The PHREG Procedure

Testing the Global Null Hypothesis

The following statistics can be used to test the global null hypothesis $H_{0}\colon {\bbeta } = \Strong{0}$. Under mild assumptions, each statistic has an asymptotic chi-square distribution with $\mi{p}$ degrees of freedom given the null hypothesis. The value $\mi{p}$ is the dimension of $\bbeta $. For clustered data, the likelihood ratio test, the score test, and the Wald test assume independence of observations within a cluster, while the robust Wald test and the robust score test do not need such an assumption.

Likelihood Ratio Test

\[  \chi ^{2}_{\mr{LR}}=2 \left[ l (\hat{\bbeta }) - l(\mb{0}) \right]  \]

Score Test

\[  \chi ^{2}_{S}= \left[\frac{ \partial l(\mb{0}) }{ \partial {\bbeta } } \right]’ \left[-\frac{\partial ^2 l(\mb{0})}{\partial \bbeta ^2} \right]^{-1} \left[ \frac{ \partial l(\mb{0}) }{ \partial {\bbeta } } \right]  \]

Wald’s Test

\[  \chi ^{2}_{W}=\hat{\bbeta }’ \left[-\frac{\partial ^2 l(\hat{\bbeta })}{\partial \bbeta ^2} \right] \hat{\bbeta }  \]

Robust Score Test

\[  \chi ^2_{\mr{RS}} = \left[\sum _ i \bL ^0_ i \right]’ \left[ \sum _ i \bL ^0_ i{\bL ^0_ i}’ \right] ^{-1} \left[\sum _ i \bL ^0_ i \right]  \]

where $\bL ^0_ i$ is the score residual of the ith subject at $\bbeta = \mb{0}$; that is, $\bL ^0_ i = \bL _ i(\mb{0},\infty )$, where the score process $\bL _ i(\bbeta ,t)$ is defined in the section Residuals.

Robust Wald’s Test

\[  \chi ^{2}_{\mr{RW}}=\hat{\bbeta }’ [\hat{\bV }_ s(\hat{\bbeta })]^{-1} \hat{\bbeta }  \]

where $\hat{\bV }_ s(\hat{\bbeta })$ is the sandwich variance estimate (see the section Robust Sandwich Variance Estimate for details).