Tests

To understand the general form of the score statistics, let be the vector of first partial derivatives of the log likelihood with respect to the parameter vector , and let be the matrix of second partial derivatives of the log likelihood with respect to . That is, is the gradient vector, and is the Hessian matrix. Let be either or the expected value of .Consider a null hypothesis H0. Let be the MLE of under H0. The chi-square score statistic for testing H0 is defined by

and it has an asymptotic distribution with r degrees of freedom under H0, where r is the number of restrictions imposed on by H0.

### Residual Chi-Square

When you use SELECTION=FORWARD, BACKWARD, or STEPWISE, the procedure calculates a residual score chi-square score statistic and reports the statistic, its degrees of freedom, and the p-value. This section describes how the statistic is calculated.

Suppose there are s explanatory effects of interest. The full model has a parameter vector

where are intercept parameters, and are slope parameters for the explanatory effects. Consider the null hypothesis where t < s. For the reduced model with t explanatory effects, let be the MLEs of the unknown intercept parameters, and let be the MLEs of the unknown slope parameters. The residual chi-square is the chi-square score statistic testing the null hypothesis H0; that is, the residual chi-square is

where .

The residual chi-square has an asymptotic chi-square distribution with s-t degrees of freedom. A special case is the global score chi-square, where the reduced model consists of the k intercepts and no explanatory effects. The global score statistic is displayed in the "Model-Fitting Information and Testing Global Null Hypothesis BETA=0" table. The table is not produced when the NOFIT option is used, but the global score statistic is displayed.

### Testing Individual Effects Not in the Model

These tests are performed in the FORWARD or STEPWISE method. In the displayed output, the tests are labeled "Score Chi-Square" in the "Analysis of Effects Not in the Model" table and in the "Summary of Stepwise (Forward) Procedure" table. This section describes how the tests are calculated.

Suppose that k intercepts and t explanatory variables (say v1, ... , vt) have been fitted to a model and that vt+1 is another explanatory variable of interest. Consider a full model with the k intercepts and t+1 explanatory variables (v1, ... ,vt,vt+1) and a reduced model with vt+1 excluded. The significance of vt+1 adjusted for v1, ... ,vt can be determined by comparing the corresponding residual chi-square with a chi-square distribution with one degree of freedom.

### Testing the Parallel Lines Assumption

For an ordinal response, PROC LOGISTIC performs a test of the parallel lines assumption. In the displayed output, this test is labeled "Score Test for the Equal Slopes Assumption" when the LINK= option is NORMIT or CLOGLOG. When LINK=LOGIT, the test is labeled as "Score Test for the Proportional Odds Assumption" in the output. This section describes the methods used to calculate the test.

For this test the number of response levels, k+1, is assumed to be strictly greater than 2. Let Y be the response variable taking values 1, ... , k, k+1. Suppose there are s explanatory variables. Consider the general cumulative model without making the parallel lines assumption

where g(.) is the link function, and is a vector of unknown parameters consisting of an intercept and s slope parameters .The parameter vector for this general cumulative model is

Under the null hypothesis of parallelism ,there is a single common slope parameter for each of the s explanatory variables. Let be the common slope parameters. Let and be the MLEs of the intercept parameters and the common slope parameters . Then, under H0, the MLE of is

and the chi-squared score statistic has an asymptotic chi-square distribution with s(k-1) degrees of freedom. This tests the parallel lines assumption by testing the equality of separate slope parameters simultaneously for all explanatory variables.

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