Modified Score Statistic

The PHREG Procedure

Modified Score Statistic

Let $\bld{U}_0$ be the n×p matrix of efficient scores and $\bld{I}_0$ be the p×p information matrix, both evaluated at ${\beta}=\bld{0}$ ; let $\bld{1}$ be a column n-vector of 1's. Let $\hat{{\beta}}_1$ be the one-step estimate of ${\beta}$ ; that is,

$\hat{{\beta}}_1=\bld{I}_0^{-1} (\bld{U}_0'\bld{1})$

the covariance matrix is estimated by $\hat{\bld{V}}({\beta}_1)=\bld{I}_0^{-1}$ .

The score statistic for testing H₀: ${\beta}$ = $\bld{0}$ can be expressed as a Wald test statistic (Therneau and Grambsch [2000]):

$(\bld{U}'_0\bld{1})' \bld{I}_0^{-1} (\bld{U}_0'\bld{1}) &=& [\bld{I}_0^{-1} ... ...d{1})] \ &=& \hat{{\beta}}_1' [\hat{\bld{V}}({\beta}_1)]^{-1} \hat{{\beta}_1}$

The modified score test statistic for testing H₀: ${\beta}$ = $\bld{0}$ is obtained by replacing $\hat{\bld{V}}({\beta}_1)$ in the score statistic by the robust sandwich estimate $\hat{\bld{V}}^s_0=\bld{D}_0'\bld{D}_0$ where $\bld{D}_0=\bld{U}_0\bld{I}_0^{-1}$ :

$\hat{{\beta}_1} (\hat{\bld{V}}^s_0) ^{-1} \hat{{\beta}_1} &=& [\bld{I}_0^{-1} (... ...{1})] \ &=& (\bld{U}'_0\bld{1})' (\bld{U}_0' \bld{U}_0)^{-1} (\bld{U}'_0\bld{1})$

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