 Research article
 Open access
 Published:
Estimating restricted mean survival time and expected lifeyears lost in the presence of competing risks within flexible parametric survival models
BMC Medical Research Methodology volume 21, Article number: 52 (2021)
Abstract
Background
RoystonParmar flexible parametric survival models (FPMs) can be fitted on either the causespecific hazards or cumulative incidence scale in the presence of competing risks. An advantage of modelling within this framework for competing risks data is the ease at which alternative predictions to the (causespecific or subdistribution) hazard ratio can be obtained. Restricted mean survival time (RMST), or restricted mean failure time (RMFT) on the mortality scale, is one such measure. This has an attractive interpretation, especially when the proportionality assumption is violated. Compared to similar measures, fewer assumptions are required and it does not require extrapolation. Furthermore, one can easily obtain the expected number of lifeyears lost, or gained, due to a particular cause of death, which is a further useful prognostic measure as introduced by Andersen.
Methods
In the presence of competing risks, prediction of RMFT and the expected lifeyears lost due to a cause of death are presented using RoystonParmar FPMs. These can be predicted for a specific covariate pattern to facilitate interpretation in observational studies at the individual level, or at the populationlevel using standardisation to obtain marginal measures. Predictions are illustrated using English colorectal data and are obtained using the Stata postestimation command, standsurv.
Results
Reporting such measures facilitate interpretation of a competing risks analysis, particularly when the proportional hazards assumption is not appropriate. Standardisation provides a useful way to obtain marginal estimates to make absolute comparisons between two covariate groups. Predictions can be made at various timepoints and presented visually for each cause of death to better understand the overall impact of different covariate groups.
Conclusions
We describe estimation of RMFT, and expected lifeyears lost partitioned by each competing cause of death after fitting a single FPM on either the logcumulative subdistribution, or causespecific hazards scale. These can be used to facilitate interpretation of a competing risks analysis when the proportionality assumption is in doubt.
Background
In observational studies of timetoevent data, researchers are often interested in decomposing the overall probability of death into component parts due to the event of interest, and competing, but mutually exclusive outcome events. For example, in cancer studies, it is of interest to partition the overall probability of death into the probability of death due to cancer and the probability of death due to other causes. These are referred to as causespecific cumulative incidence functions (CIFs) and are often chosen as the primary estimand of interest. The causespecific CIF gives the probability of dying from the cause of interest at a particular time whilst also being at risk of dying from other causes of death [1, 2]. In order to arrive at these quantities and to circumvent bias, methods that appropriately account for the competing nature of the events must be applied. The restricted mean failure time (RMFT) has been proposed as an alternative summary measure that is based on the area under the allcause probability of death up to a specific timepoint[3]. In an analogous way to the decomposition into causespecific CIFs, the RMFT can be further partitioned to give the expected number of life years lost due to a specific cause before a given timepoint. In this paper, we describe how the aforementioned measures can be obtained using a flexible parametric model (FPM) as the estimation approach by modelling covariate effects either using (1) the direct relationship with the causespecific CIF on the subdistribution hazards (SDHs) scale, or (2) modelling all causespecific hazard functions (CSHs) to obtain each causespecific CIF [4–7]. Choosing FPMs as the estimation method allows us to estimate effects conditional on covariates, and effects averaged over specific covariate distributions.
Forming contrasts to compare exposure groups is often a further key focus in many large populationbased studies. A common approach would be to report either causespecific hazard ratios (HRs), which measures the effect of an exposure group on the rate of dying from a cause of interest, or subdistribution hazard ratios (SHRs), which measures the effect of an exposure group on the risk of dying from a cause of interest, whilst assuming that the causespecific HR or SHR was constant over time. However, it is well known, for instance, that the HR for tumor size in cancer studies will vary over time since diagnosis, with stronger relative effects shortly after diagnosis [8–10]. When nonproportional hazards are present i.e. when the HR is expected to change over time, it has been argued that the HR as the target estimand is not appropriate and there are further issues in making causal inferences using HR measures due to its noncollapsibility as a relative risk measure [11]. As an alternative to the HR, estimation of the difference in restricted mean survival time (RMST), also known as the restricted mean lifetime (RMLT), as the primary estimand has been proposed [12–19]. This, in contrast to the HR, is known as a collapsible measure [11, 20]. Furthermore, this single summary measure can still be presented when relaxing the assumption of proportional hazards within the modelbuilding process. These can either be presented as conditional differences, which is the average covariate effect on the individual, or marginal differences, which is the average covariate effect on the population [21].
In the presence of competing risks, Andersen [3] introduces the analogue to the RMST measure for the CIF which gives the (total) number of years lost before a prespecified time, i.e. RMFT, and demonstrates how this can be partitioned to give the expected number of lifeyears lost due to each cause of death [22]. In his approach, he estimates RMFT and expected number of lifeyears lost using regression models with pseudoobservations [3, 23]. These models only allow prediction for specific quantities of interest and only at single timepoints. Therefore separate models must be fitted to estimate, for example, either the causespecific CIF or RMFT, when it may be of interest to obtain both and at various timepoints. For instance, to allow comparability and to obtain the entire picture of the impact of different groups on outcome, it has been suggested that differences in RMST, RMFT and therefore, expected number of lifeyears lost, should be reported alongside their respective survival, or cumulative incidence functions [24]. Alternatively, the RoystonParmar FPM approach for estimating RMST, which is extended for competing risks to estimate partitioned RMFT, as introduced in this paper, can be used [25]. In contrast to more popular approaches, such as the Cox model, a parametric estimate of the baseline hazard function is obtained as part of the full likelihood function. This is estimated using restricted cubic splines (RCS), allowing easy prediction of absolute comparisons between key quantities of interest. What’s more, standard errors for predictions can be estimated via the delta method, which offers computational advantages in larger data compared to approaches for nonparametric and semiparametric methods which use bootstrapping, or jackknife resampling methods [26]. Further advantages include the easy inclusion of timedependent effects using interactions with RCS for relaxing the proportional hazards assumption. Estimating both the baseline effects, and timedependent effects to model departures from the baseline using splines allows a unified approach for estimating all required parameters in order to obtain predictions of all quantities of interest. Therefore, we introduce in this paper how RMFT as the chosen estimand can be estimated using FPMs in the presence of competing risks on either the CSHs or cumulative incidence scale as the estimator [5, 7]. This extends on previous work by Royston and Parmar where estimation in the presence of competing risks is not considered [16]. This approach allows the researcher to obtain differences in effect between exposure groups either conditional on a set of covariates, or averaged over a covariate distribution, also known as marginal estimates. Furthermore, both marginal and conditional estimates can be obtained from the same model where the prediction of marginal estimates using standardisation is proposed [27, 28]. We, therefore, further demonstrate how difference in marginal estimates of RMFT as the chosen estimand for the comparison between covariate groups can be obtained within FPMs for competing risks.
We begin with a brief review of competing risks in the Overview of competing risks section and highlight particular interest in the causespecific CIF. This is followed by an introduction of RMFT as the chosen estimand in Overview of restricted mean survival time for competing risks section along with other useful measure such as expected lifeyears lost. The Flexible parametric survival models section details FPM approaches for estimation in the presence of competing risks. In the Estimation section, we show how absolute differences between RMFT and expected number of lifeyears lost are calculated to assess the impact of a covariate. We further demonstrate how these models can be used for easily obtaining marginal estimates and associated contrasts using standardisation in the Standardisation for marginal differences section. For illustration of these various measures, English colorectal cancer data obtained from National Cancer Registration and Analysis Service (NCRAS) is analysed in the Results: colorectal cancer survival in England section where comparisons between the most and least deprived colorectal cancer patients are made, accompanied by Stata code for estimation in Appendix: Stata code for obtaining predictions. Finally, the paper is concluded with a discussion on the use and estimation of RMST in the presence of competing risks within FPMs. Although we specifically consider application to cancer studies, where the event of interest is death from cancer, the methods are generalizable to other timetoevent data and therapeutic areas.
Methods
Overview of competing risks
In the presence of competing risks, an individual is at risk of failing from more than one event where the occurrence of one event means that others cannot occur. In the context of a cancer survival study, this is when a patient can die from a multitude of other causes as well as the cancer itself. However, if the patient dies from one of these other causes, it means that the time at which the patient would have died from cancer is never observed. One of the key quantities, and often the chosen estimand of interest within this framework, is the causespecific CIF [1].
Causespecific CIF
Let T be a nonnegative random variable for the time to death from any cause. Furthermore, let D denote the cause of death in the presence of k=1,…,K competing risks, where D=1,…,K. It follows that the estimand, causespecific CIF, F_{k}(t), is defined as,
This is interpreted as the probability of dying from cause k by time t whilst also being at risk of dying from other competing causes of death. Note here that the causespecific CIF is an improper distribution function since the integral of F_{k}(t) at infinity is always less than 1 [3].
The target estimand, the causespecific CIF, can be calculated using either all k CSH functions, or by utilising the onetoone relationship between the causespecific SDH function. These are briefly introduced below.
Causespecific hazards
The CSHs, \(h^{cs}_{k}(t)\), give the instantaneous mortality rate from a particular cause k given that the patient is still alive at time t in the presence of all the other causes of death such that,
It follows that the target estimand, the causespecific CIF, can be calculated as a function of all k CSH functions,
where \(S(t) = \exp {\left (\sum _{k=1}^{K} \int _{0}^{t} h_{k}^{cs}(u)\text {d}u\right)}\) is the allcause survival function.
Subdistribution hazards
Alternatively, Gray [29] introduces the SDH function for cause \(k, h_{k}^{sd}(t)\), which offers a direct onetoone relationship with the causespecific CIF estimand. This has the following mathematical formulation,
which is interpreted as the instantaneous “sub”rate of failure at time t from cause k amongst those who are still alive, or have died from any of the other K−1 competing causes excluding cause k [30].
This is not defined as a typical epidemiological rate since the riskset includes those that are either still alive or have died from a competing cause of death. However, if individuals do not experience the competing event, then the SDH rate and the CSH rate are both equivalent [31]. It should be noted that, due to the nature of the riskset in the definition of a SDH, it is very difficult to interpret [30, 32, 33].
The causespecific CIF estimand can be directly obtained from the SDH for cause k using the standard survival transformation of the cumulative SDH function for cause \(k, H^{sd}_{k}(t)\), such that,
This shows that a onetoone correspondence is maintained between the SDH function for a specific cause of death and the causespecific CIF.
The choice of which scale to model on depends entirely on the research question to be answered which would relate to other quantities specific to the modelling approach that may be of interest. For instance, if primary interest is in aetiological outcome, then the estimand of interest would be on the CSH rates. For interest in prognostic outcome, one may wish to quantify effects on the risk of dying from a specific cause of death. In this case, the estimand of interest would be the causespecific CIF, which can be obtained as function of all CSHs, or through the SDH for cause k. Further discussion on this topic is provided elsewhere [4, 34].
Overview of restricted mean survival time for competing risks
The RMST measure quantifies the average survival, or time lived, of a patient from time 0 up to a predefined timepoint, t^{∗}. In the absence of competing risks, the RMST before t=t^{∗},μ(t^{∗}), of a random variable T is equal to the expectation of min(T,t^{∗}). RMST, in the absence of covariates, can be expressed as the estimand,
where S(t) is the allcause survival function. If time is measured in years, this is the average lifeyears lived before time t^{∗}. The choice of t^{∗} should be predetermined and clinically motivated, and will vary by, for example, cancer types [15, 16]. This is also often chosen at maximum followup time [13, 35].
In addition to this, Andersen [3] proposes calculation of the expected number of years lost before time t^{∗} such that the estimand can be defined as,
Expected loss in life due to a cause of death
In the presence of competing risks, Andersen [3] shows that the (total) number of years lost, L(0,t^{∗}), can be decomposed into the number of years lost due to each cause k [22]. It follows that since,
then the RMST in Eq. 6 can be expressed as a function of each causespecific CIF through the following integral,
Equation 7 can also be written as a sum of the integral of each causespecific CIF such that,
which may also be referred to as restricted mean failure time (RMFT). It follows that RMFT can be partitioned where we have the estimand,
which gives the expected number of years lost due to cause k before time t^{∗}.
Flexible parametric survival models
For competing risks data, many adopt the causespecific Cox proportional hazards model, or the Fine & Gray approach as the chosen estimator for the estimands introduced in the Overview of competing risks and Overview of restricted mean survival time for competing risks sections. Here, we propose the use of FPMs as the chosen estimator in order to obtain the estimand of interest. FPMs are increasing in popularity since the baseline SDH or CSH function is estimated as part of a fully specified likelihood function and allows the estimation of various estimands from a single model [5, 7]. These models were introduced for standard survival data (in the absence of competing risks) on various scales by Royston and Parmar [9] using a general link function, g(·), to better capture and represent the behaviour of real world data. To increase flexibility and more accurately capture complex shapes of the cumulative hazard function, Royston and Parmar [9] proposed the use of RCS (see Appendix A). Under the assumption of proportional hazards, Rutherford et. al [36] showed in simulations that FPMs more accurately capture complex shapes of hazard functions. They further illustrated that unbiased estimates of the HRs were obtained. Given a vector of M knots, m, and a vector of M−1 parameters, γ, with a RCS function, s(ln(t);γ,m) we have that,
where, β, is a vector of coefficient parameters and, x, is a vector of covariates.
Equation 12 can also be easily extended for timedependent effects to model nonproportionality by fitting interactions between the associated covariates and the spline functions. Using this interaction, a new set of knots, m_{e}, are introduced, which represent the e^{th} timedependent effect with associated parameters α_{e}. If there are e=1,⋯,E timedependent effects, Eq. 12 can be extended such that,
Nonproportional hazards are a common occurrence in studies with long followup time, or, in the context of cancer studies, when the effect of covariates (e.g tumor size, or treatment) on cancerrelated mortality varies over time [8–10, 19]. FPMs, extended for timedependent effects as in the Equation above, have also been shown to accurately capture complex shapes of the hazard function with timedependent effects i.e. where there is nonproportionality in the hazards [37]. This result is consistent with what was shown by Rutherford et. al. for FPMs without timedependent effects i.e. proportional hazards, as mentioned above [36]. Further technical details on FPMs for standard survival data in the absence of competing risks can be found elsewhere [9, 25, 38].
The models described in Eqs. 12 and 13 can be fitted on either CSHs scale [7], where G_{k}(t∣x)=S_{k}(t∣x), or cumulative incidence scale [5, 6], where G_{k}(t∣x)=1−F_{k}(t∣x), based on different link functions, g(·). The relationship of these with the causespecific CIF are defined in the Causespecific hazards and Subdistribution hazards sections. Therefore, it follows that, using a complementary loglog link function, the corresponding logcumulative CSHs FPM (otherwise referred to as a causespecific FPM), is,
and can be fitted in a similar way to the standard FPM. Alternatively, models for all k causes can be fitted simultaneously by restructuring the data as described by Hinchliffe et. al. [7].
The logcumulative SDHs FPM for cause k (also known as the flexible parametric cumulative incidence model, or FPCIM), on the other hand is defined as,
and can be fitted using the approach outlined using either the full likelihood function as described by Mozumder et. al. [5] or by using timedependent censoring weights, similar to the FineGray model, as detailed by Lambert et. al. [6]. As previously mentioned, alternative link functions are also available for models on either scale. See for example, Lambert et. al. [6].
Estimation
Causespecific cumulative incidence function
If modelling on the cumulative incidence scale using SDHs, after fitting the FPCIM in Eq. 15, the causespecific CIF is obtained by the following,
Alternatively, when modelling on the CSHs scale, after fitting the causespecific FPM in Eq. 14, and as shown in Eq. 3, the integral below must be evaluated in order to obtain the causespecific CIF,
where the predicted CSH function is,
and the predicted allcause survival function is,
However, as the above integral is not of closed form, numerical approximation techniques must be used. Here, the GaussLegendre quadrature approximation method is used [39]. Details of this method is provided in Appendix B. Therefore, after fitting the causespecific FPM for each k causes, the predicted causespecific CIF at t_{1},⋯,t different timepoints over an interval [0,t] is approximated by applying Gaussian quadrature rules with W(u)=1 such that,
where, \(\widehat {f}_{k}^{*}(t)\), is the “sub”density function such that,
Restricted mean failure time and expected number of lifeyears lost due to each cause of death
If RMFT is the chosen target estimand of interest, this can be predicted as the integral under the allcause CIF such that,
where the predicted expected number of lifeyears lost before time t^{∗} due to each cause k is,
Again, as above in Eq. 17, as the integral is of closedform, we use the GaussLegendre quadrature approximation technique to numerically evaluate,
It follows that the RMST can also be obtained by,
Conditional differences
In populationbased studies, i.e. nonrandomised studies, it may be of interest to make absolute or relative comparisons between different covariate groups. As an alternative summary measure, or estimand, to the HR, we can calculate the difference in RMST between two covariate groups, or the difference in expected loss in life due to different causes [19]. Let X be a binary covariate that denote the group of interest and Z be the set of measured covariates with a specific covariate pattern z_{j}. To estimate the average number of life years gained in group X=0 compared to group X=1, we have that,
Alternatively, we can also estimate the expected reduction in the loss (or gain) in life due to cause k by,
Partitioning in this way is particularly useful if covariates act differently on different causes of death. For example, those from a particular covariate group may lose (or gain) some lifeyears due to a specific cause of death in comparison to another covariate group.
Absolute measures of gains or losses in years of life are presented above as potential estimates of interest. To obtain relative measures, the ratio between the RMST estimates, or expected loss in life due to cause k for the two covariate groups are calculated. Extension can also be made for comparisons on a unit increase in a continuous covariate Z, and for timedependent effects.
Standardisation for marginal differences
Regression standardisation is part of the estimator that can be used to obtain marginal predictions for different covariate groups at each observation given a set of measured confounders [27, 28]. Here, we apply standardisation to RMST and causespecific CIFs estimates obtained from a flexible parametric competing risks survival model. In this case, it is of interest to compare the average lifeyears lived before time t^{∗} between two different groups [17, 18]. This is done by obtaining marginal estimates which are calculated as an average over every individual in the observed dataset. This enables comparisons that solely focus on the differences between the two groups of interest by forcing the same covariate distribution over multiple confounders. If all exposures and confounders are measured at baseline, this is essentially equivalent to the Gformula [40]. For example, to compare males and females, estimates must be standardised by age in order to force the same age distribution for both males and females. Extension can be made for multiple covariates and other potential confounders. This is calculated using an average of RMST estimates for each patient to summarise the risk for a certain covariate group. For instance, let X be an indicator variable that denotes the group of interest and Z be the set of measured covariates. Then the predicted RMST estimate for the i^{th} individual, where i=1,…,N, is,
where X is fixed to a specific value, x, and Z is the observed covariate pattern, z_{i}, for the i^{th} individual. We can then average over the marginal distribution of Z for all the predicted restricted mean life estimates obtained for each individual i such that,
This allows us to calculate marginal differences between covariate groups. For example, between group X=0 and group X=1, the marginal difference in RMST is,
In recent literature, some have advocated the use of RMST as a causal measure [41, 42]. For a causal interpretation, the consideration of additional assumptions are required and by adjusting for all appropriate confounders, these measures can be extended and interpreted as causal effects and thus, used as an estimand [21]. This is because, as shown above, they provide marginal comparisons averaged over the same covariate distribution by using standardisation. Standardisation, otherwise referred to as Gcomputation, has also been highlighted by Gran et al. [43] as an approach for obtaining useful summary causaleffect measures in more complicated multistate models. However, this is beyond the scope of the paper and estimation of causal effects are not explicitly discussed here. Note also that we only consider timefixed confounders and that there are additional complexities when considering timedependent riskgroups [44].
Results: colorectal cancer survival in England
Data
Data was obtained from the National Cancer Registration and Analysis Service (NCRAS) to illustrate the estimation of various measures introduced in the Overview of restricted mean survival time for competing risks section. The data consist of English colorectal (ICD10: C18, C19 and C20) male and female cancer patients aged between 45 and 90 years old. Patients are diagnosed on or after 1998 are included with followup restricted to either 10 years or censored at 31 Dec 2013, whichever comes first. Analysis is further restricted to patients from the most or least deprived groups as defined by the upper and lower quintiles of the English index of multiple deprivation 2010 (IMD 2010). These groups are selected to simplify analysis and to make for easy illustration of presenting different metrics to allow comparisons between the two groups. The final data consisted a total of 159,022 individuals of which 48,845 die from cancer, 7,987 from cardiovascular disease (CVD) and 32,133 from other causes. In Appendix C, summary statistics on the age distribution, and number of patients in each deprivation and sex groups are provided.
Model
For demonstration purposes, predictions are obtained after fitting an FPCIM simultaneously for all k causes of death and standard errors for confidence intervals (CIs) are obtained using the delta method. However, predictions are also available after fitting causespecific FPMs. This paper focusses on the various estimands we can obtain from such models, namely, the RMST measure and expected lifeyears lost.
Models are fitted simultaneously for all k causes of death using the approach of Lambert et al. [6] and Geskus [45]. This fits the model after restructuring the data and applying timedependent weights that are obtained parametrically to the censoring distribution of the competing causes of death. Alternatively, using the approach described by Jeong and Fine [46], models can be fitted on individuallevel data using the full likelihood function [47]. Models for each of the causes of death include sex, IMD 2010 deprivation group (upper and lower quintile only) and a nonlinear effect of continuous age using RCS with 3 DF centred at 45 years old at diagnosis. Timedependent effects to relax the proportionality assumptions are included for sex, nonlinear age and deprivation group with 2 DF and 3 DF are used for the baseline RCS function. In order to evaluate whether assuming nonproportional (subdistribution) hazards was more sensible, and is more consistent with the data, a likelihood ratio test was performed. This compared the FPCIM with timedependent effects to relax the proportionality assumption to the one without that assumed proportional SDHs. The likelihood ratio test statistic was 752.94 and the associated pvalue was less than 0.0001. This shows that relaxing the proportionality assumption leads to a statistically significant improvement in model fit. Note that this is an illustrative model and we therefore omit formal evalutation of the model performance. When evaluating the model in practice, we recommend conducting a sensitivity analysis, particularly in the selection of the number of knots. This can be done by comparing the Akaike information criterion and the Bayesian information criterion as an informal guide to selecting the appropriate number of knots and covariates [6].
Analysis of data with conditional estimates
Causespecific cumulative incidence functions
Causespecific CIFs are presented in Fig. 1 for male colorectal cancer patients. The probability of dying from cancer at 10 years from diagnosis for the most deprived male patients is approximately 36.5% (95% CI: 35.5%, 37.5%) for those aged 50 years old at diagnosis. This slightly increases to approximately 40.5% (95% CI: 39.8%, 41.1%) for those aged 80 years old at diagnosis. However, the largest change is in the probability of dying from other causes and CVD which have an increasing contribution to the probability of dying from any cause for older male patients from the most (and least) deprived groups. For instance, the probability of dying from any cause by 10 years from diagnosis for the most deprived 50 year old male patients at diagnosis is 53.6% of which 17.1% is due to other causes and CVD. In contrast, the allcause probability of death for the most deprived male patients aged 80 years old diagnosis is much higher at 92.5%. However, although the probability of dying due to cancer has only increased from 36.5% to 42.5%, much of the overall probability of dying is due to other causes (38.4%) and CVD (13.6%).
Absolute CIF differences between the most and least deprived male patients aged 50, 65 and 80 years old at diagnosis are presented on the third row of Fig. 1. This shows that, for 50 year olds, the difference between CIFs for the most and least deprived groups are similar for deaths due to cancer and other causes. There is very little difference between the two deprivation groups for deaths due to CVD, however, this is due to a generally very low probability of death due to CVD. On the other hand, for older male patients, the difference in the probability of dying from other causes and CVD between the most and least deprived is larger and increases over time. This leads to a greater disparity in the probability of dying from other causes and CVD between the most and least deprived patients compared to the difference in the probability of dying due to cancer. Furthermore, after approximately 1 year from diagnosis for 65 year olds, and 2 years for 80 year olds, the difference in the probability of dying due to cancer for the most deprived compared to the least deprived patients reduces. This change in difference between the most and least deprived is greatest for the 80 year old male patients with cancerspecific CIF difference reducing from approximately 4.6% (95% CI: 4.2%, 5.0%) at 1 year from diagnosis to 3.2% (95% CI: 2.6%, 3.7%) by 10 years from diagnosis.
Restricted mean failure time and expected number of lifeyears lost due to a particular cause of death
As discussed in the Overview of restricted mean survival time for competing risks section, as a useful summary measure, the RMST estimate can be obtained. This is equivalent to the white area of the associated stacked plot in Fig. 1 up to t^{∗} for a particular covariate pattern. Conversely, the area of the stacked areas give an estimate of the RMFT. The area of each of the partitioned stacks for each of the respective causes of death yield the expected life years lost due to cancer, CVD and other causes. These are presented for the most and least deprived 50, 65 and 80 year old male patients in Fig. 2. Each of the stacks represent the average lifeyears lived in total and lifeyears lost due to a specific cause. The plots here present lifeyears lost and lived before different points in time up to 10 years from diagnosis. However, particular interest here is in the lifeyears lived, or lost, before 10 years from diagnosis. For example, total average lifeyears lived before 10 years from diagnosis for the most deprived 50 year old male patients is 3.99 years (95% CI: 3.84 years, 4.14 years). Of the 6.01 years of the total lifeyears lost, 2.72 years (95% CI: 2.60 years, 2.85 years) are due to cancer, 0.07 years (95% CI: 0.06 years, 0.09 years) are due to CVD and 1.19 (95% CI: 1.11 years, 1.28 years) due to other causes.
Table 1 presents differences in lifeyears lost due to each cause of death before 10 years from diagnosis between the most and least deprived groups for 50, 65 and 80 year olds, along with their associated 95% CIs. The absolute estimates of expected lifeyears lost for the most and least deprived patients at the individual ages are also presented. This provides us with an understanding of how many additional lifeyears most deprived patients are expected to lose due to a specific cause of death in comparison to the least deprived patients. For instance, at 10 years from diagnosis, 50 year old male patients from the most deprived group lose an additional 0.32 (95% CI: 0.28, 0.36) lifeyears due to cancer, 0.01 (95% CI: 0.01, 0.02) lifeyears due to CVD and 0.33 (95% CI: 0.30, 0.36) lifeyears due to CVD compared to the least deprived group. For older male patients aged 80 years old, there is a greater disparity in lifeyears lost due to CVD (0.16 lifeyears) and other causes (0.76 lifeyears) between the most and least deprived.
Analysis of data with marginal estimates
When interest is in the covariate effects of particular groups, for example, between deprivation groups, it is useful to obtain standardised estimates as described in the Standardisation for marginal differences section. By marginalising over the same covariate distribution, fairer comparisons can be made between particular covariate groups of interest. In this example, we standardise by age and sex in order to summarise the differences in survival between patients from the most and least deprived groups.
Causespecific probability of death for the Most deprived compared to the least deprived
Figure 3 illustrates standardised CIFs stacked for each cause of death and Fig. 4 presents absolute risk differences for each cause between the least and most deprived patients. As illustrated in Fig. 3, patients from the most deprived group have a higher probability of dying from any cause (73.8%) compared to those from the least deprived group (63.3%). However, when partitioned into the different causes of death, the difference in total mortality between the most and least deprived groups is mostly due to other causes and CVD as indicated by the area proportions. The causespecific marginal risk difference between the most and least deprived are presented in Fig. 4 along with their respective 95% CIs. As can be seen here, the largest difference in risk is due to other causes and the largest difference in risk between the least and most deprived groups is due to other causes at 10 years from diagnosis (6.3%; 95% CI: 5.8%, 6.9%). Generally, the disparity in the probability of dying from other causes or CVD between the most and least deprived patients continues to increase over followup time. However, the cancerspecific risk difference between the most and least deprived increases only for the first 2 years. After this point, the disparity in the probability of dying due to cancer between the most and least deprived begins to decrease.
Expected number of lifeyears lost for the Most deprived compared to the least deprived
In Fig. 2, the expected lifeyears lost and total average lifeyears lived were presented for each cause of death before various timepoints, t^{∗}. By obtaining marginal estimates through standardisation over age and sex, we can focus on specific comparisons between the least and most deprived patients. The marginal expected lifeyears lived for each cause of death and total average lifeyears lived before each time, t^{∗}, are similarly illustrated in Fig. 5. If t^{∗}=10, then we have that the total average lifeyears lived before 10 years from diagnosis for the most deprived patients is 4.39 (95% CI: 3.78, 5.00). Of the 5.61 total expected lifeyears lost, 3.03 (95% CI: 2.66, 3.46) years are lost due to cancer, 0.46 (95% CI: 0.27, 0.81) years due to CVD and 2.11 (95% CI: 1.76, 2.53) years due to other causes. By obtaining marginal estimates of expected lifeyears lost, we are able to directly compare both deprivation groups and determine the additional lifeyears lost for patients that are the most deprived standardised by age and sex. Thus, where t^{∗}=10, we have that the additional lifeyears lost due to cancer, CVD and other causes before 10 years from diagnosis for the most deprived patients is 0.31 (95% CI: 0.25, 0.37), 0.05 (95% CI: 0.02, 0.08) and 0.44 (95% CI: 0.33, 0.54) lifeyears respectively.
Discussion
This paper presents novel estimation of RMLT and expected lifeyears lost from within the flexible parametric survival modelling framework in the presence of competing risks. This can be done either on the CSHs or cumulative incidence scale and allows easy incorporation of timedependent effects to relax the proportionality assumption. These also offer additional advantages over the more popular Cox PH and Fine and Gray models [5, 7]. In particular, we illustrate how one can easily obtain comparative predictions based on the expected number of lifeyears lost due to a specific cause of death in addition to other useful estimands, such as absolute differences in the cumulative incidence functions. A common approach for obtaining marginal estimates uses inverse probability weighted estimating equations. However, different estimators need to be calculated subject to whether it is of interest to obtain marginal or nonmarginal/conditional estimates [48, 49]. On the other hand, marginal estimates using standardisation are easily obtained in addition to conditional estimates within the FPM approach from a single model. FPMs in both a standard survival analysis and for competing risks data offer numerous advantages in prediction, specifically, through its estimation of the baseline hazard function using RCS and easy inclusion of timedependent effects. In spite of this, it is also important to consider limitations that are often highlighted. One such limitation is the problem of choosing the appropriate number of knots for the underlying baseline hazard function using RCS, and for when including timedependent effects when relaxing the proportional hazards assumption. However, a number of extensive simulation studies have been carried out evaluating how many knots are required in order to accurately capture (both simple and complex i.e. timedependent) shapes of the baseline hazard function. For instance, Bower et. al. [37] and Syriopoulou et. al. [50] both conclude predictions are not sensitive to the choice in the number of knots, provided that a sufficient number of degrees of freedom are used. In other words, too few degrees of freedom may be too simple to accurately capture the effect, and too many will lead to overfitting. As a guideline, 5 degrees of freedom to capture baseline effects and 3 degrees of freedom for any timedependent effects are suggested as a starting point. However, it is further suggested that for each individual study, sensitivity analyses are carried out in order to assess model fit and robustness to the choice in degrees of freedom [37, 50]. Syriopoulou et. al [50] also reach similar conclusions with extension to marginal modelbased estimates when obtaining predictions using standardisation. Alternatively, a penalised approach for choosing the appropriate number of degrees of freedom for RCS can be used [51]. The interpretation of the RMLT measure also has some notable limitations. Although communication in terms of changes in lifeyears lost to clinicians and patients rather than probabilities is attractive, applying an upper bound, t^{∗}, to the time interval may add some difficulty in understanding of the measure. This is because, RMLT for an arbitrary choice of t^{∗} can only be used to estimate the average risk within a restricted time period for a group of patients. Furthermore, it should be highlighted that the expected lifeyears lost makes comparison with an immortal cohort where patients are alive for the whole interval from 0 to time t^{∗}. A similar “unrestricted” measure that do not compare to an immortal cohort can be estimated within the relative survival framework based on extrapolation of the excess hazard rate. This is usually referred to as the number of life years lost, or the loss in expectation of life and is calculated based on a comparison of the lifeexpectancy of cancer patients to a comparable population group who are assumed to be cancerfree [52–54]. However, this relies on the assumption that this extrapolation is appropriate which is not made for the RMLT estimate. In addition to the above, due to the dependence of the interpretation of RMST on followup time, comparison between different studies, for example, between countries, becomes difficult. It has also been further shown that the difference in RMST between two covariate groups depends on the outcome rates within each group. Therefore, it is recommended that differences in RMST, RMFT and expected number of lifeyears lost, are reported alongside their respective survival, or cumulative incidence functions, in order to allow comparability and to obtain the entire picture of the impact of different groups on outcome [24]. This further points to additional advantages of estimation of RMFT within the flexible parametric modelling framework, as these additional measures are easily obtained from the same model.
Conclusions
The RMLT measure is presented as a useful summary measure with an attractive interpretation which can aid in the analysis of competing risks data. As discussed by others, it is also useful to present estimated causespecific CIFs alongside CSHs [6, 34]. We propose FPMs as the chosen estimator as it allows easy estimation of various estimands from a single model providing both conditional and marginal estimates. Note that, although not discussed here, if appropriate confounders are adjusted for, one can also infer causal effects between two groups using standardisation. However, one must also consider the additional complexities and issues in interpretation with the inclusion of timedependent riskgroups [44]. Furthermore, the RMLT measure can be easily extended for obtaining conditional estimates, for example, the average lifeyears lived before t^{∗} years given survival to time t_{0} from diagnosis. Example Stata code for the model and prediction of measures provided in this paper is outlined in Appendix: Stata code for obtaining predictions.
Appendix A: Restricted cubic spline variables
Given a vector of M knots, m and a vector of M−1 parameters, γ, with M−1 degrees of freedom (df), the restricted cubic spline function, s(ln(t);γ,m), is defined as,
Where z_{1},⋯,z_{(M−1)} are the basis functions of the restricted cubic splines and are defined as,
where,
and
Usually, M knots are placed at equally spaced centiles of the distribution of the uncensored logsurvival times including two boundary knots at the 0^{th} and 100^{th} centiles.
Appendix B: Gaussian quadrature
With the general Gaussian quadrature rule, the integral of any polynomial function, g(u), over the interval [−1,1] can be evaluated. This performs best for integrals that can be approximated by a polynomial function of degree 2m−1, where m is a predetermined number of points, otherwise known as nodes, or abscissae. Hence, this integral can be evaluated for,
where, W(u), is a known weighting function. Here, the integral, e.g. the causespecific cumulative incidence function, is calculated using GaussLegendre quadrature, with W(u)=1. With this, based on a set of predefined number of nodes, \(u^{\prime }_{i}\), and associated Lagrange polynomials of degree m,P_{m}(u), weights, \(w^{\prime }_{i}\), for i=1,…,m, are obtained such that,
and are provided by Abramowitz and Stegun [55]. Therefore, Eq. 35 is approximated by,
However, for survival data, functions are evaluated over an interval [0,t]. Therefore, to apply the Gaussian quadrature rule in Eq. 35, integrals over the interval [0,t] must be changed to an interval over [−1,1] such that,
Therefore, a function evaluated at t_{1},…,t different timepoints over an interval [0,t] is approximated by applying Gaussian quadrature rules with W(u)=1 such that,
Appendix C: Additional summary statistics
Table 2 provide summary statistics on the distribution of key covariates of interest for inclusion in analysis i.e. sex, deprivation group (least/most deprived) and age, by cause of death, and in total.
Figure 6 represents the causespecific cumulative incidence functions estimates obtained by the nonparametric AalenJohansen estimator. This summarises the probability of dying from each cause of death by sex and deprivation groups.
Figure 7 illustrates the allcause survival probabilities obtained by the nonparametric KaplanMeier estimator. This summarises the allcause probability of survival by sex and deprivation groups.
Stata code for obtaining predictions
This appendix outlines Stata code used to obtain predictions presented in the paper. Some userdefined Stata commands are required which can be installed from the Boston College Statistical Software Components (SSC) archive by calling,
The following must be installed before running the code:

stpm2: To fit the flexible parametric models described in Flexible parametric survival models section.

rcsgen: To generate the restricted cubic spline functions.

stcrprep: To restructure data and calculate timedependent censoring weights in order to fit models on the subdistribution hazards scale using standard Stata commands.
To obtain marginal (and nonmarginal) estimates using standardisation, the standsurv command must be installed. This will be released on SSC soon, however, in the meantime, it can be installed by running,
Preparing the data for analysis
To prepare the data for a survival analysis in Stata, we must first run the stset command. We identify the variable that records survival time (in days), exit2, the indicator variable for cause of death, cod, where death from cancer = 1, CVD = 2 and other causes = 3 and finally the variable for date of diagnosis, dx. The scale option is used to transform the survival time into years from days and we use the exit option to restrict followup time to 10 years from diagnosis and censor those still alive at 2014. In order to ensure that the death indicator, _d, generated after stset matches the death indicator for cause of death, we create a new cause of death indicator, cod2, so that those who die either after 10 years from diagnosis or 2014 are administratively censored. Finally, to generate restricted cubic spline variables for the nonlinear effect of age centred at 45 years old at diagnosis, we use rcsgen. For 3 degrees of freedom, 3 new age spline variables are created, rcsage1 − rcsage3, and we store knot positions and matrix for orthogonalization which are required for postestimation predictions at specific ages.
To restructure the data and calculate the timedependent censoring weights so that we may fit a model on the subdistribution hazards scale, we use stcrprep[56]. Here, we specify wtstpm2 to estimate the censoring distribution using a RoystonParmar flexible parametric model with covariates included in the censcov option. The data is restructured based on the variable failcode, which splits the data according to the cause of interest. This is used to fit identify for which cause the model is to be fitted for. For clarity, we create dummy variables for each of the causes of death from failcode and generate _cancer, _cvd and _other. Another indicator variable, event, is also created to identify at which split time interval, or row, death (from any cause) is observed for that patient. To incorporate the calculated weights from stcrprep, we must stset the data again with tstart and tstop. These are also provided by stcrprep and give the times at which an individual starts and stops being at risk.
Model
The model described in Flexible parametric survival models section can be fitted in two ways after preparing the data. We can either fit separate models for each of the causes of death, or fit a single model to cancer, CVD and other causes simultaneously. Here, we demonstrate for the latter to make illustration of the code for obtaining predictions postestimation easier. However, in order to fit the equivalent single model with coefficients comparable to the models fitted individually to each of the causes of death, the knot locations on the causespecific survival time distributions must be stored. These are stored in global macros for each of the causes of death.
Here we define a global macro of the list of covariates to be included in the single model. As the data is stacked, interactions need to be created between the covariates and the indicator variable for each cause of death. See Lunn and McNeil[57] for further details. The baseline coefficient, i.e. the constant in the causespecific model, is calculated in _cancer, _cvd and _other. We therefore fit a model for each of the causes of death simultaneously without a constant using nocons and the baseline splines using rcsbaseoff. Instead, the baseline splines are specified as timedependent splines for the coefficient that corresponds to the constant in its respective model for that particular cause of death. These were stored in the global macro bknotstvc_opt. Since knots are specified according to the time scale, rather than the logtime scale, the knscale(time) option is used.
Predictions
Although standsurv was written for obtaining marginalised predictions, it can also be used to obtain nonmarginalised estimates. This is done by simply specifying the entire covariate pattern so that the predictions are not averaged over any covariate distribution. To obtain predictions at a specific age, we need to calculate the spline variables at that particular age centred at 45 years old with the same knot locations and projection matrix as before. The spline variables are stored in the local macros c1, c2 and c3. An example is given below when the cause of interest is cancer and we want to make comparisons between the most and least deprived male patients aged either 50, 65, or 80 years old at diagnosis.
As we do not average over each observation, we must tell standsurv to only take the first observation in the stacked data to calculate nonmarginalised predictions. This is done using if _n == 1. The failure option is used to obtain the cumulative incidence functions that is specified in each at option. To calculate the difference between at1 and at2, we use contrast(difference).
Since we are making predictions at particular covariate patterns for each of the causes separately, specifying rmft gives us estimates of the expected lifeyears lost due to a particular cause of death. To calculate RMLT, we need to take the sum of all of the at options, where the expected lifeyears lost due to cancer, CVD and other causes is specified in each. We do this by creating our own contrast in a userdefined mata function which can be called in the option userfunction. An example of this is also given below.
In order to obtain marginalised estimates, in each at option, only the covariate pattern for the group of interest need to be given. For the covariate distribution that we want to average over, as we have created interactions between the covariates and the causes of death, these must be mapped to each covariate e.g. sex_cancer = sex. The others are excluded from the at option for the other causes of death. In this case, because we want to average over covariates that we wish to standardise by, we need to identify the row for each patient in the stacked data that corresponds to the failure time of that individual. This is done by creating the indicator variable first and using it as an if condition in standsurv. As before, we give an example for specifying macros for use in the at options for deaths due to cancer.
The causespecific CIF differences are thus calculated as follows,
As highlighted above, we can write userfunctions to define our own contrasts. Below is an example for when interest is in calculating the difference in RMLT between the most and least deprived patients.
Availability of data and materials
The data that support the findings of this study are available from Public Health England (https://www.gov.uk/government/publications/accessingpublichealthenglanddata/aboutthepheodrandaccessingdata), but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available.
Abbreviations
 CSH:

Causespecific hazards
 CIF:

Cumulative incidence function
 CI:

Confidence interval
 CVD:

Cardiovascular disease
 DF:

Degrees of freedom
 FPM:

Flexible parametric model
 FPCIM:

Flexible parametric cumulative incidence model
 HR:

Hazard ratio
 LYL:

Lifeyears lost
 RCS:

Restricted cubic splines
 RMFT:

Restricted mean failure time
 RMLT:

Restricted mean lifetime
 RMST:

Restricted mean survival time
 SDH:

Subdistribution hazards
 SHR:

Subdistribution hazard ratio
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This project was funded by Cancer Research UK [Grant Number C1483/A18262] and supported the research presented in this article. Cancer Research UK had no role in the methods development, study design, data collection, analysis and interpretation of the data, or in the writing of the article. The article processing charge for open access was funded by Cancer Research UK as a member of the 2020/21 Charity Open Access Fund (COAF).
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Contributions of each author to the work detailed in the manuscript are as follows. All authors, SIM, MJR and PCL, equally contributed towards the conception and design, acquisition of data, and interpretation of data. Specifically, SIM carried out the analysis of the data and prepared the manuscript. PCL and MJR supervised the project and contributed comments and suggestions throughout. All authors were involved in the drafting and submission of this research article. The author(s) read and approved the final manuscript.
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Mozumder, S.I., Rutherford, M.J. & Lambert, P.C. Estimating restricted mean survival time and expected lifeyears lost in the presence of competing risks within flexible parametric survival models. BMC Med Res Methodol 21, 52 (2021). https://doi.org/10.1186/s12874021012130
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DOI: https://doi.org/10.1186/s12874021012130