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 $H_0$. Let $\hat{\btheta }$ be the MLE of $\btheta $ under $H_0$. The chi-square score statistic for testing $H_0$ 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 $H_0$, where r is the number of restrictions imposed on $\btheta $ by $H_0$.