The PHREG Procedure

Testing the Global Null Hypothesis

Likelihood Ratio Test
Score Test
Wald’s Test
Robust Score Test
Robust Wald’s Test

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).