Score Statistics and Tests

The LOGISTIC Procedure

Score Statistics and Tests

To understand the general form of the score statistics, let $U({\gamma})$ be the vector of first partial derivatives of the log likelihood with respect to the parameter vector ${\gamma}$ , and let $H({\gamma})$ be the matrix of second partial derivatives of the log likelihood with respect to ${\gamma}$ . That is, $U({\gamma})$ is the gradient vector, and $H({\gamma})$ is the Hessian matrix. Let $I({\gamma})$ be either $-H({\gamma})$ or the expected value of $-H({\gamma})$ .Consider a null hypothesis H₀. Let $\hat{{\gamma}}_0$ be the MLE of ${\gamma}$ under H₀. The chi-square score statistic for testing H₀ is defined by

$U'(\hat{{\gamma}}_0)I^{-1}(\hat{{\gamma}}_0)U (\hat{{\gamma}}_0)$

and it has an asymptotic $\chi^2$ distribution with r degrees of freedom under H₀, where r is the number of restrictions imposed on ${\gamma}$ 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

${\gamma}=(\alpha_1, ... ,\alpha_k,\beta_1, ... ,\beta_s)'$

where $\alpha_1, ... ,\alpha_{k}$ are intercept parameters, and $\beta_1, ... , \beta_s$ are slope parameters for the explanatory effects. Consider the null hypothesis $H_0\colon \beta_{t+1}= ... =\beta_s=0$ where t < s. For the reduced model with t explanatory effects, let $\hat{\alpha}_1, ... , \hat{\alpha}_k$ be the MLEs of the unknown intercept parameters, and let $\hat{\beta}_1, ... , \hat{\beta}_t$ 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

$U'(\hat{{\gamma}}_0)I^{-1}(\hat{{\gamma}}_0)U (\hat{{\gamma}}_0)$

where $\hat{{\gamma}}_0=(\hat{\alpha}_1, ... ,\hat{\alpha}_k, \hat{\beta}_1, ... ,\hat{\beta}_t,0, ... ,0)'$ .

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, ... , v_t) have been fitted to a model and that v_t+1 is another explanatory variable of interest. Consider a full model with the k intercepts and t+1 explanatory variables (v₁, ... ,v_t,v_t+1) and a reduced model with v_t+1 excluded. The significance of v_t+1 adjusted for v₁, ... ,v_t 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

$g({Pr}(Y\leq i|{x}))= (1,x'){\gamma}_i, 1 \leq i \leq k$

where g(.) is the link function, and ${{\gamma}}_i=(\alpha_i, \beta_{i1}, ... , \beta_{is})^'$ is a vector of unknown parameters consisting of an intercept $\alpha_i$ and s slope parameters $\beta_{i1}, ... , \beta_{is}$ .The parameter vector for this general cumulative model is

${\gamma}=({\gamma}'_1, ... ,{\gamma}'_k)'$

Under the null hypothesis of parallelism $H_0\colon \beta_{1m}=\beta_{2m}= ... =\beta_{km}, 1 \leq m \leq s$ ,there is a single common slope parameter for each of the s explanatory variables. Let $\beta_1, ... ,\beta_s$ be the common slope parameters. Let $\hat{\alpha}_1, ... , \hat{\alpha}_k$ and $\hat{\beta}_1, ... , \hat{\beta}_s$ be the MLEs of the intercept parameters and the common slope parameters . Then, under H₀, the MLE of ${\gamma}$ is

$\hat{{\gamma}_0}=(\hat{{\gamma}}'_1, ... ,\hat{{\gamma}}'_k)' {with} \hat{{\gamma}}_i=(\hat{\alpha}_i,\hat{\beta}_1, ... , \hat{\beta}_s)' 1 \leq i \leq k$

and the chi-squared score statistic $U'(\hat{{\gamma}}_0)I^{-1}(\hat{{\gamma}}_0)U(\hat{{\gamma}}_0)$ 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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