The PHREG Procedure

Type 3 Tests

The following statistics can be used to test the null hypothesis $H_{0L}\colon {\mb {L}\bbeta } = \Strong{0}$, where $\mb {L}$ is a matrix of known coefficients. Under mild assumptions, each of the following statistics has an asymptotic chi-square distribution with $\mi {p}$ degrees of freedom, where p is the rank of $\mb {L}$. Let $\tilde{\bbeta }_{\mb {L}}$ be the maximum likelihood of $\bbeta $ under the null hypothesis $H_{0\mb {L}}$; that is,

\[  l(\tilde{\bbeta }_{\mb {L}}) = \max _{\mb {L}\bbeta =0}l(\bbeta )  \]

Likelihood Ratio Statistic

\[  \chi ^{2}_{\mr {LR}}=2 \left[ l (\hat{\bbeta }) - l(\tilde{\bbeta }_{\mb {L}}) \right]  \]

Score Statistic

\[  \chi ^{2}_{S}= \left[\frac{ \partial l(\tilde{\bbeta }_{\mb {L}}) }{ \partial {\bbeta } } \right]’ \left[-\frac{\partial ^2 l(\tilde{\bbeta }_{\mb {L}})}{\partial \bbeta ^2} \right]^{-1} \left[ \frac{ \partial l(\tilde{\bbeta }_{\mb {L}}) }{ \partial {\bbeta } } \right]  \]

Wald’s Statistic

\[  \chi ^{2}_{W}=\left( \mb {L}\hat{\bbeta } \right) ’ \left[ \mb {L}\hat{\mb {V}}(\hat{\bbeta })\mb {L}’ \right] ^{-1} \left( \mb {L}\hat{\bbeta } \right)  \]

where $\hat{\bV }(\hat{\bbeta })$ is the estimated covariance matrix, which can be the model-based covariance matrix $\left[-\frac{\partial ^2 l(\hat{\bbeta })}{\partial \bbeta ^2} \right]^{-1}$ or the sandwich covariance matrix $V_ S(\hat{\bbeta })$ (see the section Robust Sandwich Variance Estimate for details).