The SURVEYLOGISTIC Procedure

Score Statistics and Tests

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

${\mb {g}}’(\hat{\btheta })\mb {I}^{-1}(\hat{\btheta }){\mb {g}} (\hat{\btheta })$

It has an asymptotic $\chi ^2$ distribution with r degrees of freedom under , where r is the number of restrictions imposed on $\btheta$ by .