Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 34 / 3

Assume independent individuals and independent censoring, with the censoring law not involving the event-rate parameters. If $T_j$ is the survival time and $C_j$ its censoring time, we observe $x_j=\min (T_j,C_j)$ and $v_j=1_{\{T_j\le C_j\}}$. The exponential distribution givesAn observed event contributes $f_j(x_j)$, while a right-censored observation contributes $S_j(x_j)$, because it only says the event has not occurred by $x_j$. Multiplying these survival likelihood contributions and omitting censoring factors independent of the parameters givesWithout independent censoring the censoring factors cannot generally be discarded in this way.

Let $D_k={\sum }_{j:g_j=k}{v}_j$ be the observed event count and $T_k={\sum }_{j:g_j=k}{x}_j$ the total person-time in group $k$. Both groups are present; suppose each has $T_k>0$. The grouped log-likelihood isWhen $D_k>0$, its derivative is $D_k/{\beta }^{(k)}-T_k$ and its second derivative is $-D_k/({\beta }^{(k)}{)}^2<0$. Therefore the maximum-likelihood estimates areBoth events and censored observations contribute to $T_k$. If $D_k=0$, the likelihood decreases with the positive rate: its supremum is at the boundary ${\beta }^{(k)}\downarrow 0$, rather than at a finite strictly positive estimate. Writing $D_k/T_k=0$ then describes the closure estimate.

Under the null hypothesis of a common rate, the same maximization givesLet $D=D_0+D_1$ and $T=T_0+T_1$. The maximized linear terms cancel, since $T_k{\hat{\beta }}^{(k)}=D_k$ and $T\tilde{\beta }=D$. Thus the likelihood-ratio test usesInterpret a zero-event summand by continuity as zero. If $D=0$, both likelihood suprema are equal and $W=0$: no rate comparison is informed by observed events.

Under positive common hazard, independent censoring and the usual increasing-information conditions in both groups, Wilks theorem gives $W\Rightarrow {\chi }_1^2$, because the unrestricted model has two rate parameters and the null has one. Reject for large $W$; at 5%, the usual cutoff is approximately $3.84$. The chi-squared calibration is asymptotic and can be poor with sparse event counts or boundary estimates.

Set $\alpha =\log \{{\beta }^{(1)}/{\beta }^{(0)}\}$ and absorb the common multiplier into $h_0(t)=m(t){\beta }^{(0)}$. The model becomesIt is a Cox proportional-hazards model; the unknown baseline does not need a parametric form. The equality hypothesis is precisely $\alpha =0$. The absolute multipliers and $m$ have a scale ambiguity, but their hazard ratio $e^{\alpha }$ is identifiable when both groups supply information.

For an untied event at $a_k$, let $R_k$ be the risk set just before the event, and let $i_k$ be the subject who fails. The conditional event probability is proportional to that subject's hazard, so the common baseline cancels:Multiply these factors to form the Cox partial likelihood. One can maximize it and compare its maximum with its value at $\alpha =0$ using a partial likelihood-ratio test, or use the corresponding score test.

Explicitly, with group risk-set sizes $r_{k0},r_{k1}$ and observed event indicator $d_{k1}$ for group 1, the score at zero and its information areUnder the null the event label has exactly this Bernoulli distribution conditional on its current risk set. Hence $U^2/V$ is the usual one-degree-of-freedom log-rank test statistic, with an asymptotic chi-squared distribution when $V>0$ and enough informative events are observed. Censoring can differ between groups, provided it is independent of event time within the conditioning model. If there are recorded ties, use the appropriate tied-risk-set version: for $d_k$ events the null variance contribution is $d_k(r_k-d_k)r_{k0}{r}_{k1}/[r_k^2(r_k-1)]$.