The PHREG Procedure

Testing the Global Null Hypothesis

The following statistics can be used to test the global null hypothesis upper H 0 colon bold-italic beta equals bold 0. Under mild assumptions, each statistic has an asymptotic chi-square distribution with p degrees of freedom given the null hypothesis. The value p is the dimension of bold-italic beta. 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 Subscript normal upper L normal upper R Superscript 2 Baseline equals 2 left-bracket l left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis minus l left-parenthesis bold 0 right-parenthesis right-bracket

Score Test

chi Subscript upper S Superscript 2 Baseline equals left-bracket StartFraction partial-differential l left-parenthesis bold 0 right-parenthesis Over partial-differential bold-italic beta EndFraction right-bracket prime left-bracket minus StartFraction partial-differential squared l left-parenthesis bold 0 right-parenthesis Over partial-differential bold-italic beta squared EndFraction right-bracket Superscript negative 1 Baseline left-bracket StartFraction partial-differential l left-parenthesis bold 0 right-parenthesis Over partial-differential bold-italic beta EndFraction right-bracket

Wald’s Test

chi Subscript upper W Superscript 2 Baseline equals ModifyingAbove bold-italic beta With caret prime left-bracket minus StartFraction partial-differential squared l left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis Over partial-differential bold-italic beta squared EndFraction right-bracket ModifyingAbove bold-italic beta With caret

Robust Score Test

chi Subscript normal upper R normal upper S Superscript 2 Baseline equals left-bracket sigma-summation Underscript i Endscripts bold upper L Subscript i Superscript 0 Baseline right-bracket prime left-bracket sigma-summation Underscript i Endscripts bold upper L Subscript i Superscript 0 Baseline bold upper L Subscript i Superscript 0 Baseline prime right-bracket Superscript negative 1 Baseline left-bracket sigma-summation Underscript i Endscripts bold upper L Subscript i Superscript 0 Baseline right-bracket

where bold upper L Subscript i Superscript 0 is the score residual of the ith subject at bold-italic beta equals bold 0; that is, bold upper L Subscript i Superscript 0 Baseline equals bold upper L Subscript i Baseline left-parenthesis bold 0 comma normal infinity right-parenthesis, where the score process bold upper L Subscript i Baseline left-parenthesis bold-italic beta comma t right-parenthesis is defined in the section Residuals.

Robust Wald’s Test

chi Subscript normal upper R normal upper W Superscript 2 Baseline equals ModifyingAbove bold-italic beta With caret prime left-bracket ModifyingAbove bold upper V With caret Subscript s Baseline left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis right-bracket Superscript negative 1 Baseline ModifyingAbove bold-italic beta With caret

where ModifyingAbove bold upper V With caret Subscript s Baseline left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis is the sandwich variance estimate. For more information, see the section Robust Sandwich Variance Estimate.

Last updated: December 09, 2022