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An at-risk process indicates or counts which individuals belong to the risk set at each time.

Under the null hypothesis of equal event-time distributions, every member of the combined risk set has the same instantaneous chance of being the next event. Conditional on one event at $a_j$ and on the two risk-set sizes,so

The risk set at event time $a_j$ contains individuals still under observation and event-free immediately before $a_j$. For group $k$ its size isThe use of $\ge$ keeps the individual who experiences the event at $a_j$ in the risk set just before that event.

The variance of an estimated hazard increment is large when its group has few individuals in the risk set. The log-rank weights are near zero when either $r_j^{(0)}$ or $r_j^{(1)}$ is small and are largest when both groups retain substantial information. They therefore suppress noisy late-event comparisons and weight each observed-minus-expected event by its available information. Unit weights would instead give equal influence to unstable increments from depleted risk sets.

Letand let $C$ be the common partial likelihood contribution from the first $n-3$ observations. Since $t_{n-2}>x_{n-2}$, the three possible complete-data tail orderings and their partial likelihoods areTheir sum isWhen individual $n-2$ is right-censored at $x_{n-2}$, that individual leaves the risk set before the event at $x_{n-1}$, so the directly calculated Cox partial likelihood is also $C,a/(a+b)$. Summing over the unobserved compatible event orderings therefore reproduces the censored-data partial likelihood.

For each observed event $i$ with $v_i=1$, let $R_i=\{j:x_j\ge x_i\}$ be its risk set. The Cox partial likelihood isMaximize it to obtain $\hat{\beta }$, estimate its variance from the observed partial information, and test $H_0:\beta =0$ using a partial likelihood-ratio test, Wald test, or score test. A positive fitted coefficient means the group with $z=1$ has the larger hazard.

A semiparametric proportional hazards model specifieswith finite-dimensional parameter $\beta$ but an unspecified baseline hazard $h_0$. A partial likelihood uses a component of the data likelihood that depends on $\beta$ while eliminating the nuisance function. In the Cox proportional-hazards model, conditioning on which member of each risk set experiences the event produces the Cox partial likelihood.

Construct the two countries' period-specific risk sets with the same left truncation and right censoring rules, then compare their event counts by a Log-rank test. This is a nonparametric comparison because it does not specify the shape of either country's hazard function.