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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
. Let
be the MLE of
under
. The chi-square score statistic for testing
is defined by
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and it has an asymptotic distribution with
degrees of freedom under
, where
is the number of restrictions imposed on
by
.
When you use SELECTION=FORWARD, BACKWARD, or STEPWISE, the procedure calculates a residual 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 explanatory effects of interest. The full cumulative response model has a parameter vector
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where are intercept parameters, and
are the common slope parameters for the
explanatory effects. The full generalized logit model has a parameter vector
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where is the slope parameter for the
th effect in the
th logit.
Consider the null hypothesis , where
for the cumulative response model, and
, for the generalized logit model. For the reduced model with
explanatory effects, let
be the MLEs of the unknown intercept parameters, let
be the MLEs of the unknown slope parameters, and let
, be those for the generalized logit model. The residual chi-square is the chi-square score statistic testing the null hypothesis
; that is, the residual chi-square is
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where for the cumulative response model , and for the generalized logit model
, where
denotes a vector of
zeros.
The residual chi-square has an asymptotic chi-square distribution with degrees of freedom (
for the generalized logit model). A special case is the global score chi-square, where the reduced model consists of the
intercepts and no explanatory effects. The global score statistic is displayed in the "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.
These tests are performed when you specify SELECTION=FORWARD or STEPWISE, and are displayed when the DETAILS option is specified. 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) Selection" table. This section describes how the tests are calculated.
Suppose that intercepts and
explanatory variables (say
) have been fit to a model and that
is another explanatory variable of interest. Consider a full model with the
intercepts and
explanatory variables (
) and a reduced model with
excluded. The significance of
adjusted for
can be determined by comparing the corresponding residual chi-square with a chi-square distribution with one degree of freedom (
degrees of freedom for the generalized logit model).
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. For small sample sizes, this test might be too liberal (Stokes, Davis, and Koch; 2000). This section describes the methods used to calculate the test.
For this test the number of response levels, , is assumed to be strictly greater than 2. Let
be the response variable taking values
. Suppose there are
explanatory variables. Consider the general cumulative model without making the parallel lines assumption
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where is the link function, and
is a vector of unknown parameters consisting of an intercept
and
slope parameters
. The parameter vector for this general cumulative model is
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Under the null hypothesis of parallelism , there is a single common slope parameter for each of the
explanatory variables. Let
be the common slope parameters. Let
and
be the MLEs of the intercept parameters and the common slope parameters. Then, under
, the MLE of
is
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and the chi-square score statistic has an asymptotic chi-square distribution with
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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Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.