Parametric survival distributions
EasyHTA fits parametric survival models using maximum likelihood estimation via the flexsurv package (Jackson, 2016). Seven distributions are supported: all six recommended by NICE DSU Technical Support Document 14 (Latimer, 2011), plus the gamma. Nine optional Royston-Parmar spline models add more flexible hazard shapes. For the reconstruction step that precedes model fitting, see The Guyot algorithm.
Supported distributions
| Distribution | Parameters | Hazard shape | Notes |
|---|---|---|---|
| Exponential | rate | Constant | Useful as a baseline or sensitivity model; appropriate when a constant hazard is plausible. |
| Weibull | shape, scale | Monotone increasing, monotone decreasing, or constant (exponential as a special case) | Simple, and appropriate when a monotone hazard is plausible. |
| Log-normal | meanlog, sdlog | Unimodal: initial increase followed by later decline | Appropriate when a unimodal hazard is plausible (early rise, later decline). |
| Log-logistic | shape, scale | Decreasing if shape ≤ 1; unimodal if shape > 1 | Heavier tails than log-normal, so long-tail behaviour must be justified. |
| Gompertz | shape, rate | Monotone increasing, monotone decreasing, or constant (shape > 0, < 0, or = 0) | Can represent exponentially changing hazards; background mortality should be assessed or incorporated separately rather than implicitly absorbed. |
| Gamma | shape, rate | Monotone increasing if shape > 1, monotone decreasing if shape < 1, constant if shape = 1 (exponential as a special case); tends to a constant hazard in the tail | Its asymptotically constant hazard should be justified against the plausible long-term hazard. Nested within the generalised gamma. |
| Generalised gamma | mu, sigma, Q | Flexible; nests Weibull, gamma, and log-normal as special cases | Useful flexible reference model; nested special cases can be compared via constrained likelihood comparisons where statistically valid. |
Model selection
EasyHTA computes AIC and BIC for all fitted models. Lower values indicate better fit relative to model complexity. AIC and BIC are internal fit diagnostics against the observed data only; they do not validate the plausibility of the extrapolated tail. Model choice should also weigh hazard plausibility, external evidence, and uncertainty in the extrapolated period. You can record a written justification alongside the chosen distribution, so this reasoning travels with the analysis.
Royston-Parmar spline models
Spline models are optional and off by default. Turn on Include spline models when you reconstruct a curve, or choose Fit spline models on the fitting step of a curve you have already fitted; the reconstruction does not run again. Spline models are available in the standard partitioned survival template and are not offered for cure-model analyses.
A Royston-Parmar spline models a transformation of the survival function as a smooth function of log time, using a restricted cubic spline with boundary knots at the earliest and latest observed event times and a chosen number of internal knots (Royston and Parmar, 2002). EasyHTA fits nine models: three scales, each with one, two or three internal knots.
| Scale | What the spline describes | Special case |
|---|---|---|
| Hazard | The log cumulative hazard | With no internal knots and a straight line in log time, this is the Weibull model |
| Odds | The log cumulative odds of having had the event by that time | With no internal knots and a straight line in log time, this is the log-logistic model |
| Normal | The probit (normal quantile) of the probability of having had the event by that time | With no internal knots and a straight line in log time, this is the log-normal model |
Knot placement
Internal knots are placed at quantiles of the observed event times, so they sit where the data is densest. The event times of a reconstructed curve are themselves estimated from the curve’s coordinates, so the knot positions depend on the reconstruction: two reconstructions of the same Kaplan-Meier curve can place knots slightly differently. The choice between spline models is therefore less stable than it looks, and the covariance of a spline fit describes uncertainty conditional on one reconstruction. The Assumptions Log records the knot locations in years for any selected spline.
Number of knots
One to three internal knots matches the number of events a typical reconstructed trial curve carries. Each extra knot adds a parameter that the events between two adjacent knots have to support, and with few events a flexible curve starts to follow the steps of the reconstructed data instead of the hazard. The fitting step therefore marks a spline with fewer than 10 observed events in any interval between adjacent knots as Low events. The flag is advisory: the model is still fitted and can still be selected.
Selection
- Best AIC and Best BIC are awarded among the seven standard distributions. Spline models sit in their own group, with AIC and BIC shown for fit rather than as a ranking.
- A written justification is required to confirm a spline model. A spline follows the data closely, so the reason it is credible for the extrapolated period has to be stated.
- The hazard diagnostic compares each fitted hazard with general-population mortality for the entered start age and life table. A flexible model extrapolates the hazard from the final events, and nothing in the fit stops that hazard falling below background mortality. The Technical tab plots every fitted hazard against the population hazard and marks any fit that drops below it with the first whole year it does. On the fitting step, a spline that drops below it carries a badge.
- Consistent arms. Comparing arms on different spline scales, or a spline against a standard family, makes different statements about how the treatment effect changes over time. EasyHTA warns when the choices for the two arms differ in this way, and adds a note for smaller mismatches, such as the same scale with a different number of knots.
- Reporting. The exported comparison figure and survival curves show the seven standard distributions plus the selected spline model. In the app, the other spline fits are listed in the extrapolation table and can be shown on the comparison chart from its legend, and the fit statistics sheet of the Excel export lists them all.
A spline model that does not converge keeps its place in the group, marked Did not converge. In a probabilistic sensitivity analysis, a draw whose survival curve rises is corrected to a non-increasing curve, and the share of such draws is reported with the results. A high share means the spline is poorly identified by the data.
Parameter estimation
All models are fitted by maximum likelihood estimation. Where a fit produces a usable variance–covariance matrix, it drives EasyHTA’s own probabilistic sensitivity analysis, sampling correlated parameter draws from the fitted covariance. The same matrix is included in the Excel export for a receiving model, provided the parameter scale (flexsurv estimates positive parameters on the log scale), ordering, and any back-transformations are handled correctly there.
TSD14 alignment
- All six standard distributions recommended in TSD14 are available, alongside the gamma
- The in-app comparison table shows AIC and BIC for every fitted distribution; log-likelihood is also available in the Excel export
- Visual inspection overlays (fitted curves vs the reconstructed KM) are shown in the model-fitting results views
- The variance–covariance matrix drives EasyHTA’s own PSA and is also exported for use in a receiving model
- Analysis metadata (export date, reconstruction method) is recorded in the
Metadatasheet of the Excel export - For complex hazards, non-proportional hazards, cure assumptions, or situations where background mortality materially affects extrapolation, analysts should also consider methods discussed in NICE DSU TSD21 alongside TSD14. Royston-Parmar spline models, one of the flexible approaches TSD21 covers, are available in-app (see Royston-Parmar spline models)
References
Guyot P, Ades AE, Ouwens MJNM, Welton NJ (2012). Enhanced secondary analysis of survival data: reconstructing the data from published Kaplan–Meier survival curves. BMC Medical Research Methodology, 12, 9. doi:10.1186/1471-2288-12-9
Jackson C (2016). flexsurv: A Platform for Parametric Survival Modeling in R. Journal of Statistical Software, 70(8), 1–33. doi:10.18637/jss.v070.i08
Royston P, Parmar MKB (2002). Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with application to prognostic modelling and estimation of treatment effects. Statistics in Medicine, 21(15), 2175–2197. doi:10.1002/sim.1203
Latimer NR (2011, last updated 2013). NICE DSU Technical Support Document 14: Undertaking survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: NICE DSU. sheffield.ac.uk/nice-dsu
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