The SURVEYPHREG Procedure

Testing the Global Null Hypothesis

The following statistics can be used to test the global null hypothesis $H_{0}\colon {\bbeta }=\Strong{0}$ . Let d be the number of clusters (or observations) minus the number of strata (or one) and p be the number of estimable parameters in the analysis model.

The likelihood ratio test is expressed as

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

where L( $\cdot$ ) denotes the partial pseudo-likelihood described in Partial Likelihood Function for the Cox Model. The p-value is computed by using a chi-square distribution with p degrees of freedom. The usual assumptions required for a likelihood ratio test do not hold for the pseudo-likelihood that is used by PROC SURVEYPHREG, leading to other methods for testing the global null hypothesis, such as the Wald test discussed below.

Wald’s test is expressed as

$W_{F}=\left( \frac{d-p+1}{dp} \right) \hat{\bbeta }’ \left[\widehat{\mb {V}}(\hat{\bbeta }) \right]^{-1} \hat{\bbeta }$

The p-value is computed by using an F distribution with degrees of freedom. For the Taylor series linearization method, the DF=PARMADJ option in the MODEL statement computes the p-value by using an F distribution with degrees of freedom.

If you specify the DF=NONE option in the MODEL statement, then the procedure computes

$W_{\chi ^{2}}=\hat{\bbeta }’ \left[ \widehat{\mb {V}}(\hat{\bbeta }) \right]^{-1} \hat{\bbeta }$

and the p-value is computed by using a chi-square distribution with p degrees of freedom.