Broadening the candidate set of parametric models when extrapolating survival: the case for models with U-shaped hazards.

doi:10.21203/rs.3.rs-5303870/v1

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Broadening the candidate set of parametric models when extrapolating survival: the case for models with U-shaped hazards.

https://doi.org/10.21203/rs.3.rs-5303870/v1

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In health technology assessment (HTA), extrapolation of time-to-event data is common to estimate the benefit of a new health technology beyond the observed period of data. The regular set of parametric models commonly used for extrapolation does not include models which assume a U-shaped hazard rate, that is initially decreasing and then increasing hazard rate.

We compared the visual and statistical fit and prediction of models which assume a U-shaped hazard rate (Chen, bathtub and Rayleigh) to the regular set of parametric models (exponential, log-normal, log-logistic, Weibull, generalised gamma, Gompertz) across a range of settings and data types, including hip arthroplasty, functional tricuspid regurgitation and knee osteoarthritis.

U-shaped hazard models outperformed or matched standard parametric models in visual fit, goodness of fit statistics and long-term predictions when compared to extended follow-up.

Bathtub models should feature routinely in HTA submissions involving extrapolation of survival data, allowing for exploration of a wider range of scenarios and potentially more accurate predictions, resulting in better informed valuation and decision making for emerging health technologies.

Survival Analysis

Extrapolation

Health Technology Assessment

Parametric models

In health technology assessment (HTA) it is often necessary to model the observed data and or to extend analysis beyond maximum follow up to estimate outcomes in the longer term and satisfy the time horizon favoured by decision making authorities; frequently a life-time horizon is required. For time-to-event outcomes, such as overall survival, various parametric models are fitted to data using widely available in software programmes such as Stata and R.

Commonly six “standard” distributions are explored (Weibull, Gompertz, loglogistic, lognormal, exponential, generalised gamma)¹. These models can represent a wide range of hazard behaviours (Fig. 1). The exponential is the least flexible, modelling a constant hazard rate. The Weibull or Gompertz can model a monotonic hazard rate, either always increasing or always decreasing. Finally, the lognormal and loglogistic models tend to provide a decreasing hazard over time, sometimes after an initial period of an increasing hazard rate. There are no models in this standard set of models which allow for an initial decreasing hazard rate followed by increasing hazard rate over time. Such models do exist and may be referred to as “bathtub” models due to their “U” shaped hazard. A failure to include these models in candidate sets results in a failure to explore all potential future scenarios in areas of high uncertainty and may bias methods of model averaging. In this paper we explore the potential benefits of expanding the standard set of parametric models to include a 3-parameter bathtub² and a 2-parameter bathtub (Chen 2000 ³) models that here we term “bathtub” and “Chen” respectively. The Chen model (Supplement: section 1) can generate non-linearly increasing hazard as well as “U shaped” bathtub hazard. We also briefly consider Rayleigh models (Peace and Tsai 2009 ⁴; Supplement: section 2) that generate linearly increasing hazard. The “stgenreg” user written Stata package (Crowther and Lambert 2013 ⁵) enables exploration of such models that are not pre-specified in software packages that sometimes may generate better fit to the observed data than obtained using standard models.

A bathtub model with three parameters is described by Collet², whilst Chen³ describes a two parameter model that can generate a bathtub or non-linearly increasing hazard (Supplement: Section 1). Their survival and hazard functions are shown in equations 1–4.

Bathtub 3 parameter hazard function

$$\:h\left(t\right)=at+\frac{b}{(1+gt)}\:\:\:\:\:\:\left(1\right)$$

Bathtub 3 parameter survival function

$$\:\text{S}\left(\text{t}\right)=\text{e}\text{x}\text{p}\left(-\frac{{\text{a}\text{t}}^{2}}{2}\text{}\right)\cdot\:{\left(1+\text{g}\text{t}\right)}^{-\:\frac{b}{g}}\text{}\:\:\:\:\:\:\left(2\right)$$

Chen hazard function

$$\:h\left(t\right)=\lambda\:\beta\:{t}^{\beta\:-1}{e}^{{t}^{\beta\:}}\:\:\:\:\:\:\left(3\right)$$

Chen survival function

$$\:\text{e}\text{x}\text{p}(\lambda\:-\lambda\:\text{exp}\left({t}^{\beta\:}\right)\:\:\:\:\:\:(4)$$

Data sources

We fit a range of survival models to data from a wide range of populations and medical interventions with study designs including RCTs, cohort, and registry studies. We lead with a clear example using re-created data from Benfari et al. ⁶ where standard parametric models are outperformed by models with increasing hazard rates and we provide a wide range of cases where these increasing hazard models show good fit motivating their inclusion for regular consideration when extrapolating survival in a HTA setting.

RCTs included the left atrial appendage study of Whitlock et al., 2021⁷ and the tricuspid valve replacement study TRISCEND II⁸. Cohort studies included the Benfari et al., 2019⁶, Essayagh et al., 2021⁹, and Tribouilloy et al 2024¹⁰ investigations of patients with isolated functional tricuspid regurgitation, and the Bae et al.,¹¹ Korean study of microfracture treatment for osteoarthritic knees. Registry studies included: The National Joint Registry of England and Wales reports 9¹² and 20¹³ and Swedish Hip Arthroplasty Register 2018 and 2023^{14, 15} on failure after Primary Total Hip Replacement (THR) procedures (registry data sources include very substantial numbers of individuals followed for many years so that the trajectory of the time to event plots is internally consistent and less changeable compared to that in smaller studies). Relevant details from the included studies are briefly summarised in Table 1.

Analyses

Published Kaplan-Meier (KM) plots were digitised using Digitzelt (http://www.digitizelt.xyz) and IPD reconstructed using the method of Guyot et al. ¹⁶. In registry data where KM plots encompassed thousands of events that could not all be captured by digitisation; we proportionately reduced the patient numbers to handleable size, preserving the underlying survival distribution.

All models were fitted using the streg or stgenreg command in Stata 18 (REF) or using stgenreg command of Crowther and Lambert⁵. The 2-parameter³ and 3-parameter ² bathtub models and Rayleigh models ⁴ were generated using following stgenreg commands:

Rayleigh 1-parameter model: stgenreg, haz(2:*exp([ln_lambda2]):*#t)nodes(30)

Rayleigh 2-parameter model: stgenreg, haz([lambda1] :+ 2:*[lambda2]:*#t)nodes(30)

Collet bathtub model (3 parameters): stgenreg, haz(([alpha]:*#t:+ [beta]:/(1:+[gamma]:*#t)))nodes(30)

Chen model (2 parameters): stgenreg, loghazard( [ln_lambda] :+ [ln_beta] :+ (exp([ln_beta]) :- 1):* log(#t) :+ exp(exp([ln_beta]) :* log(#t)))

Models were assessed for goodness of fit using AIC and BIC values, visual fit, and clinical plausibility in extrapolation.

Table 1

Overview of data sources used
Data source	Description	Sample size/events	Length of follow-up
Benfari et al., 2019^{∞, 6}	Mortality of patients with four different levels of functional tricuspid regurgitation (FTR): Trivial, mild, moderate and severe.: Trivial .Mild Moderate Severe	N 4329/1795 deaths N 4178/ 2173 deaths N 2255/ 1371 deaths N 745/ 502 deaths	Maximum follow-up 10 yrs
Essayagh et al., 2021^{∞, 9}	Mortality under medical management of patients with four different levels of functional tricuspid regurgitation. Trivial . Mild .Moderate Severe	N 2288/ 20% dead at 10 yrs N 1858/ 35% dead at 10 yrs N 767/ 53% dead at 10 yrs N 157/ 81% dead at 10 yrs	Maximum follow-up 10 yrs
National Joint Registry for England and Wales Report 20 ¹³	Cumulative revision of primary total hip replacement (THR) in patients with osteoarthritis. Cumulative revision of primary total hip replacement (THR) with cemented metal on polythene devices. Cumulative revision of primary total hip replacement (THR) with cemented ceramic on polythene devices. Cumulative re-revision of revised primary total hip replacement (THR) Cumulative revision of primary total hip replacement (THR) with uncemented metal on polythene devices.	1,273,746/ ~7.3% required revision 358,641/ ~6.6% required revision 59,975/ ~ 5% required revision 43,682/ ~20% required re-revision 205,001/ ~8.5% required revision	Maximum follow-up 18 yrs
National Joint Registry for England and Wales Report 9*^{, 12}	Cumulative revision of primary total hip replacement (THR) with cemented metal on polythene devices. Cumulative revision of primary total hip replacement (THR) with cemented ceramic on polythene devices. Cumulative revision of primary total hip replacement (THR) with uncemented metal on polythene devices.	125,285/ ~2.4% required revision 13,871/ ~2% required revision 59,993/ ~3.2% required revision	Maximum follow-up 8 yrs
Whitlock et al., 2021 ⁷ RCT	Cumulative incidence of stroke or embolism after: No occlusion of left atrial appendage preformed during heart surgery. Occlusion of left atrial appendage preformed during heart surgery.	2391/ 168 events 2379/ 114 events	Maximum follow-up 6 yrs
Tribouilloy et al., 2024 ¹⁰	Cumulative mortality of patients with functional tricuspid regurgitation and co-morbidities, mean age 75 ± 12 years	715/ 175 deaths	Maximum follow-up 5 yrs
Bae et al., 2013 ¹¹	Time to the requirement for total knee replacement (TKR) after microfracture procedure for osteoarthritis of the knee. Mean age 61.3 years, most patients female.	134 knees / 51 TKR	Maximum follow-up 12 yrs
TRISCEND II RCT as reported by US FDA⁸	All-cause Mortality: .After percutaneous replacement of tricuspid valve using the EVOQUE device OR .Optimal Medical Therapy	259/ 14.8% dead at 18 mos 113/ 23.4% dead at 18 mos	Maximum follow-up 18 mos
yrs years; mos months ∞ Patient comorbidities differed between FTR levels * The National Joint Registry truncates Kaplan Meier plots at the time it is judged that uncertainty is unacceptably high.

Benfari et al.,2019 study of mortality in patients with four levels of functional tricuspid regurgitation⁶

Figure 2 shows the poor fit of regular parametric models to the follow-up from Benfari et al. for trivial FTR, suggesting the need for an alternative approach.

In Fig. 3a, bathtub and Chen models have been fitted separately to all four FTR groups from both Benfari et al. For the Benfari et al. trivial group (best prognosis), it is clear that the Chen and bathtub models provide a superior visual fit. This is supported by the goodness of fit statistics which show a significant improvement over the regular parametric models (Table 2).

This is maintained across all four levels of FTR where the bathtub and Chen models provided almost identical visual fit (Fig. 3) and at least one of these models provided the best AIC/BIC score. Either bathtub or Chen models provided the best AIC and BIC scores (Table 2) and neither model strayed from the 95% CIs of the KM plots. In contrast standard parametric models provided poorer AIC and BIC scores, poorer visual fit and almost all models strayed outside the KM 95% CIs (Supplementary section 2). Chen models were very similar to bathtub over 10 years observation time but following further extrapolation Chen models predicted more favourable longer-term survival (Supplement: section 3).

Essayagh et al., 2021 study of mortality in patients with four levels of FTR under medical management ⁹

KM plot 95% CIs were wider than in the larger Benfari et al. study. For all four levels of FTR bathtub and Chen models provided similar and the best models according to visual fit (Fig. 3b (right)). Either bathtub or Chen models provided the best AIC and BIC scores (Table 2) and both models did not stray appreciably outside the KM 95% CIs. Standard models presented poorer visual fit. (Supplement: section 4)

Table 2

Information criteria (AIC BIC) scores for parametric models; Tricuspid Regurgitation, microfracture and atrial occlusion studies
Model	AIC	BIC	Model	AIC	BIC	Model	AIC	BIC	Model	AIC	BIC
Benfari et al., Trivial functional regurgitation			Essayagh et al., Trivial functional regurgitation			Tribouilloy et al., functional regurgitation			Bae et al., microfracture of the knee
bathtub	11391.6	11410.72	bathtub	2273.44	2290.66	ggamma	1182.412	1196.128	ggamma	227.83	236.53
Chen	11457.57	11470.31	Chen	2281.45	2292.93	bathtub	1184.632	1198.349	Gompertz	227.83	233.63
ggamma	11489.82	11508.94	Weibull	2281.51	2292.99	lognormal	1185.011	1194.156	Weibull	228.84	234.64
Weibull	11550.12	11562.86	ggamma	2281.76	2298.98	loglogistic	1194.666	1203.81	Chen	228.30	234.09
loglogistic	11643.51	11656.26	exponential	2282.99	2288.73	Weibull	1196.236	1205.381	Rayleigh 2	229.11	234.90
Gompertz	11733.51	11746.25	loglogistic	2283.69	2295.17	Gompertz	1198.722	1207.866	Rayleigh 1	229.52	232.41
lognormal	11745.43	11758.18	gompertz	2284.83	2296.31	exponential	1208.135	1212.707	loglogistic	232.44	238.24
exponential	11769.46	11775.83	Rayleigh 2	2284.85	2296.33	Chen	1208.419	1217.564	lognormal	237.58	243.38
Rayleigh 2	11741.83	11754.58	lognormal	2294.47	2305.96	Rayleigh 1	1501.44	1506.012	exponential	272.94	275.84
Rayleigh 1	14945.58	14951.95	Rayleigh 1	2580.30	2586.04	Rayleigh 2	NA	NA	bathtub	NA	NA
Benfari.et al Mild functional regurgitation			Essayagh. et al Mild functional regurgitation			TRISCEND II control arm
bathtub	11903.75	11922.76	Chen	2790.99	2802.04	lognormal	294.6	301.4
Chen	11981.74	11994.42	Ggamma	2859.81	2876.39	ggamma	294.7	304.9
lognormal	12203.39	12216.07	Weibull	2874.22	2885.27	bathtub	294.9	305.1
ggamma	12015.36	12034.37	loglogistic	2892.36	2903.41	loglogistic	296.5	303.3
Weibull	12062.17	12074.84	bathtub	2893.42	2910.00	Chen	296.7	303.5
loglogistic	12158.08	12170.76	exponential	2906.42	2911.95	Weibull	296.9	303.7
Gompertz	12382.84	12395.51	gompertz	2908.39	2919.44	Gompertz	297.0	303.8
exponential	12505.97	12512.31	Rayleigh 2	2908.39	2919.45	exponential	309.4	312.8
Rayleigh 2	12416.85	12429.52	lognormal	3016.19	3027.24	Rayleigh 1	385.8	389.2
Rayleigh 1	16861.08	16867.41	Rayleigh 1	3528.87	3534.40	Rayleigh 2
Benfari.et al. Moderate functional regurgitation			Essayagh et al., Moderate functional regurgitation			Whitlock et al., left atrial appendage No occlusion
Chen	7019.15	7030.59	bathtub	1587.68	1601.61	Chen	1847	1859
bathtub	7019.35	7036.51	ggamma	1592.95	1606.88	bathtub	1851	1869
ggamma	7046.38	7063.54	Weibull	1593.39	1602.68	lognormal	1901	1912
Weibull	7072.59	7084.03	Chen	1594.52	1603.80	Weibull	1905	1917
loglogistic	7165.33	7176.77	loglogistic	1600.24	1609.53	loglogistic	1905	1917
lognormal	7214.13	7225.57	Gompertz	1602.95	1612.24	ggamma	1907	1924
Gompertz	7255.78	7267.22	Rayleigh 2	1603.59	1612.87	Gompertz	2117	2129
Rayleigh 2	7272.49	7283.93	exponential	1604.87	1609.52	Rayleigh 2	2159	2171
exponential	7322.90	7328.62	lognormal	1610.09	1619.37	exponential	2199	2205
Rayleigh 1	10139.16	10144.88	Rayleigh 1	1990.27	1994.92	Rayleigh 1	3007	3013
Benfari.et al) Severe functional regurgitation			Essayagh.et al. Severe functional regurgitation			Whitlock et al., left atrial appendage occlusion
bathtub	2442.81	2456.65	bathtub	457.1133	466.2821	Chen	1353.035	1364.584
ggamma	2466.09	2479.93	loglogistic	457.9641	464.0766	ggamma	1378.3	1341.1
lognormal	2469.57	2478.80	lognormal	454.7669	460.8794	bathtub	1381.676	1398.999
Weibull	2470.93	2480.16	Ggamma	456.7364	465.9051	lognormal	1400.537	1412.086
loglogistic	2485.78	2495.01	Weibull	459.2854	465.3979	loglogistic	1406.278	1417.827
Chen	2487.90	2497.12	gompertz	459.9671	466.0796	Weibull	1406.585	1418.134
Gompertz	2515.70	2524.92	Rayleigh 2	461.6414	467.7539	Gompertz	1542.111	1553.659
exponential	2565.68	2570.30	exponential	466.8290	469.8853	exponential	1651.162	1656.937
Rayleigh 1	3710.03	3714.64	Chen	467.3864	473.4988	Rayleigh 1	2266.306	2272.08
Rayleigh 2	NA	NA	Rayleigh 1	648.6065	651.6627	Rayleigh 2	NA	NA
NA = Not applicable model did not converge

Tribouilloy et al 2024 of mortality in a French cohort with isolated functional tricuspid regurgitation¹⁰

Chen and bathtub models generated very similar good visual fit (Fig. 4b (middle)). Generalised gamma and bathtub provided the best AIC BIC scores, while Chen provided poor AIC BIC scores (Table 2). With extrapolation to 25 years the generalised gamma model predicted clinically implausible 50% survival and other standard parametric models similarly predicted substantial proportion of patients surviving beyond 25 years (Supplementary: section 3) generating overoptimistic survival curves for an elderly population with serious comorbidities. Only bathtub and Chen models generated plausible long-term extrapolations and both generated U shaped hazard plots (Supplement: section 5).

Bae et al., 2013; Korean Cohort with degenerative osteoarthritic knee treated by microfracture¹¹

After ten years of follow-up, around 50% of microfracture-treated knees required TKA in this cohort. Over the observation period all parametric models except lognormal loglogistic and exponential generated reasonably good visual fit with similar AIC BIC scores; Chen, Gompertz and Rayleigh models generated well-fitting models (Fig. 4a (left), Supplement: section 7). On extrapolation of models to 25 years there were considerable differences between models with only Gompertz and Chen models predicting 100% failure within 20 years. The hazard for failure of other models barely changed over the observation period and beyond (Supplement: section 7) despite ageing of the population and inevitable wear and tear of the microfracture-treated knee. The more clinically plausible scenario would seem to be an increasing hazard that gets steeper with increasing age of patient, progressive wear and tear, and the exacerbation of osteoarthritis through time. This scenario is best satisfied by the Rayleigh, Gompertz and Chen models.

Occlusion versus no-occlusion of left atrial appendage (RCT; Whitlock et al.,2021)

A bathtub model for the no-occlusion arm (Fig. 4c (right)) generated superior AIC/BIC scores relative to standard parametric models and better visual fit; all standard models for the no-occlusion arm generated poor visual fit (Supplement: section 8). The Chen model AIC BIC scores were superior to those for 3-parameter bathtub despite its poor visual fit, and the hazard plot differed from the bathtub (3-parameter) hazard. The Chen model appears unsuited to substantial and rapid initial accumulation of events as found in both arms of this RCT. For the intervention arm (occlusion) the 3-parameter bathtub model again generates a good visual fit and Chen 2-parameter model a poor fit (Fig. 4c) while several standard models (e.g. Weibull) generated good visual fit for the intervention arm (Supplement: section 8).

Total Hip Replacement (THR) (Registry studies)

Revision after THR failure varies according to many characteristics of both THR device and patient age, gender, and other demographics of patient populations. The National Joint Registry Annual report 20 (2023) ¹³ itemises many KM plots for cumulative failure of THR. We analysed failure for THR recipients with osteoarthritis, for recipients of cemented metal on polythene and of ceramic on polythene devices, and for re-revision after first THR revision. In each of these four examples bathtub and Chen models provided the best visual fit and almost identical models (Fig. 5) accompanied by the lowest AIC/BIC scores (Table 3). Standard models provided poorer visual fit (Supplement: section 9) and poorer AIC/BIC scores.

Table 3

Goodness of fit statistics for models fitted to THR data
Model	AIC	BIC	Model	AIC	BIC
THR Osteoarthritis			THR cemented ceramic on polythene
Chen	300.2661	308.4213	Chen	296.4028	305.1967
bathtub	301.9992	314.232	bathtub	296.7181	309.9089
exponential	302.1448	306.2224	Gompertz	298.4029	307.1967
Gompertz	303.7814	311.9367	exponential	298.5097	302.9066
Rayleigh 2	303.8298	311.9851	Rayleigh 2	298.8838	307.6777
Weibull	303.8698	312.0251	Weibull	300.4538	309.2477
loglogistic	304.1212	312.2765	loglogistic	300.6615	309.4553
ggamma	305.6258	317.8587	ggamma	302.2484	315.4392
lognormal	307.6922	315.8475	lognormal	303.9896	312.7835
Rayleigh 1	328.2803	332.358	Rayleigh 1	314.5399	318.9368
THR re-revision			THR cemented metal on polythene
Chen	346.0576	352.6442	Chen	316.8134	325.2426
bathtub	348.9028	358.7827	bathtub	318.0951	330.7389
Weibull	349.3799	355.9665	exponential	319.2385	323.4531
loglogistic	349.5192	356.1058	Gompertz	319.798	328.2272
lognormal	349.6722	356.2588	Rayleigh 2	319.9957	328.4249
ggamma	351.3068	361.1867	Weibull	321.2365	329.6658
Gompertz	360.9285	367.5151	loglogistic	321.5145	329.9437
exponential	368.2898	371.5831	ggamma	322.9653	335.6091
Rayleigh 1	471.5511	474.8444	lognormal	325.9746	334.4038
Rayleigh 2	NA	NA	Rayleigh 1	339.8937	344.1083
NA = Not applicable model did not converge

Triscend II RCT: tricuspid valve replacement with EVOQUE system for patients with tricuspid regurgitation.⁸

The US FDA presented KM plots for mortality in the TRISCEND II trial (see Supplementary: section 6). For the medical treatment arm bathtub and Chen models generated almost identical good visual fit for the cumulative incidence of death. Weibull and generalised gamma models delivered marginally lower IC values than bathtub and Chen models but on extrapolation produced less plausible mortality curves for such an aged population with extensive comorbidities. The bathtub g parameter is almost zero so that bathtub and Rayleigh 2 models predict virtually identical curves for survival and for hazard. All models fitted to the intervention arm generated clearly implausible survival curves. Longer follow up with more patients is required for modelling.

Are bathtub and Chen model predictions supported by longer term follow up Registry findings?

We compared extrapolated bathtub and Chen model predictions for Registry cumulative failure rates with that reported for later registry analyses. For this we used the National Joint registry 9th annual report (NJR9) for England and Wales ¹² with the 20th annual report^{13 (NJR20)}. The NJR9 and NJR20 KM plots for frequently used devices show increasing revision rates over time (Fig. 6). Performance of cemented MOP and cemented COP devices was reasonably consistent across these time spans, but in the case of uncemented MOP revision performance appears to have improved with time.

Using NJR9, ¹²) Chen and bathtub models of 125,285 cemented MOP recipients suggests predict between 6% and 7% cumulative revision rate at 18 years, consistent with 6.6% reported in NJR20, when about 2.94 times as many recipients (N = 368,641) were available for analysis (Fig. 6a top).

Chen and bathtub extrapolations for 59,983 uncemented MOP recipients predict 6% and 12% cumulative revision respectively. In this case, the NJR20 (N = 205,001) revision rate is 8.5%, suggesting an average of these two models outperforms a single extrapolation (Fig. 6b middle).

Chen and bathtub models of 13,871 cemented COP recipients from NJR9 predict about 5% and 6% cumulative revision respectively corresponding very closely to the 18-year follow up reported in NJR20 (Fig. 6c bottom).

Similar results were found in analyses conducted using the Swedish Arthroplasty Registry (Supplement: section 10).

This paper has demonstrated the utility of the Chen and bathtub models when extrapolating time-to-event data across a range of settings. Their distinct forms show that there are situations where they outperform or provide substantially different extrapolations compared to the routine models. Broadening the set of candidate models might allow for more accurate predictions, or at least exploration of a wider set of alternative scenarios exploring potential uncertainty.

Hence, models with U-shaped hazard rates such as Chen or bathtub, alongside other alternative models, such as Rayleigh-1 or -2 parameter models should be included in the routine set of models considered when extrapolating these outcomes in a HTA setting. The improved accuracy of predicting long-term outcomes can result in fairer valuations of health technologies for technology developers and healthcare providers, and better decisions being made. This is particularly relevant for first-in-class technologies, where there exists no data on long-term effects of any similar technology and therefore the probability of a future adverse outcome is higher.

U-shaped models should also be considered in applications of model averaging, which has already been shown to be beneficial in some cases¹⁷. In cases where little is known about patient outcomes beyond an observed follow-up period, it can only be considered fair to include candidate models where the hazard rate increases over time when you are also including models which assume the opposite (e.g. log-normal and log-logistic). Of course, models should be scrutinised for plausibility which may rule out certain functional forms, and so we do not see any downsides to including the U-shaped models described in this paper within the set of candidate models.

These results raise the question of why these models are not already included within HTA submissions. This may be because U-shaped models are not referred to in the NICE Technical Support Documents which relate to survival analysis. ^{18, 19} Given their tendency for an increasing hazard rate, we anticipate that their extrapolations will be more pessimistic compared to the routine parametric models. Hence, it is unlikely that submissions to NICE from pharmaceutical companies will introduce these models without prompting, where more optimistic extrapolations are typically preferred, allowing the quality adjusted life-year gains to be collected over a longer time-period.

These models could also be used in a mixture cure model (MCM) setting, which would permit an increasing hazard rate for a subgroup of the population, whilst the remaining population are considered cured or are subject to a background event rate. MCMs are increasingly relevant as emerging technologies such as gene and CAR-T therapies show potential to transform patient outcomes. Use of bathtub or Chen models in this setting might be more palatable to pharmaceutical companies, as the increasing hazard rate would not apply to “cured” patients and so they will not necessarily produce the most pessimistic extrapolations.

A strength of this work is how the benefits of U-shaped models have been shown to apply beyond a single case-study or disease area. We have has used publicly available data, and provided code where helpful, maximising the transparency, reproducibility and ease of implementation of the models described.

A limitation of this research is that we have not considered flexible models such as fractional polynomials or restricted cubic splines. Whilst these can fit very well to the data, they are in danger of overfitting and providing implausible extrapolations, or of relying on key assumptions to be made beyond the observed period of follow-up that can make them equivalent to an underlying parametric model. The flexibility of splines may sometimes result in a U-shaped hazard rate but may also take on many other forms, hence they cannot be relied upon to provide balance to models with n-shaped hazard rates. These models also rely on specification of combinations of terms, scales and knot locations in order to be optimised making a meaningful comparison more difficult, particularly across the range of different types of dataset in this paper. Standard parametric models have less subjectivity in how they are fitted making a comparison more straightforward, however future work could compare the fit and extrapolations of U-shaped models to these flexible approaches.

This study has shown how models with U-shaped hazard rates can match or outperform models routinely used in HTA when fitting to observed data and predicting future survival rates in a range of fields.

These models should be included in the regular set of parametric models considered when extrapolating time-to-event data.

Bathtub and Chen models do not necessarily generate the same predictions despite having U-shaped hazard rate, they may differ slightly or greatly depending on the data and should both be considered.

Ethics approval and consent to participate: Not applicable
Consent for publication: Not applicable
Availability of data and materials: Not applicable
Competing interests: None to declare
Funding: No funding was received specific to this work however DG is supported by NIHR award 14/25/05
Authors' contributions: MC had the original research idea and performed all modelling. DG produced the manuscript and contributed to developing the research idea.
Clinical trial number: Not applicable
Acknowledgements: None

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No competing interests reported.

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Broadening the candidate set of parametric models when extrapolating survival: the case for models with U-shaped hazards.

Status:

Version 1

Abstract

Figures

BACKGROUND

METHODS

Data sources

Analyses

RESULTS

Benfari et al.,2019 study of mortality in patients with four levels of functional tricuspid regurgitation⁶

Bae et al., 2013; Korean Cohort with degenerative osteoarthritic knee treated by microfracture¹¹

Occlusion versus no-occlusion of left atrial appendage (RCT; Whitlock et al.,2021)

Total Hip Replacement (THR) (Registry studies)

Are bathtub and Chen model predictions supported by longer term follow up Registry findings?

Discussion

Conclusion

Declarations

References

Additional Declarations

Supplementary Files

Status:

Version 1

Broadening the candidate set of parametric models when extrapolating survival: the case for models with U-shaped hazards.

Status:

Version 1

Abstract

Figures

BACKGROUND

METHODS

Data sources

Analyses

RESULTS

Benfari et al.,2019 study of mortality in patients with four levels of functional tricuspid regurgitation6

Bae et al., 2013; Korean Cohort with degenerative osteoarthritic knee treated by microfracture11

Occlusion versus no-occlusion of left atrial appendage (RCT; Whitlock et al.,2021)

Total Hip Replacement (THR) (Registry studies)

Are bathtub and Chen model predictions supported by longer term follow up Registry findings?

Discussion

Conclusion

Declarations

References

Additional Declarations

Supplementary Files

Status:

Version 1

Benfari et al.,2019 study of mortality in patients with four levels of functional tricuspid regurgitation⁶

Bae et al., 2013; Korean Cohort with degenerative osteoarthritic knee treated by microfracture¹¹