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An accelerated life family consists of survival distributions related by positive time scaling: $T_z$ has the distribution of $c_z{T}_0$. Therefore the survivor function is $S_z(t)=S_0(t/c_z)$ and, for densities, the hazard function is $h_z(t)=c_z^{-1}{h}_0(t/c_z)$. This need not be a constant hazard ratio.

Suppose an unobserved class has probabilities $\pi$ and $1-\pi$, with hazard functions $q$ and $q+r$, and set $Q(t)={\int }_0^{t}q$, $R(t)={\int }_0^{t}r$. For an independently right-censored observation $(x,v)$, its survival-parameter likelihood contribution isThis sums the two class-specific likelihoods before taking a logarithm. All hazards must be nonnegative. A common parameter-free censoring mechanism factors out; class-specific censoring instead alters the mixture weights and cannot silently be discarded.

// Target: probability-and-statistics.bigb

If entry is uniform on calendar interval $[{\tau }_a,{\tau }_b]$ and follow-up stops at ${\tau }_c$, the censoring duration is uniform on $[{\tau }_c-{\tau }_b,{\tau }_c-{\tau }_a]$. Its hazard function diverges at the longest possible follow-up. It is independent censoring when entry date is independent of prognosis after the required conditioning.

In a proportional hazards model $h(t\mid z)=h_0(t)e^{{\beta }^{T}z}$, the baseline survival function is $S_0(t)=\exp \{-{\int }_0^t{h}_0(u)du\}$, corresponding to the reference covariate vector $z=0$. Integrating the hazard function gives $S(t\mid z)=S_0(t{)}^{e^{{\beta }^{T}z}}$. Estimated hazard ratios alone therefore do not determine absolute survival probabilities: the baseline hazard or baseline survival function must also be estimated.

A cell-cycle age density counts biological cells per unit time spent in one specified phase of the cell cycle. Its integral over phase age is the population in that phase. Phase age resets on entry into the next phase, so it differs from chronological age since birth. A phase-exit hazard function supplies loss from the phase and incoming flux to the next phase.

A counting process records accumulated events. It is a nonnegative integer-valued Stochastic process $N_t$, usually adapted and càdlàg, with $N_0=0$ and nondecreasing sample paths. Nonexplosion means $N_t<\infty$ at every finite time. In a simple counting process, every jump is one; broader counting conventions also allow several events at the same instant.

A rate-$\lambda$ Poisson process has independent stationary increments, mean $\lambda t$ and variance $\lambda t$. A one-event process $N_t=1_{\{T\le t\}}$, with $T>0$, instead records whether its single event has occurred. It is a counting process without the Poisson process's unrestricted repeated arrivals.

The path $N_t$ is the observed count. A counting-process intensity describes its conditional event rate, and a compensator accumulates that predictable rate or, more generally, a predictable event measure. These objects are not the same: the count has integer jumps, whereas a continuous compensator can increase between observed events. In Survival analysis, the one-event convention naturally combines a Hazard function with an at-risk process.

A predictable intensity $\lambda (t)$ is the derivative of an absolutely continuous compensator of a counting process: $N(t)-{\int }_0^t\lambda (u)du$ is a local martingale. It represents the conditional event rate given the immediately preceding history. In survival analysis the individual intensity is an at-risk indicator multiplied by its hazard function.

Suppose $T\ge 0$ has an absolutely continuous probability distribution, with survival function $S(0)=1$. On an interval where $S(t)>0$, its cumulative hazard isHere $h$ is the hazard function; its integral is dimensionless because the units of rate and time cancel. The identity follows by integrating the derivative of $\log S$, so mere continuity of the survival function is not enough: the absolute continuity assumption supplies a probability density function. If $S$ reaches zero, interpret $-\log S$ as $+\infty$. The normalization $S(0)=1$ also matters; an initial probability mass at zero would require an initial term $-\log S(0)$.

Accumulating hazard functions adds on the $H$ scale and multiplies survival factors on the $S$ scale. For $S(a)>0$ and $b>a$,For instance, a cumulative hazard increment of $2$ gives event probability $1-e^{-2}\approx 0.864665$, rather than a probability of $2$. With $h(t)=0.2$ per year, $H(5)=1$ and $S(5)=e^{-1}\approx 0.367879$. With the increasing hazard function $h(t)=0.04t$ per year when $t$ is expressed in years, $H(t)=0.02t^2$, so $H(5)=0.5$ gives $S(5)=e^{-0.5}\approx 0.606531$. Its coefficient has units of inverse years squared.

The constant hazard function produces an exponential distribution; the increasing example is a Weibull distribution. Their survival functions both decrease, although one hazard function stays constant and the other increases. At year ten both have $H(10)=2$ and $S(10)=e^{-2}$ despite different earlier histories. The accumulation identity is described in Lu Tian's Stanford notes, Section 1.1.

For a discrete hazard $q_j$ at each possible event time $t_j$, the exact counterpart is $S(t)={\prod }_{t_j\le t}(1-q_j)$. Taking its negative logarithm sums $-\log (1-q_j)$, rather than $q_j$. Therefore exponentiating a Nelson–Aalen estimator does not give exactly the Kaplan–Meier estimator; small-jump comparison of cumulative hazard estimators quantifies the difference.

A cure model mixes an individual class with identically zero hazard function and a susceptible class with survivor function $S_{\ast }$. For $0\le \pi <1$, its representation as a proportional frailty model with $\mathbb{E}U=1$ is $U=0$ with probability $\pi$ and $U=1/(1-\pi )$ otherwise, with baseline hazard $h_0=(1-\pi )h_{\ast }$. The frailty law is a mixture of two Dirac measures, rather than an ordinary density. Its Laplace transform evaluated at $H_0$ gives the displayed survivor function. The limiting survival equals $\pi$ when $S_{\ast }(t)\to 0$; otherwise the plateau also includes susceptible individuals who never experience the event.

Let the survival time $T$ have an exponential distribution with rate $\lambda =0.2$ per year. Its hazard function is $h(t)=0.2$ per year, its cumulative hazard is $H(t)=0.2t$ for $t$ in years, and its survival function is $S(t)=e^{-0.2t}$. For a subject still event-free after five years, the conditional probability of an occurrence within the next two years isThe original population's probability of an occurrence between those same endpoints is instead $S(5)-S(7)=e^{-1}-e^{-1.4}\approx 0.121282$. Dividing by $S(5)\approx 0.367879$ recovers the conditional answer. The memoryless property makes the two-year conditional answer independent of the preceding five years.

For just $\Delta =0.01$ year, the exact conditional event probability is $1-e^{-0.002}\approx 0.001998$, close to the instantaneous approximation $h(5)\Delta =0.002$. For two years that approximation gives $0.4$, whereas the exact answer is about $0.329680$. At the single exact time five, the event probability is zero. These are three different quantities: an instantaneous hazard function with inverse-time units, a short-interval approximation, and an exact finite-interval probability.

Let a frailty random variable have a gamma distribution with mean one and variance $v>0$. Its shape and rate are both $1/v$. For conditional hazard function $U{a}^z{h}_0(t)$, $z\in \{0,1\}$, averaging the conditional survivor function gives $(1+v{a}^z{H}_0(t){)}^{-1/v}$. Differentiating its logarithm yields the population hazard function $a^z{h}_0(t)/(1+v{a}^z{H}_0(t))$ and hence the displayed hazard ratio. Writing $x=v{H}_0(t)$ gives $r-1=(a-1)/(1+ax)$. Therefore the population hazard ratio moves monotonically from $a$ towards one as exposure $x$ increases, even though the individual conditional hazard ratio remains $a$. The limit one requires $H_0(t)\to \infty$ for fixed positive $v$, or $v\to \infty$ for fixed positive $H_0(t)$; it is not a consequence of elapsed time alone. This is survival selection through preferential removal of larger frailties.

Germán Rodríguez works on statistical methods for population research. His Princeton teaching and research site connects generalized linear models, survival analysis and demographic questions about event timing. The site's biographical note identifies his training in biostatistics and his work in population research.^[ref]

His Survival Models by Germán Rodríguez explains how survival functions and hazard functions encode the same event-time distribution in different ways. With right censoring, an event observed at $t$ contributes density $f(t)$, whereas an observation ending without an event contributes $S(t)$. Writing the contribution as $S(t)h(t{)}^{\delta }$, with event indicator $\delta \in \{0,1\}$, makes that distinction explicit. These likelihood conventions let incomplete follow-up inform estimation without treating a censored time as an observed failure.^[ref]

A positive, exponentially increasing hazard function $h(t)=q{e}^{\beta t}$ gives the survivor functionIts mean is ${\beta }^{-1}{e}^{q/\beta }{E}_1(q/\beta )$, where $E_1$ is the exponential integral. Unlike the exponential distribution, its future mean waiting time is not the reciprocal of its current hazard function.

A positive Gumbel auxiliary function makes $\overline{F}(t+a(t)x)/\overline{F}(t)\to e^{-x}$ as $t$ approaches the right endpoint of a distribution, for every real $x$. Under the Von Mises conditions for extreme values, the reciprocal hazard function is a valid choice.

For a finite mixture with positive weights $p_i$ and rates ${\lambda }_i>0$, the survivor function is $S(t)={\sum }_i{p}_i{e}^{-{\lambda }_{i}t}$ and the hazard function is the surviving-population mean of the component rates. Write $w_i(t)=p_i{e}^{-{\lambda }_{i}t}/S(t)$. Differentiation gives $w_i^{\prime }=w_i(h-{\lambda }_i)$ and therefore $h^{\prime }=h^2-{\sum }_i{w}_i{\lambda }_i^2=-\mathrm{Var}(\Lambda \mid T>t)$. The hazard is decreasing, strictly so when distinct rates have positive weights. For finitely many rates it tends to the smallest rate with positive weight. This is survival selection even though each individual component has a constant hazard.

For a survival time with an absolutely continuous probability distribution, probability density function $f$, and survival function $S$, the hazard function iswhere $S(t)>0$, with the derivative identity holding almost everywhere. At points where the local limit exists, equivalentlyThe hazard function measures an instantaneous conditional rate. If time is in years, its units are inverse years, and its value can exceed one. A probability for a short interval is approximately $h(t)\Delta$, not $h(t)$; for an absolutely continuous probability distribution, the probability of an occurrence at the single exact time $t$ is zero. The discrete hazard, by contrast, is a conditional probability at a possible event time.

The factorization $f(t)=h(t)S(t)$ distinguishes a probability density function for the original population from a hazard function for those still event-free. With an exponential distribution of rate $0.2$ per year, $h(5)=0.2$ per year but $f(5)=0.2e^{-1}\approx 0.073576$ per year: the latter also accounts for the probability of reaching year five. The exponential survival risk over an interval example computes finite-horizon probabilities from this rate.

At distinct observed event times $t_1<t_2<\cdots$, let $n_j$ be the risk set size immediately before $t_j$, and let $d_j$ count events at $t_j$. The Kaplan–Meier estimator of the survival function isThe empty product is one. Each factor estimates a conditional survival probability among those still observed and event-free; their product joins the successive steps. This is nonparametric statistics: no specified shape for the survival distribution is imposed. Use independent censoring and the appropriate risk sets to interpret the result as an estimate for the intended population.

The Kaplan–Meier estimator is a right-continuous function with downward jumps at event times. Right censoring changes later denominators without causing a jump. A tick on the curve marks right censoring, rather than a failure. For a tied group of $d_j$ events, use the one factor $(n_j-d_j)/n_j$; subjects right-censored at that recorded time remain in $n_j$ for this factor. The fraction $d_j/n_j$ estimates a discrete hazard, not a continuous hazard function in inverse-time units. With no censoring, the factors telescope to the fraction of the original cohort whose observed survival times exceed $t$, the complement of its empirical distribution function.

Edward L. Kaplan and Paul Meier published ^[ref]. Their paper addresses lifetime estimation when observation ends before failure and explicitly calls attention to the assumption of independence between lifetime and potential observation loss. The date 1958 is the original publication year; the publisher's later online-posting date is not the discovery date.

In his ^[ref], Kaplan traces Meier's interest to cancer-duration data in 1952 and his own to the lifetimes of vacuum tubes in undersea telephone-cable equipment a year later. The journal encouraged them to combine their manuscripts. The shared mathematical question was how to estimate a survival function when some lifetimes are still running or disappear from observation.

For independently left-truncated and right-censored survival data, let $E_i$ and $Y_i$ be entry and exit times on the same clock. At each distinct event time $t_j$, the risk set is $R_j=\{i:E_i<t_j\le Y_i\}$, with size $r_j$, and $d_j$ events occur. The Kaplan–Meier estimator isBefore entry a subject contributes neither an event nor time at risk. Thus adding observed entrants can increase successive risk sets; pretending that all subjects were observed from time zero distorts the estimated hazard function. The estimator identifies the survival function over ages covered by observation; normalization from birth requires observation arbitrarily close to time zero or additional knowledge of earlier survival.

For a nonnegative continuous event time with survival function S, hazard function h and H the cumulative hazard, split the expectation at c. The censored contribution is S(c)H(c). The event contribution is ${\int }_0^{c}H(t)f(t)dt=1-S(c)-S(c)H(c)$ by integration by parts, since $dH=hdt$ and $f=hS$. Adding gives the displayed identity. Linearity extends it to the expected observed event count in a cohort, without requiring independence between subjects.

The quantities $n_i(\tau ,t)$ are cell-cycle age densities: $n_i(\tau ,t)d\tau$ counts biological cells whose time since entering phase $i$ lies in $[\tau ,\tau +d\tau ]$. Here $\tau$ is time spent in the current phase, rather than time since birth. In this two-stage approximation to the cell cycle, a biological cell passes from G1 phase to G2 phase, then completes cell division and produces two daughters entering G1 phase.

The time derivative measures change at fixed phase age. The age derivative transports biological cells towards larger $\tau$ at unit speed, because $d\tau /dt=1$. Thus each equation is an age-structured population equation. The nonnegative hazard function ${\Gamma }_i(\tau )$ means that, conditional on still being in phase $i$ at age $\tau$, the probability of leaving during the next $dt$ is ${\Gamma }_i(\tau )dt+o(dt)$. Its product with $n_i$ is the loss from that phase-age class; it is not an additional death term.

The boundary condition at $\tau =0$ specifies the incoming flux. Every G1 phase exit contributes one entrant to G2 phase, whereas every G2 phase exit contributes two entrants to G1 phase. The initial conditions specify the phase-age composition at $t=0$. Consequently integration over phase age gives the useful population balanceEach completed cell division increases the total number of biological cells by one.