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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.

Write $S(t)=\mathbb{P}(T>t)$ for the survival function; this is the function denoted $F_T$ in the survival notation. For a proper continuous event-time distribution with hazard function $h$, its cumulative hazard function is $H(t)={\int }_0^{t}h(s)ds=-\log S(t)$. The probability integral transform makes $S(T)$ uniform on $(0,1)$, including when $S$ has flat intervals: those intervals have zero event probability. ConsequentlyThus the cumulative hazard probability transformation gives $\boxed{H(T)\sim \mathrm{Exponential}(1)}$. No strict monotonicity of $H$ is needed on intervals that carry no event mass. The usual proper continuous survival law is essential; an atom of subjects who never fail would instead require separate treatment of the mass at infinity.

For a continuous nonnegative event time with hazard function $h$, the cumulative hazard function isIts derivative is the instantaneous hazard wherever the derivative exists.

Insert $f_i=h_i{F}_i$ and $F_i=e^{-H_i}$ into the survival likelihood. Each individual's log contribution simplifies toThereforeFailures contribute a log hazard function term as well as exposure, whereas both failures and censored subjects contribute the negative cumulative hazard function over their observed follow-up.

For individual $i$, let ${\pi }_i={\pi }_i(\psi )$, $q_i(t)$ be the known baseline hazard function, and $r_i(t;\varphi )$ the extra hazard. PutThe class-specific survival functions are $e^{-Q_i(t)}$ and $e^{-Q_i(t)-R_i(t)}$. Consequently the unconditional survivor and failure density areMixture probabilities must be summed before taking a logarithm; a weighted average of class log-likelihoods would be a different, complete-data calculation. Substitution into the right-censored survival likelihood givesEquivalently, this additive excess-hazard mixture likelihood has contributionHere $q_i$ and $q_i+r_i$ must be nonnegative hazards. The factor $-Q_i(x_i)$ can be dropped when maximizing over $\varphi ,\psi$ because $q_i$ is known. Factoring out censoring presumes a common parameter-free censoring mechanism independent of failure and latent class, conditional on the observed covariates; unmodelled class-specific censoring would change these mixture contributions.

For $t\ge 0$ and $\theta >0$, the exponential distribution has probability density function $f(t)=\theta e^{-\theta t}$. Integrating the probability density function gives the survivor function $S(t)=P(T>t)=e^{-\theta t}$. The hazard function divides instantaneous failure density by the probability of having survived, and the integrated hazard accumulates this rate. ThusThe probability density function is zero for $t<0$, and the survivor function is one there.

Use the paper's notation $F$ for the survivor function, not for a cumulative distribution function. For a nonnegative continuous event time,The hazard function is defined where survival is positive, and the last expression is the integrated hazard. These definitions assume the usual absolutely continuous survival distribution so that the displayed density and hazard derivatives exist almost everywhere.

Normalize the probability density function using $u=t^c$:The corresponding Weibull distribution has survival functionSince the distribution is continuous, the choices $>$ and $\ge$ have the same probabilities. The conditional probability is consequentlyFor $s,t>0$ and $c>1$, the supplied inequality, applied to $s/t$, gives $(s+t{)}^c>t^c+s^c$. Taking exponentials yieldsThis expresses the increasing hazard function $h_c(t)=c{t}^{c-1}$: surviving to a later age shortens the remaining lifetime in this conditional distribution. At $c=1$ equality holds, as expected from the memoryless property of the exponential distribution.

The paper uses $F$ for the survivor function, rather than for the cumulative distribution function. For a nonnegative survival time $T$ with an absolutely continuous probability distribution, defineWhere $F(t)>0$, the hazard function is the instantaneous event rate conditional on survival:The integrated hazard is $H(t)={\int }_0^{t}h(u)du$. Since $h=-F^{\prime }/F$, integration gives $H(t)=\log F(0)-\log F(t)$. With the usual time origin and no event at zero, $F(0)=1$, soConversely, differentiating $F=e^{-H}$ gives $F^{\prime }=-hF$, so these relationships determine each of the survivor function, hazard function and integrated hazard from either of the others. A positive limiting survivor function is possible if the integrated hazard remains finite; a proper finite event time instead has $F(t)\to 0$.

Differentiating the log-likelihood again givesThus the Observed Fisher information for the rate is proportional to the total number of observed events. At a positive interior maximum-likelihood estimator, the large-sample variance estimate and standard error arePrecision is therefore driven by the number of observed events rather than by the number recruited alone. Heavy right censoring can leave a large study with little information about its hazard function. Under the regularity conditions for the Fisher information, the expected information is $\mathbb{E}[D]/{\theta }^2$.

A right-censored observation is not wholly uninformative: its exposure contributes to $E$, changes the maximum-likelihood estimator, and affects the expected number of events under a sampling design. Its contribution $-\theta x_i$ happens to have zero second derivative in this rate parameterization. For example, if no events occur, the likelihood still declines as $e^{-\theta E}$ and disfavors large rates, despite its zero curvature. The usual interior variance approximation is then invalid.

In a frailty model, an individual's unobserved susceptibility to an event is represented by a nonnegative frailty random variable multiplying that individual's hazard function. Conditional on its value, subjects with the same measured covariates can therefore have different event rates. The frailty is a persistent latent characteristic of the individual, not newly sampled noise at each time.

Here the conditional hazard function is $u{e}^{\beta z}{h}_0(t)$, with baseline hazard $h_0$. A multiplier $u>1$ raises susceptibility and $u<1$ lowers it. Normalization $\mathbb{E}U=1$ fixes the frailty scale: without it, multiplying every frailty by a constant and dividing the baseline hazard by that constant would give the same individual hazards. The variance of $U$ measures unobserved heterogeneity. An atom at $U=0$, if allowed, would give a permanently zero conditional hazard.

The hazard ratio between groups is $e^{\beta }$ for individuals compared at the same frailty. It need not equal the population hazard ratio, because survival selection changes the frailty distribution among those still alive or event-free.

Condition on a frailty random variable value $U=u$. Integrating the conditional hazard function gives $u{e}^{\beta z}{H}_0(t)$, so the conditional survivor function isAverage this conditional probability over the common frailty density $g$:By definition the Laplace transform of $g$ is $\tilde{g}(s)={\int }_0^{\infty }{e}^{-su}g(u)du$. HenceThis averages survivor functions, not hazards. Differentiating its logarithm, when justified, givesThus the population hazard function uses the mean frailty among current survivors, explaining why a conditional proportional hazards model may cease to be proportional after frailty is integrated out.