Cumulative Hazard

Cumulative Hazard in Survival Analysis and Markov Models

Several steps that turn trial survival data into model inputs, from checking the shape of a curve to setting cycle probabilities, work on the cumulative hazard scale, written $H(t)$, even when a report only shows survival. The cumulative hazard is the running total of the hazard rate and maps directly onto survival. It is estimated from censored data with the Nelson-Aalen estimator, and log-cumulative hazard plots help to choose parametric curves and judge proportional hazards, as described in Technical Support Document 14 (TSD 14) of the NICE Decision Support Unit. Differences in $H(t)$ become transition probabilities for a Markov model, and $H(t)$ itself is not a probability and can exceed 1.

How the cumulative hazard connects hazard and survival

The hazard describes the event rate at each instant among people still event free, and the cumulative hazard is its running total from the time origin. Because survival depends on the whole history of the hazard, the cumulative hazard is the quantity that maps directly onto the survival function. For a continuous event time the relationship is as follows.

$$H(t) = \int_0^t h(u)\,du, \qquad S(t) = \exp\{-H(t)\}, \qquad H(t) = -\log S(t)$$

where $H(t)$ is the cumulative hazard at time $t$, $h(u)$ is the hazard at time $u$, $S(t)$ is the probability of being event free beyond $t$, and $\log$ is the natural logarithm. The hazard is the slope of $H(t)$, so a cumulative hazard rising in a straight line implies a constant hazard, one that bends upwards implies a rising hazard, and one that flattens implies a falling hazard. For the exponential model with constant hazard $\lambda$, the cumulative hazard is simply $\lambda t$.

$H(t)$ starts at 0, never falls and has no upper limit, growing without bound as survival approaches zero. It passes 1 exactly when survival falls below $\exp(-1)$, about 0.368. Only when $H(t)$ is small is it close to the probability of having had the event, ${1 - S(t)}$.

Estimating the cumulative hazard from censored trial data

Trial data arrive as event and censoring times, so the cumulative hazard is first estimated without assuming any distribution. The Nelson-Aalen estimator adds, at each event time, the number of events divided by the number still at risk. Stata documents the estimator and its variance as follows.

$$\hat{H}(t) = \sum_{j:\ t_j \le t} \frac{d_j}{n_j}, \qquad \widehat{\mathrm{Var}}\{\hat{H}(t)\} = \sum_{j:\ t_j \le t} \frac{d_j}{n_j^2}$$

where $\hat{H}(t)$ is the estimated cumulative hazard at time $t$, $t_j$ are the distinct event times, $d_j$ is the number of events at $t_j$ and $n_j$ is the number at risk just before $t_j$. Censored patients leave the risk set without adding a step, which is how censoring is handled; this is valid only if censoring is non-informative, meaning that patients who are censored have the same risk of the event as those who remain under follow-up, an assumption TSD 14 states for the standard methods. Stata draws the estimate with sts graph, cumhaz; in R, the survfit function of the survival package returns it, offers the Fleming-Harrington correction for tied event times, and can derive survival from the cumulative hazard.

The estimator is closely related to minus the log of the Kaplan-Meier estimator. The two agree when each step is small but part company when few patients remain. One death among two at risk adds 0.5 to the Nelson-Aalen estimate but 0.693 to minus the log of the Kaplan-Meier estimate. A worked Nelson-Aalen calculation appears on the Cox-Snell residual page.

Log-cumulative hazard plots for choosing survival curves

TSD 14, written by Latimer in 2011 and last updated in March 2013, sets out how parametric survival models should be chosen when patient-level trial data are available. Its first step is a log-cumulative hazard plot (or suitable residual plots, such as the Cox-Snell plot mentioned above), which shows whether observed hazards look constant, monotonic or non-monotonic, whether proportional hazards is reasonable, and where the hazard changes. The standard plot shows the log of minus the log of survival against log time, which is $\log H(t)$ against $\log t$.

The plot works because the Weibull model becomes a straight line on this scale. NICE DSU TSD 21 writes survival as $S(t) = \exp(-\lambda t^{\gamma})$, which gives the following log cumulative hazard.

$$\log H(t) = \log\{-\log S(t)\} = \log \lambda + \gamma \log t$$

where $\lambda$ is the Weibull scale parameter and $\gamma$ its shape parameter, so the intercept is $\log \lambda$ and the gradient is $\gamma$. A straight line suggests a Weibull model, and a straight line with gradient 1 suggests an exponential model, since $\gamma = 1$ gives a constant hazard. Under proportional hazards one arm's hazard is a fixed multiple of the other's, so its cumulative hazard is the same multiple and the two lines are parallel, separated vertically by the log of the hazard ratio.

TSD 14 turns these readings into a model selection sequence. If the plots are roughly straight, those models are fitted and assessed further; if the arms' plots are parallel, proportional hazards models are considered, and if not, each arm is fitted separately; if the plots are not straight, piecewise, time-varying or more flexible models are considered. Variations on the plot test the Gompertz, log-normal and log-logistic distributions. In its illustrative example, a gradient below 1 after about five weeks led TSD 14 to judge an exponential model unlikely to suit one arm. Its review of past NICE appraisals found these plots underused, and it described the appraisal of pemetrexed for lung cancer (TA181), in which the assessment group saw each arm's cumulative hazard settle into a steady linear rise at some point after the end of treatment, a sign of a constant risk of death, and combined the area under the Kaplan-Meier curve with the area under an exponential model fitted to the tail of the Kaplan-Meier curve.

The same scale underlies flexible parametric models. The Royston-Parmar model replaces the straight line $\log \lambda + \gamma \log t$ with a spline in log time, and with no internal knots it reduces to the Weibull model. TSD 21 notes that beyond the final knot the log cumulative hazard is again linear in log time, so the extrapolated tail is similar to a Weibull. TSD 14 adds a limit that applies to all these checks, namely that they describe fit to the observed data only and say nothing about whether the extrapolated portion of a curve is plausible.

Turning cumulative hazards into transition probabilities

A Markov cohort model needs, for each cycle, the probability that a person event free at the start has the event by the end. Because survival is the exponential of minus the cumulative hazard, that probability depends only on how much the cumulative hazard rises across the cycle. The conversion is as follows.

$$p(t) = 1 - \frac{S(t)}{S(t-\Delta)} = 1 - \exp\{-[H(t) - H(t-\Delta)]\}$$

where $p(t)$ is the probability of the event during the cycle ending at time $t$, $\Delta$ is the cycle length, and $H(t) - H(t-\Delta)$ is the cumulative hazard accumulated within the cycle. Time $t$ is measured from the origin of the fitted curve, such as randomisation, so in a cohort model it is the time since model start at each cycle end. When risk depends instead on time spent in a state, the clock restarts on entry and the model needs tunnel states; the DARTH tutorial on time-dependent cohort models by Alarid-Escudero and colleagues takes this route, using differences in a Weibull cumulative hazard across tunnel cycles. With a rising hazard the cumulative hazard climbs faster in later cycles, so the same fixed cycle length carries a larger probability as time passes, as the worked example below shows.

For a single exit the formula is exact. When a state has several competing exits, a cause-specific cumulative hazard cannot be converted on its own through ${1 - \exp(-H)}$, because people removed by another exit cannot go on to have the event; the competing risks and transition probability pages set out the joint calculation. Under proportional hazards the treated cumulative hazard is the hazard ratio times the control one, so the treated cycle probability is ${1 - (1 - p_C)^{\mathrm{HR}}}$, where $p_C$ is the control cycle probability. TSD 14 cautions that this hazard ratio should come from the same parametric model, fitted with treatment as a covariate, and not from a Cox model or a different distribution.

Parameterisation also needs checking before a fitted curve is moved into a spreadsheet. TSD 21 writes the Weibull cumulative hazard as $\lambda t^{\gamma}$, whereas the DARTH tutorial writes $(\lambda t)^{\gamma}$, so the two scale parameters differ for the same curve, as the parametric survival model page notes across software.

Worked example of an illustrative Weibull curve in six annual cycles

The numbers in this example are illustrative and do not come from a real trial. Suppose a Weibull curve fitted to overall survival in a control arm has cumulative hazard $H(t) = 0.10\, t^{1.5}$, with $t$ in years, and a cohort model uses annual cycles. For year 4, for instance, the cumulative hazard is as follows.

$$H(4) = 0.10 \times 4^{1.5} = 0.10 \times 8 = 0.8$$

where $H(4)$ is the cumulative hazard at the end of year 4, so survival to year 4 is $\exp(-0.8)$, which is 0.4493. The cycle probability for year 3 uses the rise in the cumulative hazard across that year, as follows.

$$H(3) - H(2) = 0.5196 - 0.2828 = 0.2368$$

where $H(3)$ and $H(2)$ are the cumulative hazards at the end of years 3 and 2, so the probability of death in year 3 among those alive at its start is ${1 - \exp(-0.2368)}$, which is 0.2108. The table repeats the steps for every cycle.

1. Check the trace. Multiplying the six values of ${1 - p(t)}$ gives 0.2300, the survival read directly from the curve at year 6, so the cohort model reproduces the fitted curve exactly.

2. Read the cumulative hazard correctly. $H(t)$ passes 1 between years 4 and 5, when survival falls below 0.368. The value of 1.118 at year 5 is a cumulative hazard, not a probability of 112%.

3. See what the shortcuts would do. Using each increment of $H(t)$ directly as the cycle probability gives year 6 survival of 0.1786 instead of 0.2300, so deaths are overstated. Using the last column as a per-cycle probability is far worse, because it applies the risk accumulated since time zero to a single cycle, giving 0.2464 in year 2 instead of 0.1671.

4. Check the shape on the log scale. $\log H(1)$ is $-2.3026$ and $\log H(4)$ is $-0.2231$, so the gradient against log time is as follows.

$$\frac{-0.2231 - (-2.3026)}{1.3863} = \frac{2.0795}{1.3863} = 1.50$$

where 1.3863 is $\log 4 - \log 1$. A straight line with gradient 1.5, above 1, is the signature of a rising hazard; an exponential curve would give a gradient of 1.

5. Apply a treatment effect. With an illustrative hazard ratio of 0.70 under proportional hazards, the treated cumulative hazard rises by ${0.70 \times 0.2368}$ in year 3, about 0.1658, so the treated cycle probability is ${1 - \exp(-0.1658)}$, about 0.1527. On the log-cumulative hazard plot the treated line sits parallel to the control line, lower by about 0.357, the log of 0.70.

Cumulative hazard, survival and cumulative incidence compared

Three quantities describe accumulated risk, and only two of them are probabilities. Confusing them is a common source of error when survival evidence is converted for models.

Misreadings that distort survival inputs

Treating the cumulative hazard as a probability, or its steps as hazards, is a common source of error. The first two misreadings below bias results in a known direction; the third mistakes noise for signal.

Reading $H(t)$ as the proportion who have had the event overstates that proportion; values above 1 make the error obvious, but a value of 0.5 misleads quietly, since the matching probability is about 0.39. Using increments of $H(t)$ as cycle probabilities overstates events in every cycle, and the gap grows with cycle length and risk. The steps of a Nelson-Aalen estimate are not hazard estimates either, and Stata's documentation notes that the increments do not themselves estimate the hazard, which it obtains by smoothing them, and large steps at the right of the curve often reflect a handful of patients at risk rather than a change in hazard. A straight log-cumulative hazard plot within follow-up also leaves the extrapolated tail untested, for the reason TSD 14 gives above.

Sources

• Alarid-Escudero F, Krijkamp E, Enns EA, Yang A, Hunink MGM, Pechlivanoglou P, Jalal H. A tutorial on time-dependent cohort state-transition models in R using a cost-effectiveness analysis example. Medical Decision Making. 2023;43(1):21-41. https://doi.org/10.1177/0272989X221121747 (accepted manuscript checked: https://pmc.ncbi.nlm.nih.gov/articles/PMC9844995/, section 4.2, equations 2 to 6)
• Collett D. Modelling Survival Data in Medical Research. 3rd ed. Boca Raton: CRC Press; 2015.
• Jackson C. flexsurv: Flexible parametric survival and multi-state models. R package version 2.3.2. 2024. Help page flexsurvspline. https://search.r-project.org/CRAN/refmans/flexsurv/html/flexsurvspline.html
• Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011, last updated March 2013. Sections 2.9, 3.2, 3.5, 4.1.5, 4.4 and 6.1. https://doi.org/10.15131/shef.data.33745843
• Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020 (updated March 2022). Sections 3.2.2 and 4.3.1. https://doi.org/10.15131/shef.data.33849118
• StataCorp. Stata Survival Analysis Reference Manual, Release 19. Entry sts. College Station, TX: Stata Press; 2025. https://www.stata.com/manuals/ststs.pdf
• Therneau TM. survival: Survival analysis. R package version 3.7-0. 2024. Help page survfit.formula. https://search.r-project.org/CRAN/refmans/survival/html/survfit.formula.html