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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 $d_k(r_k-d_k)r_{k0}{r}_{k1}/[r_k^2(r_k-1)]$.

Keep the counting process and at-risk process notation from the introductory derivation. Because the individual hazards add, the aggregate conditional intensity isThe known individual contributions must be averaged over the current risk set, rather than over the original cohort. DefineThen $dN(t)/Y(t)$ estimates $\{{\overline{h}}_0(t)+h_1(t)\}dt$. Subtract the known average contribution to obtain the offset-adjusted cumulative hazard estimatorTake the quotient to be zero when $Y=0$. If $M(t)=N(t)-{\int }_0^t\lambda (u)du$, direct substitution givesThis establishes the target and the mean-zero estimation noise, under the usual integrability conditions. It estimates $H_1(t)$ while there is a nonempty risk set; no data identify its continuation after follow-up has ended. With common known $h_{0i}=h_0$ and individuals at risk throughout $[0,t]$, it reduces to ${\hat{H}}_1(t)=\hat{H}(t)-H_0(t)$.

The estimate may decrease between events because the known offset is subtracted continuously. That is not an algebraic error: it is an unconstrained estimating-equation estimator, not automatically a nonnegative monotone cumulative hazard estimate. If such shape constraints are imposed, the fitting procedure must explicitly account for them. The working hazard model itself must satisfy $h_{0i}+h_1\ge 0$.

Under the null hypothesis of equal survival distributions, the two groups have the same event hazard. Conditional on the current risk set and one event, the event is in A with probability $r_j^A/(r_j^A+r_j^B)$ and in B with probability $r_j^B/(r_j^A+r_j^B)$. Consequently, whenever both groups are represented,Thus a weighted sum of differences in estimated hazard increments is centered under the null. Its sign reflects which group has more events relative to its numbers at risk. The weights may depend on event time and the preceding risk sets, but must not be chosen after seeing which group failed; otherwise this conditional-centering argument fails. Positive weights emphasize selected time regions without reversing an increment's direction. Such tests can have weak power against crossing hazards whose positive and negative contributions cancel.

Let $I_j=d_j^A$, so $d_j^B=1-I_j$. Its conditional null variance is $r_j^A{r}_j^B/(r_j^A+r_j^B{)}^2$. Since the unweighted hazard difference is $I_j(1/r_j^A+1/r_j^B)-1/r_j^B$, the conditional variance of its weighted contribution is ${\omega }_j^2/(r_j^A{r}_j^B)$. Successive centered contributions form martingale differences. Their variance contributions add, providing a variance standardization for the test.

With the specified log-rank weight, put $r_j=r_j^A+r_j^B$. ThenTherefore the displayed sum is exactly the unstandardized log-rank statisticIts two-sided large-sample log-rank test uses $U^2/V\Rightarrow {\chi }_1^2$ when $V>0$ and information is sufficient. If one group has no remaining subjects at some event time, its event contributes zero information. Use the final observed-minus-expected expression to define the contribution as zero; the original divided expression has an undefined $0/0$ and the log-rank weight becomes zero there. Thus strict positivity of the proposed weights implicitly restricts the comparison to event times when both groups are at risk. Independent censoring and a valid conditional common-hazard model are required throughout.

For individual $i$, let $N_i(t)$ count its observed event by time $t$ and let $Y_i(t)$ indicate membership of the risk set immediately before $t$. Set $N={\sum }_i{N}_i$ and $Y={\sum }_i{Y}_i$. Under a common hazard function $h(t)$ and independent censoring, a currently at-risk subject has event probability $h(t)dt+o(dt)$ in the next short interval. Summing over the current risk set gives the conditional intensityEquivalently, the event count decomposes aswhere $M$ is the mean-zero counting-process martingale obtained by subtracting its conditional compensator. Estimating the unknown hazard increment by the observed event increment divided by the risk set size gives the Nelson–Aalen estimatorHere $a_j$ are the distinct observed event times and $r_j=Y(a_j)$ includes the subject who is about to fail. A censored observation affects subsequent risk sets, but causes no event jump. This is the requested no-ties estimator; tied event counts would produce $d_j/r_j$.

The derivation also identifies the estimation error:Thus it estimates $H(t)$ on the time range where individuals remain at risk. It is not exactly unbiased for the entire cumulative hazard after the last risk set disappears. Its usual Nelson–Aalen variance estimator is ${\sum }_{a_j\le t}{r}_j^{-2}$, since the compensator variance increment of the no-ties event process is $YdH$.

Let $R(t^{\ast })$ be the risk set immediately before $t^{\ast }$, conditional on the preceding history. Only its members can experience the next event. In a short interval $[t^{\ast },t^{\ast }+dt)$, the conditional probability that subject $j$ fails is $h_0(t^{\ast })e^{\beta z_j}dt+o(dt)$, whereas the probability of one event from the risk set is $h_0(t^{\ast }){\sum }_{i\in R(t^{\ast })}{e}^{\beta z_i}dt+o(dt)$. Dividing and taking the small-interval limit givesIt is zero outside the risk set. The common baseline hazard cancels. This is the conditional event-label probability underlying the Cox partial likelihood. Conditioning at an exact continuous event time is understood by this limiting conditional-intensity argument, rather than as division by the zero unconditional probability of an event at a fixed time. We use the standard simple event process with no simultaneous events and independent censoring.

Average the covariate over its possible event subjects using the conditional probabilities from part (a). Define the risk-weighted sumswith $z_i^0=1$. ThenThis is the mean covariate of the next event subject, conditional on the current risk set and an event. It is generally not the unweighted average over the original sample. For $\beta =0$ it reduces to the unweighted current risk-set mean.

For the distinct observed event times $a_k$, let $i_k=\pi (a_k)$ and let $R_k$ be the risk set immediately before that event. The Cox partial likelihood isThe different risk sets already incorporate each event and any intervening censoring. Differentiating this log-likelihood givesTherefore a finite interior maximum-likelihood estimate from the Cox partial likelihood satisfies the first-order conditionMoreoverEach summand is the conditional event-covariate variance, so the partial likelihood is log-concave. If some informative risk set has unequal covariates, the curvature is strictly negative and an existing finite root is the unique maximizer.

The finite-interior qualification is necessary. With two at-risk subjects having covariates zero and one, suppose the subject with covariate one is the sole observed event and the other is subsequently censored. Then $L_p(\beta )=e^{\beta }/(1+e^{\beta })$ increases strictly and has its supremum only as $\beta \to +\infty$; the score $1/(1+e^{\beta })$ has no finite zero. If every event risk set has identical covariates, the coefficient is instead unidentifiable and the score is identically zero. The displayed fitted-score equation applies to the regular case in which the proportional-hazards estimate is finite.

Among subjects alive at any given time and with the same relevant covariates or observed history, those censored at that time have the same subsequent failure distribution as those who remain under observation. Equivalently, the observed risk set represents the failure hazard of comparable subjects still alive, rather than selectively retaining lower- or higher-risk subjects. This is a statement about their future conditional risk, not that the observed failure-time distribution among censored people equals the distribution among people who eventually fail: their different observation patterns make that comparison inappropriate.

Right censoring occurs when a subject's exact failure time $T$ is unobserved but it is known that $T>C$, where $C$ is the censoring time. The observed pair is $U=\min (T,C)$ and $\Delta =1_{\{T\le C\}}$. For example, a living participant at study closure contributes a survival lower bound rather than a recorded death at closure. Right censoring differs from simply omitting the subject: time spent in the risk set remains informative.

Let $f_j=\mathbb{P}(T=t_j)$ and $h_j=\mathbb{P}(T=t_j\mid T>t_{j-1})$. Since $S(t_{j-1})=\mathbb{P}(T>t_{j-1})$, the discrete hazard satisfies $h_j=f_j/S(t_{j-1})$. HenceStarting from $S(0)=1$ and iterating, the survival function isUnder independent censoring, the risk set just before $t_j$ gives the binomial conditional failure likelihood $h_j^{d_j}(1-h_j{)}^{n_j-d_j}$, whose maximizer is ${\hat{h}}_j=d_j/n_j$. This gives the Kaplan–Meier estimatorFailures at exactly $t$ are included because the target is $\mathbb{P}(T>t)$. Censored subjects contribute to earlier risk sets and leave before later ones; they do not create a failure factor. The usual convention includes a subject censored at a tied failure time in the risk set for that failure. Estimation beyond the last observed failure is limited by the available follow-up.

The last lung-cancer failure before five years is at 4.85, where the displayed net survival estimate is 0.392. The other-death net survival estimate remains 0.498 from 4.096 until 5.065. Thus the two supplied curves imply net five-year death estimates of $1-0.392=0.608$ for lung cancer and $1-0.498=0.502$ for other causes. Their sum is 1.110, already showing that they cannot be probabilities of two disjoint actual causes of death.

In competing risks, let ${\lambda }_1(t)$ and ${\lambda }_2(t)$ be the cause-specific hazards while alive, with cumulative hazards ${\Lambda }_k(t)={\int }_0^t{\lambda }_k(u)du$. Treating the other cause as censored estimates the net survival $S_k^{\mathrm{net}}(t)=e^{-{\Lambda }_k(t)}$. The actual probability of death from cause $k$ by time $t$ is the cumulative incidence functionSince $S(u)\le e^{-{\Lambda }_k(u)}$, the net probability $1-S_k^{\mathrm{net}}(t)$ generally overstates the cumulative incidence function: death from the other cause prevents a later death of cause $k$. This discrepancy exists even for independent latent cause-specific failure times. Moreover, Kaplan–Meier estimates are not generally exactly unbiased in finite samples even for their proper target. The plotted complements are not unbiased estimators of $\mathbb{P}(T\le 5,D=k)$, and are not even consistent for that target in general.

For the desired probabilities, use the Aalen–Johansen estimator ${\hat{F}}_k(t)={\sum }_{u\le t}\hat{S}(u-)d_k(u)/n(u)$, with all-cause survival $\hat{S}(u)={\prod }_{v\le u}[1-(d_1(v)+d_2(v))/n(v)]$ and independent study censoring. Removing competing deaths from subsequent risk sets is correct for estimating cause-specific hazards; the error is interpreting the separate net survival complements as cumulative incidence functions. A hypothetical elimination-of-a-cause interpretation of net survival would require additional causal assumptions.