= Partial likelihood-ratio test
{title2=$2(\ell_p(\widehat\beta_{\rm full})-\ell_p(\widehat\beta_{\rm null}))$}
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 form
$$
T=2\{\ell_p(\widehat\beta_{\rm full})-\ell_p(\widehat\beta_{\rm null})\}.
$$
Nesting makes $T\ge0$. 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, $T\xrightarrow{d}\chi_q^2$. 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.
Back to article page
