The PHREG Procedure

Testing the Global Null Hypothesis

Subsections:

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).