Bathtub Hazard

Bathtub Hazard in Survival Models After Risky Procedures

In health economic models a bathtub hazard commonly appears after a risky intervention such as surgery or transplantation, where the death rate is highest soon after the procedure. The shape matters because cost-effectiveness analyses usually need survival over a lifetime, and the hazard rate assumed for each period drives the life-years, QALYs and costs that a model produces. Single standard survival distributions struggle with a hazard that falls and then rises, and NICE Decision Support Unit guidance describes several alternatives. In the illustrative example further down, a constant hazard overstates mean survival by more than three years.

Three phases and the two clocks behind them

The bathtub label is best known from reliability engineering. The NIST engineering statistics handbook describes product failure rates that start high and fall rapidly, stay roughly constant for most of the useful life, and then rise with wear-out. In patient cohorts the same phases arise for clinical reasons.

Demiris, Lunn and Sharples, writing on cost-effectiveness studies in cardiothoracic transplantation, describe hazards that are high immediately after transplantation because of the procedure and associated immunological events, fall steeply to a period of relative stability, and later rise with ageing and chronic complications. They add that such hazards are common in surgical applications.

Demiris and colleagues read the shape as competing risks that act additively, with an early component tied to time since the procedure and a late component tied to attained age and long-term complications. Selection offers a second, compatible reading, since TSD 21 notes that cohort mortality can fall as sicker patients die and healthier patients survive, so a falling early hazard can reflect a changing case-mix as well as recovery from the procedure.

Why single standard distributions struggle to follow the curve

Every survival model rests on the link between the hazard and the survival function. A restriction on the shape of the hazard therefore becomes a restriction on projected survival, as the identities below show.

$$h(t) = \frac{f(t)}{S(t)}, \qquad S(t) = \exp\left(-\int_0^t h(u)\,du\right)$$

where $h(t)$ is the hazard at time $t$, $f(t)$ is the probability density of the event time, $S(t)$ is the probability of surviving beyond $t$, and the integral is the cumulative hazard up to $t$.

NICE DSU Technical Support Document (TSD) 14 lists the exponential, Weibull, Gompertz, log-logistic, log normal and generalised gamma models as the standard set that should be considered for extrapolating trial data. TSD 21 summarises what each can represent. The exponential model assumes a constant hazard, and the Weibull and Gompertz models assume a hazard that increases or decreases monotonically with no turning point. The Weibull hazard shows why.

$$h(t) = \lambda \gamma t^{\gamma - 1}$$

where $\lambda$ is the scale parameter, $\gamma$ is the shape parameter and $t$ is time; the hazard falls throughout when $\gamma < 1$, is constant when $\gamma = 1$ and rises throughout when $\gamma > 1$. The log-logistic and log normal models can produce a hazard that falls, or one that rises to a single peak and then falls, which is a unimodal hazard and the reverse of a bathtub. TSD 21 notes that the generalised gamma can also represent a hazard that first decreases and then increases, so it is the one standard model that can produce a simple bathtub.

Three practical problems remain even with the generalised gamma. Its three parameters govern the early fall and the late rise jointly, so the depth of the post-procedure peak constrains the slope of the projected rise. The late rise usually lies beyond trial follow-up, so the data say little about it. Clinical hazards also often have more than one turning point. The cancer trial example in TSD 21 rises after entry, falls as sicker patients die, and rises again in the very long term with age-related mortality, and TSD 21 concludes that none of the standard parametric models can accurately model survival with two or more turning points. A bathtub in a hazard plot is a signal to consider a more flexible model or a way of adding the late rise from outside the trial.

Modelling routes used in HTA

TSD 21 describes several families of models that can represent hazards with complex shapes, and each produces the bathtub in a different way. The choice matters most for the extrapolated period, because the late rise usually lies beyond trial follow-up.

Spline-based flexible parametric models

On the log cumulative hazard scale a Weibull model is a straight line in log time. A flexible parametric model replaces that line with a restricted cubic spline of log time, as TSD 21 sets out.

$$\log H(t) = s(\log t \mid \gamma, k_0)$$

where $H(t)$ is the cumulative hazard, $s(\cdot)$ is the spline function, $k_0$ is the vector of knot locations and $\gamma$ is the vector of spline coefficients. With enough knots the fitted hazard can fall after the procedure and flatten within the observed data. TSD 21 notes that projection beyond the data is governed by the linear term beyond the final knot, which on this scale is the same form as a Weibull, so a spline model does not by itself create an age-related rise that the trial did not observe.

Piecewise models

A piecewise model fits separate hazards to separate time intervals, so an early interval can carry the post-procedure peak and later intervals a lower or rising hazard. TSD 14 describes piecewise constant models, with exponential models fitted to different periods, as a simple way to model a variable hazard, while noting that they are less useful for the extrapolated part of the curve. TSD 21 adds that cut points chosen by eye can be arbitrary, that jumps at the cut points can look clinically implausible, and that the final segment, which drives the extrapolation, is often based on few patients.

Additive hazards and background mortality

The competing-risk reading leads to models in which the overall hazard is a sum of components. TSD 21 describes poly-hazard models of this kind and reports that Demiris and colleagues proposed Weibull components in order to capture a bathtub hazard. With two components the hazard is as follows.

$$h(t) = \lambda_1 \gamma_1 t^{\gamma_1 - 1} + \lambda_2 \gamma_2 t^{\gamma_2 - 1}$$

where $\lambda_1$ and $\lambda_2$ are scale parameters and $\gamma_1$ and $\gamma_2$ are shape parameters of the two components. With $\gamma_1 < 1$ the first term falls over time and with $\gamma_2 > 1$ the second rises, so their sum falls at first and rises later.

The relative survival or excess hazard framework uses the same additive idea but fixes one component from population data.

$$h(t) = h^{*}(t) + \lambda(t)$$

where $h(t)$ is the all-cause hazard, $h^{*}(t)$ is the background hazard taken from population mortality rates by age, sex and calendar year, and $\lambda(t)$ is the excess hazard attributed to the disease or procedure. When the excess hazard declines after the early period, the rising background hazard supplies the late arm of the bathtub. TSD 21 recommends incorporating background mortality because it helps avoid extremely implausible projections.

Mixture and cure models

A mixture cure model combines a cured fraction with an uncured fraction, and TSD 21 notes that combining their hazard functions lets cure models produce overall hazards with complex shapes. Other mixtures can also give a bathtub through selection, when a short-lived high-risk subgroup dominates the early hazard and dies off, leaving a group whose hazard rises with age, as in a Gompertz model. The choice of components matters, since Demiris and colleagues cite reliability work showing that a mixture of two Weibull distributions cannot give a bathtub-shaped hazard, whereas a sum of Weibull hazards can. TSD 21 advises extreme caution with mixture models, which often fail to converge.

Worked example with a three-month peak and a late rise

The figures in this example are illustrative. A cohort undergoes a procedure with a hazard of death of 0.40 per year for the first three months, 0.05 per year from three months to five years, 0.08 per year from five to fifteen years as ageing begins to dominate, and 0.20 per year after fifteen years. Within each interval the hazard is constant, so survival and expected life-years can be calculated exactly.

For an interval of length $L_j$ with constant hazard $h_j$, survival and expected life-years follow from the cumulative hazard.

$$S(t_j) = S(t_{j-1}) \exp(-h_j L_j), \qquad A_j = S(t_{j-1}) \frac{1 - \exp(-h_j L_j)}{h_j}$$

where $t_{j-1}$ and $t_j$ are the start and end of interval $j$, $S(t_j)$ is the proportion alive at the end of the interval and $A_j$ is the expected life-years lived in the interval per person entering the cohort. For the open final interval $A_j = S(t_{j-1}) / h_j$.

1. Mean survival under the bathtub hazard. Adding the last column gives 0.2379 + 3.8257 + 4.9117 + 1.6031 = 10.5784, about 10.58 life-years per person.

2. A constant hazard fitted to five years of follow-up. Suppose the trial follows every patient for five years and an exponential model is fitted. Its estimated hazard is the deaths divided by the person-years at risk.

$$\hat{\lambda} = \frac{1 - 0.7136}{0.2379 + 3.8257} = \frac{0.2864}{4.0636} = 0.0705$$

where $\hat{\lambda}$ is the fitted constant hazard per year, the numerator is the proportion who die within five years and the denominator is the person-years lived in that period. The implied mean survival is 1 / 0.07049 = 14.19 life-years.

3. Comparing the two. The constant hazard predicts five-year survival of 0.7030 against 0.7136 under the bathtub hazard, so a visual check against the five-year Kaplan-Meier curve could look acceptable. Yet it predicts three-month survival of 0.9825 instead of 0.9048, placing far too few deaths in the post-procedure period, and it misses the late rise entirely. Mean survival is overstated by 3.61 life-years (14.19 against 10.58), most of it after year fifteen.

Carrying the post-peak hazard of 0.05 forward would be worse still, giving 0.2379 + 0.9048 / 0.05 = 18.33 life-years. In a cost-effectiveness model the same errors misplace the early costs of deaths and complications, and the overstated tail inflates life-years and QALYs in every arm. The incremental gain grows too when a treatment effect is carried over the inflated tail, because each extra survivor is credited with too many future years.

Choices that change the extrapolated tail

A bathtub hazard puts most of the decision-relevant uncertainty in the late arm, which a trial rarely observes. Several choices therefore deserve explicit reporting.

The time scale of each component should be stated, since an early component on time since the procedure and a late component on attained age respond differently when the starting age changes. TSD 21 notes that national mortality rates by age, sex and calendar year may be used for background mortality but will understate expected mortality if patients have more comorbidity than the general population. Piecewise cut points need justification and sensitivity analysis. In a cohort state-transition model whose clock starts at the procedure, time-dependent transition probabilities with short early cycles are enough to keep the peak from being averaged over a whole year, with the hazard-to-probability conversion applied interval by interval; tunnel states are needed only when patients enter the post-procedure state at different times.

How NICE guidance frames the assessment

The NICE manual for technology appraisal and highly specialised technologies (PMG36, 2022, last updated 13 October 2026) states that standard parametric and more flexible approaches, such as spline-based and cure models, are available for extrapolating time-to-event data. It asks that the clinical plausibility of each model's underlying hazard function be routinely assessed and that uncertainty in the extrapolated portion of the hazard be explored.

TSD 21 recommends that fitted and extrapolated hazard and survival functions should always be presented, with expected general population hazard and survival plotted alongside to judge credibility. For a bathtub hazard these plots show whether a model reproduces the early peak and whether the projected hazard ever drops below population mortality at the same age. TSD 21 observes that in nearly all cases the mortality of people with the disease would not be expected to fall below that of the general population, so such a crossing usually signals an implausible tail.

Related hazard shapes

The bathtub is one of the time-varying hazards, contrasting with a constant hazard, with monotonic hazards and with its mirror image, the unimodal hazard. It is also distinct from non-proportional hazards between treatment groups. A treatment effect can vary over time while each group's hazard keeps a simple shape, and both groups can share a bathtub under proportional hazards.

Misreadings of the bathtub shape

The most frequent error is to choose among standard models by information criteria and extrapolate without inspecting the hazard. TSD 21 points out that such criteria describe fit within follow-up and say little about extrapolation, and the worked example shows a model matching five-year survival while misplacing early deaths and omitting the late rise.

Two further errors repeat points made above. One is treating a well-fitting spline model as settling the tail, which still needs background mortality or other external evidence, and the other is assuming that a mixture of two Weibull distributions can produce the shape. Finally, empirical hazard estimates at the end of follow-up rest on few patients, so an apparent upturn or plateau late in a trial may be noise rather than the start of the age-related arm.

Sources

• Demiris N, Lunn D, Sharples LD. Survival extrapolation using the poly-Weibull model. Statistical Methods in Medical Research. 2015;24(2):287-301. doi:10.1177/0962280211419645
• 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; 2011, last updated March 2013. Sections 2 and 3.2.
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Section 4.6.25.
• NIST/SEMATECH. e-Handbook of Statistical Methods. Section 8.1.2.4, "Bathtub" curve. NIST Handbook 151. Gaithersburg, MD: National Institute of Standards and Technology; 2012. doi:10.18434/M32189
• 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; 2020, updated version March 2022. Sections 2.1, 3.2, 3.5, 3.6, 3.7, 4.3, 6.2 and 7.