Piecewise Model

Piecewise Model for Survival Curves and Extrapolation

In health economic evaluation, a piecewise model describes survival by cutting follow-up into time periods and letting each period carry its own hazard, so the risk of death or progression can be high soon after treatment, lower in a stable middle phase and different again later. The method suits trial data whose observed hazard has a shape that no single standard distribution follows, and the commonest version holds the hazard constant within each period, giving the piecewise exponential model. NICE Decision Support Unit Technical Support Document (TSD) 14 called piecewise parametric models an under-used approach in health technology assessment and a simple way of modelling a variable hazard. Because a cost-effectiveness model usually needs survival far beyond the end of a trial, the choice of cut points and of the model for the last period can move mean survival, and so the incremental cost-effectiveness ratio, more than the fit to the observed data suggests.

How cut points turn one hazard into several

A standard parametric survival model imposes one hazard shape for the whole of time. TSD 14 notes that Weibull and Gompertz hazards must be monotonic and log-logistic and log normal hazards are unimodal, while piecewise constant models do not restrict the hazard in this way. TSD 21, on flexible methods for survival analysis, sets out the general form, in which time is partitioned into intervals and each interval has its own hazard component.

$$h(t) = \lambda_j(t) \quad \text{for } t_{j-1} \le t < t_j, \qquad j = 1, \dots, J$$

where $h(t)$ is the overall hazard at time $t$, $J$ is the number of intervals, $t_0 = 0 < t_1 < \dots < t_{J-1}$ are the cut points and $t_J$ is the end of follow-up, or infinity when the last piece is extrapolated, and $\lambda_j(t)$ is the hazard used inside interval $j$, which may be a constant or any parametric form.

The survival function follows from the cumulative hazard, summed across the intervals already passed. When every interval has a constant hazard the sum is simple.

$$S(t) = \exp\{-\Lambda(t)\}, \qquad \Lambda(t) = \sum_{k=1}^{j-1} \lambda_k (t_k - t_{k-1}) + \lambda_j (t - t_{j-1})$$

where $S(t)$ is the probability of surviving beyond $t$, $\Lambda(t)$ is the cumulative hazard at $t$, $t$ lies in interval $j$, and $\lambda_k$ is the constant hazard in interval $k$. TSD 21 adds that the same curve can be read as a product of conditional survival functions, one for each interval.

The piecewise exponential model and how it is estimated

With a constant hazard in each interval, each piece is an exponential model for the people still at risk in that period. Rodríguez notes that closely spaced cut points where the hazard changes quickly, and wider intervals where it changes slowly, allow a reasonable approximation to almost any baseline hazard. For one interval, the maximum likelihood estimate is the number of events divided by the exposure time.

$$\hat{\lambda}_j = \frac{d_j}{T_j}, \qquad \widehat{\text{var}}(\hat{\lambda}_j) = \frac{d_j}{T_j^{2}}$$

where $\hat{\lambda}_j$ is the estimated hazard in interval $j$, $d_j$ is the number of events observed in that interval, and $T_j$ is the total time at risk accumulated in it by everyone who entered it, including people later censored. The variance is the large-sample estimate from the observed information.

Rodríguez also records that Holford (1980) and Laird and Olivier (1981) showed the piecewise exponential proportional hazards model to be equivalent to a Poisson regression on pseudo-observations, one for each person and interval, with the event indicator as the response and log exposure time as an offset. The likelihoods coincide even though the indicators are not truly Poisson, so standard software for rates fits the model, and covariates that change only at interval boundaries are easy to add. In a decision model the constant hazard converts into a transition probability for a cycle of fixed length.

$$p_j = 1 - e^{-\lambda_j u}$$

where $p_j$ is the probability of the event during one cycle that falls inside interval $j$, and $u$ is the cycle length in the same time units as $\lambda_j$.

Hybrid piecewise forms in NICE appraisals

The pieces need not all be exponential. TSD 21 states that the model for each period can in theory take any form, although standard parametric models tend to be used, and that an increasing hazard in the first piece can be followed by a decreasing one and then a rising one. It lists TA490 (nivolumab for head and neck cancer), where piecewise log normal models were fitted to overall survival and piecewise generalised gamma models to progression-free survival and time to treatment discontinuation, with further use in TA387 and TA421.

A common hybrid keeps the Kaplan-Meier curve for the early period and joins a parametric tail at a chosen point. TSD 21 explains that adjoining an exponential in this way was popularised by the Liverpool Reviews and Implementation Group and is known as the Liverpool approach. In TA391 a Weibull was joined to the Kaplan-Meier curve and in TA519 a log normal, and in TA268 the company joined a Gompertz model at 18 months and then used registry hazard rates from year 6. TSD 14 found piecewise modelling in 1 of the 45 appraisals it reviewed and described TA121, where the assessment group fitted one model to the first 12 months of progression-free survival data and a second model to the next 12 months.

Choosing cut points and the model for each piece

TSD 21 lists three assumptions behind any piecewise model. The points at which new models start are appropriate, the model in each section is appropriate, and each section holds enough data to fit a reliable survival model. Cut points can come from inspecting the observed hazard, from biological reasoning made in advance, or from methods that estimate random changepoints. TSD 14 notes that log-cumulative hazard plots show where marked changes in the observed hazard occur, which helps when considering different models for different periods.

TSD 14 recommends considering piecewise and other flexible methods if the standard distributions fit poorly or extrapolate implausibly, and using an appropriate distribution for the extrapolated portion. It also advises a piecewise approach in place of excluding data points from the right-hand side of the Kaplan-Meier curve unless those points are shown to be erroneous outliers. Where models are fitted separately to each treatment arm, TSD 14 advises using the same type of model in both arms, with any departure justified by clinical judgement and biological plausibility.

Worked example with three intervals of illustrative trial data

All numbers in this example are illustrative. A single-arm analysis follows 200 patients for up to 36 months, and the observed hazard appears to fall twice, so cut points are placed at 6 and 18 months. The model is then used to estimate mean survival for a lifetime model with monthly cycles.

1. Estimate the interval hazards. In the first 6 months there are 40 deaths in 1,000 person-months, so the hazard is 40 / 1,000 = 0.040 per month. Between 6 and 18 months there are 30 deaths in 1,500 person-months, giving 0.020. Between 18 and 36 months there are 12 deaths in 1,200 person-months, giving 0.010.

2. Build the survival curve. The cumulative hazard is 0.040 × 6 = 0.24 at 6 months, 0.24 + 0.020 × 12 = 0.48 at 18 months and 0.48 + 0.010 × 18 = 0.66 at 36 months. Survival is therefore $e^{-0.24} = 0.787$, $e^{-0.48} = 0.619$ and $e^{-0.66} = 0.517$.

3. Restricted mean over the trial. The area under each piece is the survival at its start multiplied by one minus the survival within the piece, divided by its hazard.

$$A_j = S(t_{j-1}) \, \frac{1 - e^{-\lambda_j (t_j - t_{j-1})}}{\lambda_j}$$

where $A_j$ is the area under the survival curve in interval $j$, measured in months. The areas are 1 × 0.2134 / 0.040 = 5.33, 0.7866 × 0.2134 / 0.020 = 8.39 and 0.6188 × 0.1647 / 0.010 = 10.19 months, so the restricted mean survival time to 36 months is 23.92 months.

4. Extrapolate from the last piece. If the hazard of 0.010 continues, the area under the curve beyond 36 months is 0.5169 / 0.010 = 51.69 months, since each person alive at 36 months can expect 1 / 0.010 = 100 further months, and mean survival is 23.92 + 51.69 = 75.6 months. Survival at 60 months is 0.517 × $e^{-0.24}$ = 0.407.

5. Uncertainty in the last piece. Only 12 events inform the hazard used for the tail. Its standard error is $\sqrt{12} / 1{,}200 = 0.00289$, so an approximate 95% confidence interval is 0.010 ± 1.96 × 0.00289, or 0.00434 to 0.01566. The tail then ranges between 0.517 / 0.01566 = 33.0 and 0.517 / 0.00434 = 119.1 months, so mean survival could lie anywhere between about 57 and 143 months on this source of uncertainty alone.

6. Move a cut point. Removing the 18-month cut leaves one piece covering 6 to 36 months, with 42 deaths in 2,700 person-months and a hazard of 42 / 2,700 = 0.01556. Survival at 36 months becomes $e^{-(0.24 + 0.01556 \times 30)} = 0.4933$, the restricted mean to 36 months is 5.33 + 18.86 = 24.19 months, the tail is 0.4933 / 0.01556 = 31.7 months and mean survival is 55.9 months.

7. Interpretation. The two versions differ by only 0.27 months within the trial, yet their mean survival differs by 19.7 months, because the extrapolation depends almost entirely on the final piece. In a monthly Markov model the death probabilities per cycle would be 0.0392, 0.0198 and 0.0100 in the three pieces, against 0.0154 after 6 months in the alternative.

What the final piece means for a cost-effectiveness decision

TSD 21 states that it is the model fitted to the latest section of the curve that is used for extrapolation, and that later sections may have few patients at risk and few events, leading to large standard errors. It adds that a piecewise model still relies on a final parametric extrapolation, which is unlikely to capture changes in the hazard beyond the data, such as those expected from ageing. In a TSD 21 illustration, three Weibull pieces fitted to three years of simulated data fitted the observed period reasonably but extrapolated poorly, with substantial bias.

TSD 21 therefore recommends that the time points be justified, that sensitivity to them be explored, and that observed and predicted hazard and survival plots be presented with confidence intervals for the models fitted to each period. Background mortality or other external information should be included in every piece and above all in the final one. For the economic model, step 6 of the example would sit naturally in the scenario analysis of structural uncertainty, and the hazard for the last piece belongs in the probabilistic analysis with its standard error.

Piecewise models beside splines, cure and landmark models

A piecewise hazard usually jumps at each cut point. TSD 21 notes that such jumps can look clinically implausible and that methods allowing a continuous hazard at the changepoints are similar in principle to smoothing with splines. A spline model joins its pieces smoothly at knots. The flexible parametric models in TSD 21 use a restricted cubic spline whose first and second derivatives are continuous at the knots, so they describe a time-varying hazard without steps.

A cure model assumes that a proportion of patients will never have the event of interest, so the survival curve reaches a plateau, which a piecewise model can only approach through a very low hazard in its final piece. Landmark analysis also conditions on a time point, but TSD 21 describes it as splitting patients into groups by their response category at the landmark time, whereas a piecewise model splits time. A Cox proportional hazards model leaves the baseline hazard unspecified, and Rodríguez places the piecewise exponential between that approach and fully parametric models.

Limitations and common misreadings

The main risks come from treating the cut points and the last piece as settled. Most of the errors below show up only after extrapolation, when the in-trial curve already looks acceptable.

1. Treating cut points as neutral. TSD 21 warns that cut points chosen by eye can be arbitrary and can influence the results markedly.
2. Reading a good in-trial fit as a good extrapolation. The fit within follow-up says little about the tail, as the example shows.
3. Using many short pieces. Each extra cut leaves fewer events in each piece, and TSD 21 notes the large standard errors that follow in later sections, a route to overfitting.
4. Letting the last piece rest on a handful of events. Its standard error should be reported, together with the numbers at risk.
5. Assuming the step hazard is the clinical truth. The jumps are a modelling convenience, and a smooth model may describe the same data equally well.

Sources

• 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; June 2011, last updated March 2013. Sections 2.7, 3.2, 4.1 and 5. https://doi.org/10.15131/shef.data.33745843
• Rodríguez G. Generalized Linear Models, lecture notes, Chapter 7, Survival Models. Princeton University; revised September 2010. Sections 7.2, 7.3 and 7.4. https://grodri.github.io/glms/notes/c7.pdf
• 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; 23 January 2020. Sections 3.2, 3.4, 3.5, 3.6, 4.3.4, 6.8 and 7.2. https://doi.org/10.15131/shef.data.33849118