The SURVEYLOGISTIC Procedure

Score Statistics and Tests

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

$\text{[math]}$

It has an asymptotic $\text{[math]}$ distribution with $\text{[math]}$ degrees of freedom under $\text{[math]}$ , where $\text{[math]}$ is the number of restrictions imposed on $\text{[math]}$ by $\text{[math]}$ .