Partial likelihood-ratio test ($2({\ell }_p({\hat{\beta }}_{\mathrm{full}})-{\ell }_p({\hat{\beta }}_{\mathrm{null}}))$)

For a Cox proportional-hazards model, let ${\ell }_p$ denote the logarithm of the Cox partial likelihood. To compare a full parameter space with a nested null model imposing $q$ independent smooth restrictions, maximize the same partial likelihood under each model and formNesting makes $T\ge 0$. Under the null hypothesis, a correctly specified regular model, independent censoring conditional on covariates, fixed parameter dimension, an interior true parameter and nonsingular limiting information, $T\stackrel{d}{\to }{\chi }_q^2$. The degrees of freedom count independent restrictions, not the number of observed failures. For example, testing that two identifiable regression coefficients both vanish gives $q=2$.

The calibration follows from the local quadratic expansion of ${\ell }_p$: the unrestricted maximizer removes the score component in the $q$ constrained directions, whose standardized squared length has a chi-squared distribution. It requires enough event information for this approximation. A variance parameter constrained to be nonnegative and tested at zero, or a parameter unidentified under the null model, violates the regular interior assumptions; an ordinary ${\chi }_q^2$ reference cannot be presumed. Model comparisons must also use the same observations and the same event-tie convention.