Survival Regression

Survival Regression in Health Economic Evaluation

Survival regression is the branch of regression analysis in which the outcome is the time until an event, such as death or disease progression, and follow-up usually ends for some patients before the event is seen. Each explanatory variable, such as treatment arm or age, receives a coefficient that says how it changes the event rate or stretches the time to the event. In health economics these models supply the adjusted treatment effects and the fitted survival curves from which a cost-effectiveness model derives life-years and QALYs, usually over a horizon far longer than the trial. The Cox model is the best known member, but lifetime extrapolation in health technology assessment normally needs a fully parametric one.

Why ordinary regression cannot be used on censored event times

A trial that reports overall survival at a data cut has a mixture of observed death times and censored times, for which only a lower limit on survival is known. Fitting linear regression to the recorded times would treat a patient censored at 8 months as if death had occurred then, and fitting logistic regression to death by a fixed date would discard the timing of events. Cox's 1972 paper set the problem as censored failure times with explanatory variables recorded for each individual.

Parametric survival regression handles censoring through the likelihood. Each observed event contributes its density and each censored observation contributes the probability of surviving beyond its censoring time.

$$L(\theta)=\prod_{i=1}^{n} f(t_i\mid x_i;\theta)^{\delta_i}\,S(t_i\mid x_i;\theta)^{1-\delta_i}=\prod_{i=1}^{n} h(t_i\mid x_i;\theta)^{\delta_i}\,S(t_i\mid x_i;\theta)$$

where $t_i$ is the observed time for patient $i$, $\delta_i$ is 1 if the event was observed and 0 if the time is censored, $x_i$ is the patient's covariate vector, $\theta$ holds the regression coefficients and distribution parameters, $f$ is the density of event times, $S$ is the survival function and $h=f/S$ is the hazard. The parameters are found by maximum likelihood estimation. The construction is valid only if censoring is uninformative, which Bradburn and colleagues state as a standing assumption of the models they describe.

Two ways a covariate can act on survival

The models differ mainly in where the covariates enter. Bradburn and colleagues divide them into proportional hazards approaches, which include the semi-parametric Cox model and fully parametric models, and accelerated failure time models.

In a proportional hazards model the covariates multiply the hazard by a constant factor at every time.

$$h(t\mid x)=h_0(t)\exp(\beta_1x_1+\dots+\beta_px_p)$$

where $h(t\mid x)$ is the hazard rate at time $t$ for a patient with covariates $x_1,\dots,x_p$, $h_0(t)$ is the baseline hazard when every covariate equals zero, and $\beta_j$ is the coefficient for covariate $x_j$, so that $\exp(\beta_j)$ is its hazard ratio. The Cox proportional hazards model leaves $h_0(t)$ unspecified and estimates it nonparametrically, while a parametric model gives it a fixed form such as exponential, Weibull or Gompertz. Bradburn and colleagues note that hazard ratios have the same interpretation from either.

In an accelerated failure time model the covariates act on the time scale.

$$\log T=\beta_0+\beta_1x_1+\dots+\beta_px_p+\varepsilon$$

where $T$ is the time to the event, $\beta_0$ is an intercept, $\beta_j$ are coefficients and $\varepsilon$ is a residual term whose distribution defines the model. The quantity $\exp(\beta_j)$ is a time ratio, and a value above 1 prolongs the time to the event. Log-logistic, log-normal and generalised gamma models belong to this family.

Bradburn and colleagues state that accelerated failure time (AFT) and proportional hazards (PH) models are the same when survival times follow a Weibull distribution. For a Weibull with survival $S(t)=\exp(-\lambda t^{p})$, multiplying the hazard by $\mathrm{HR}$ gives $\exp(-\lambda\,\mathrm{HR}\,t^{p})$, which equals $S(t/\mathrm{TR})$ when

$$\mathrm{TR}=\mathrm{HR}^{-1/p}$$

where $\lambda$ is the Weibull scale parameter, $p$ is its shape parameter, $\mathrm{HR}$ is the hazard ratio and $\mathrm{TR}$ is the time ratio. A fitted Weibull regression can therefore report either, and software differs in which it prints.

Building survival regressions for a cost-effectiveness model

NICE Decision Support Unit Technical Support Document 14 (TSD14) notes that a Cox model bases inferences only on the observed data, so it gives an accurate mean survival only when follow-up is almost complete, whereas a lifetime model needs the area under the whole curve. TSD14 describes two broad ways of fitting parametric survival regressions to trial data. One fits a single model to the whole dataset with treatment group as a covariate and assumes proportional hazards. The other splits the data and fits a model to each arm, which involves fewer assumptions but more parameters. For separate fits TSD14 regards the same distribution in both arms as sensible, letting shape and scale differ without the arms following very different distributions. If a proportional hazards approach is chosen, the hazard ratio should come from the parametric model fitted with treatment as a covariate, and TSD14 calls it theoretically incorrect to apply to the control curve a hazard ratio taken out of a different parametric model or a Cox model.

The NICE manual (PMG36, section 4.6.22) states that the proportional hazards assumption should always be assessed, and section 4.6.25 asks that the clinical plausibility of the underlying hazard functions be assessed routinely when alternative extrapolation models are compared.

Other covariates have a narrower role in a randomised trial. TSD14 states that it is generally not necessary to include them, since important covariates would be expected to balance through randomisation. Covariates matter more when external evidence is used. TSD14 recommends that, ideally, patient-level data from external sources be analysed by regression so that long-term survival can be estimated for patients with the characteristics of those in the trial.

Comparing fitted regressions

Several distributions often fit the observed period about equally well and then diverge. TSD14 recommends that the exponential, Weibull, Gompertz, log-logistic, log-normal and generalised gamma models all be considered. It warns that the negative 2 log likelihood is suitable only for nested models, such as one model with an extra covariate, and has been used wrongly in past appraisals to compare different distributions. The Akaike information criterion and the Bayesian information criterion do not require nesting and penalise extra parameters, the BIC more heavily.

These statistics measure internal validity only. TSD14 defines an appropriate model as one that fits the observed data and has a clinically and biologically plausible extrapolated portion, and it states that models meeting only one of those criteria are likely to be inappropriate. The wider extrapolation toolkit includes spline-based flexible parametric models and cure models.

Worked example with a Weibull regression for treatment and age

All numbers here are illustrative. A Weibull proportional hazards regression is fitted to overall survival with treatment ($x_T=1$ for the new treatment) and age ($x_A$, years above 65) as covariates. The estimates are scale $\lambda=0.10$ per year to the power $p$, shape $p=1.2$, $\beta_T=-0.40$ and $\beta_A=0.06$. Background mortality is ignored.

$$S(t\mid x)=\exp\!\left[-\lambda\,t^{p}\exp(\beta_Tx_T+\beta_Ax_A)\right]$$

where $S(t\mid x)$ is the probability of being alive at $t$ years for a patient with treatment indicator $x_T$ and centred age $x_A$, and the other symbols are as defined above.

1. Treatment effect on two scales. The hazard ratio is $\exp(-0.40)=0.670$. The time ratio is $\mathrm{TR}=0.670^{-1/1.2}=\exp(0.40/1.2)=\exp(0.333)=1.396$, so the new treatment stretches survival times by about 40%.

2. Five-year survival for a 65-year-old. With $t^{p}=5^{1.2}=6.899$, the comparator cumulative hazard is $H(5)=0.10\times6.899=0.690$ and survival is $\exp(-0.690)=0.502$. On treatment the cumulative hazard is $H(5)=0.690\times0.670=0.462$ and survival is $\exp(-0.462)=0.630$.

3. Mean survival for a 65-year-old. Weibull mean survival is $\Gamma(1+1/p)\,(\lambda e^{\eta})^{-1/p}$, where $\eta$ is the linear predictor. With $\Gamma(1.833)=0.9407$ and $\lambda^{-1/p}=0.10^{-1/1.2}=6.813$, the comparator mean is $\bar{T}_0=0.9407\times6.813=6.409$ years, where $\bar{T}$ denotes mean survival. The treated mean is $\bar{T}_1=6.409\times1.396=8.944$ years, a gain of 2.535 life-years.

4. A population of two ages. Suppose half the eligible population is aged 55 ($x_A=-10$) and half 75 ($x_A=10$), so the mean age is 65. The age multipliers are $\exp(-0.6)=0.549$ and $\exp(0.6)=1.822$.

For example, the comparator five-year survival at age 55 is $\exp(-0.690\times0.549)=\exp(-0.379)=0.685$, and the population average is $\bar{S}(5)=(0.685+0.285)/2=0.485$. The population mean gain is $\Delta\bar{T}=10.085-7.227\approx2.86$ life-years.

5. Interpretation. Entering the mean age into the regression overstates five-year survival and understates the mean life-year gain, 2.54 against 2.86, because survival is a non-linear function of the covariates. The population curve is the average of the individual curves, not the curve of an average patient. The population hazard ratio also drifts, from 0.670 near time zero to 0.729 at five years and 0.778 at ten years, because high-risk older patients are depleted faster in the comparator arm, even though every individual hazard ratio is exactly 0.670. TSD21 makes the same point in general terms, that the shape of the marginal hazard can be very different from the conditional hazards, and that covariate effects need careful modelling to reach an appropriate marginal estimate of all-cause survival.

Reading survival regression output for decisions

Each coefficient is conditional on the other covariates in the model, so the adjustment set belongs with every reported hazard ratio or time ratio, as the drift in the worked example shows. A decision model needs absolute outcomes, which come from the baseline curve combined with the coefficients. TSD14 recommends that mean time to event, not the median, be estimated, since the mean is what drives life-years gained and QALYs in a partitioned survival model or a state-transition model.

TSD14 states that uncertainty around the parameter estimates within the chosen models should be incorporated in probabilistic sensitivity analysis, and that more than one plausible set of models should be run as scenarios. Because the shape, scale and treatment coefficients of a fitted model are correlated, they are sampled jointly, not one at a time.

Where survival regression sits among neighbouring methods

Survival regression is a type of survival analysis and a type of regression analysis. The Kaplan-Meier estimator describes survival without covariates and is the usual yardstick for judging fit. A frailty model adds a random effect for unexplained heterogeneity, and a time-varying covariate is one whose value changes during follow-up. Relative survival models add expected background mortality, the piece missing from the worked example.

Common misreadings of survival regression

Errors with survival regression often come from moving between the hazard scale, the time scale and the population that the economic model represents. Two of them can be sized with the figures from the worked example.

• Reading a hazard ratio as a ratio of risks. In the example a hazard ratio of 0.670 lowers the five-year probability of death from 0.498 to 0.370, a relative fall of 26%, not 33%.
• Reading a Weibull AFT coefficient as a log hazard ratio, or the reverse.
• Predicting population survival from mean covariate values, which in the example understated the life-year gain by about 11%.
• Choosing the distribution on fit to the trial alone, without asking whether the extrapolated hazard is plausible.

Sources

• Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part II: multivariate data analysis, an introduction to concepts and methods. British Journal of Cancer. 2003;89(3):431-436. https://doi.org/10.1038/sj.bjc.6601119
• Cox DR. Regression models and life-tables. Journal of the Royal Statistical Society Series B (Methodological). 1972;34(2):187-202. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x
• Latimer NR. 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). https://www.sheffield.ac.uk/nice-dsu/tsds/survival-analysis
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36), sections 4.6.22 and 4.6.25. London: NICE; published 31 January 2022, last updated 13 October 2026. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• 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. https://www.sheffield.ac.uk/nice-dsu/tsds/flexible-methods-survival-analysis