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Let $A_i(s)$ be age in years at follow-up time $s$ and let $G_i\in \{0,1\}$ indicate the genetic risk factor. An age-dependent log-linear transition intensity model isHere $e^{{\theta }_0}$ is the onset baseline hazard for a 50-year-old without the factor; $e^{{\theta }_G}$ is the genetic hazard ratio, and $e^{{\theta }_A}$ is the hazard ratio per additional year of age. A smooth function of age could replace the linear age term if a constant age hazard ratio were unsuitable.

Under the stated constant multiplicative hazard ratios, the requested extrapolation is therefore the baseline rate times both multipliers:

A proportional hazards model assigns individual $i$ the hazard function $h_i(t)=h_0(t){\varphi }_i$, where $h_0$ is the baseline hazard for multiplier one and ${\varphi }_i>0$ is a time-constant hazard multiplier. In a Cox proportional-hazards model, ${\varphi }_i=e^{x_i^T\beta }$ for a covariate vector $x_i$ and coefficient vector $\beta$; ratios $h_i(t)/h_k(t)={\varphi }_i/{\varphi }_k$ are constant whenever the baseline is nonzero.

Assume independent individuals, independent censoring conditional on the covariates, and no tied events. Immediately before an event at $t$, condition on its observed risk set $R(t)$ and one event in a small time interval. The individual event probability is $h_0(t){\varphi }_{i}dt+o(dt)$, while the total is $h_0(t){\sum }_{k\in R(t)}{\varphi }_{k}dt+o(dt)$. Their limiting ratio is ${\varphi }_i/{\sum }_{k\in R(t)}{\varphi }_k$. Multiplication over observed events gives the Cox partial likelihoodThe unspecified baseline hazard cancels. This is the usual successive conditional event contribution defining the partial likelihood, not the full event-time likelihood conditioned simultaneously on all event times.

For the four individuals the successive event risk sets are $\{a,b,c,d\}$, $\{c,d\}$ and $\{d\}$. Individual $b$ leaves at its right censoring time and contributes no event numerator. ConsequentlyThe last event contributes one and carries no further relative-risk information.

The ordinary Nelson–Aalen estimator adds $d_j/r_j$ at event time $t_j$. Under a fitted proportional hazards model, replace the number at risk by the sum of fitted hazard multipliers:This is the Breslow estimator of the integrated baseline hazard, using fitted regression coefficients obtained by maximizing the Cox partial likelihood. The denominator arises because the total event intensity is $h_0(t){\sum }_{i\in R(t)}{\varphi }_i$; with every multiplier equal to one it reduces to the Nelson–Aalen estimator.

The estimator is right-continuous, so ${\hat{H}}_0(t_a)$ already includes the event at $t_a$. Subtracting it leaves only the events at $t_c$ and $t_d$, whose risk sets are $\{c,d\}$ and $\{d\}$. HenceThere is no increment at the censoring time $t_b$. If the lower endpoint had instead been $t_a-$, an additional $1/({\hat{\varphi }}_a+{\hat{\varphi }}_b+{\hat{\varphi }}_c+{\hat{\varphi }}_d)$ would be present. The final singleton event can increase the baseline hazard estimate even though its partial likelihood contribution is one.

A frailty model represents unobserved heterogeneity using a nonnegative frailty random variable $U$ that multiplies an individual's baseline hazard. With $H_0(t)={\int }_0^t{h}_0(s)ds$, the individual and population survivor functions areThus the population survivor function is the Laplace transform of the frailty distribution at $H_0(t)$; at points where differentiation is justified, $\overline{h}(t)=h_0(t)\mathbb{E}[U\mid T>t]$. The last identity follows by differentiating the integral and dividing $-{\overline{S}}^{\prime }$ by $\overline{S}$.

Here $g(u)=e^{-u}$ for $u\ge 0$, $\theta >0$, and $H_0(t)=\theta t^2/2$. Direct integration givesThis is population hazard under exponential frailty. For every fixed $u>0$ the individual hazard function $u\theta t$ increases linearly. In contrast,so the population hazard increases up to $t=\sqrt{2/\theta }$, then decreases, tending to zero like $2/t$. More vulnerable individuals leave the risk set earlier, leaving survivors with smaller frailties. This population decline is selection, not a decline in any fixed individual's hazard. If $\theta =0$, survival is identically one and both hazards vanish; the positive-rate comparison is then inapplicable.

Let $R_i=\{j:x_j\ge x_i\}$ be the risk set at an observed event time $x_i$. In the Cox proportional-hazards model, conditional on one member of this set failing then and on the preceding history, the probability that it is individual $i$ isThe baseline hazard cancels. Multiplying these conditional contributions gives the Cox partial likelihoodThe no-ties assumption avoids the need for a tied-event approximation. This construction assumes independent censoring conditional on the modelled covariates. The printed phrase calling $z$ a scalar parameter is read as a scalar observed covariate $z_i$; $\beta$ is the unknown regression parameter.

In the Cox proportional-hazards model, replace the number at risk in the Nelson–Aalen estimator by the sum of relative hazards:The total event counting process has conditional rate $W_{\beta }(u)h_0(u)$, so the estimated baseline hazard increment is $dN(u)/W_{\hat{\beta }}(u)$. Integrating gives the Breslow estimatorWhen $\hat{\beta }=0$, all relative hazards are 1 and this reduces to the Nelson–Aalen estimator. The denominator is positive at each observed event time.

Insert the Breslow estimator into the fitted cumulative intensities and interchange the finite sums:Every observed event contributes exactly one after summing its fitted contribution over its risk set. ThereforeThis identity is algebraic and holds for any finite coefficient used consistently in the relative hazards and the Breslow estimator; it does not require the score equation to be solved exactly. It can fail if the baseline hazard is estimated or smoothed by a different procedure.

In a proportional hazards model, individual $i$ hasThe exponential term is the hazard multiplier relative to the baseline hazard $h_0$. With $h_0$ unspecified, the Cox partial likelihood multiplies, over event times, the failing subject's multiplier divided by the sum of multipliers in the current risk set. After estimating $\beta$, the Breslow estimator is

A Stratified Cox model uses a separate baseline hazard $h_{0s}$ for each stratum but a common $\beta$. Its partial likelihood is the product of within-stratum partial likelihoods, and a separate integrated baseline hazard is estimated in each stratum.

In a matched pair, the only informative comparison occurs while both members remain at risk. If the exposed member fails first, the pair contributes $e^{\beta }/(1+e^{\beta })$; if the unexposed member fails first, it contributes $1/(1+e^{\beta })$. A censoring as the first recorded time gives no informative failure comparison, and any later one-person risk set contributes one. With only two observations per stratum, each stratum supplies almost no information about its arbitrary baseline hazard, so a useful common integrated baseline-hazard estimate is generally unavailable.

When a Cox proportional-hazards model uses the Breslow estimator, the sum of fitted martingale residuals is zero. Each estimated baseline hazard jump redistributes one observed event across its risk set, so summing cumulative fitted intensities equals summing observed event counts. The identity holds for any finite regression coefficient used consistently in both calculations.