 Research article
 Open Access
 Published:
Correcting for misclassification and selection effects in estimating net survival in clinical trials
BMC Medical Research Methodology volume 19, Article number: 104 (2019)
Abstract
Background
Net survival, a measure of the survival where the patients would only die from the cancer under study, may be compared between treatment groups using either “causespecific methods”, when the causes of death are known and accurate, or “populationbased methods”, when the causes are missing or inaccurate. The latter methods rely on the assumption that mortality due to other causes than cancer is the same as the expected mortality in the general population with same demographic characteristics derived from population life tables. This assumption may not hold in clinical trials where patients are likely to be quite different from the general population due to some criteria for patient selection.
Methods
In this work, we propose and assess the performance of a new flexible populationbased model to estimate longterm net survival in clinical trials and that allows for causeofdeath misclassification and for effects of selection. Comparisons were made with causespecific and other populationbased methods in a simulation study and in an application to prostate cancer clinical trial data.
Results
In estimating net survival, causespecific methods seemed to introduce important biases associated with the degree of misclassification of cancer deaths. The usual populationbased method provides also biased estimates, depending on the strength of the selection effect. Compared to these methods, the new model was able to provide more accurate estimates of net survival in longterm clinical trials.
Conclusion
Finally, the new model paves the way for new methodological developments in the field of net survival methods in multicenter clinical trials.
Background
Recent advances in treatment have extended the expected survival of cancer patients up to and even beyond ten years after diagnosis [1]. This leads to a nonnegligible risk of death due to other causes than the cancer of interest, more particularly for older patients. In this context it is of importance to account for the competing causes of death, and methods have been developed to estimate the specific survival of the cancer of interest while estimating the specific survival function(s) of the other(s) cause(s) of death [2]. In this context, estimation of cancerspecific survival can be interpreted as the survival for cancer patients in presence of other cause(s) of death. Another approach relies on the estimation of net survival, representing the survival that would be observed if the cancer under study were the only cause of death [3]. A main interest of this latter approach, is to be interpreted as the survival from cancer in the absence of other causes of death. [4].
Estimation of net survival is performed using either “causespecific methods” or “populationbased methods”. Causespecific methods consider as censored all deaths from noncancerspecific causes and estimate net survival with classical estimators such as KaplanMeier (KM) estimator or the Coxmodelbased estimator [5]. They assume independence between cancerrelated and cancerunrelated mortality and require accurate information on the causes of death. However, cause of death information do include errors and are sources of misclassifications [6, 7]. In several studies, the rate of misclassification was found to be very low (2 to 7%) up to 5 years of followup but this rate became much more important (up to 20%) with longterm followup [8]. These rate changes reflect the difficulty of identifying the cause of death over long followups [9], and therefore make inappropriate the use of methods requiring knowledge of the cause of death to estimate net survival (the same reasoning applies in the competing risk setting to estimate causespecific functions). Furthermore, causespecific methods rely on the assumption of independence between the censoring process and the occurrence of death from the cancer of interest [5, 10]. A classic example of violation of this assumption concerns age because it has an impact on both death from the cancer of interest and death from other causes (called herein “othercause mortality”), leading to biased estimations of net survival [5]. To limit this particular source of bias, PoharPerme et al. [11] proposed an approach based on the inverse probability of censoring weighting that uses the NelsonAalen estimator (each individual estimation is weighted by the inverse probability of the expected survival [12]).
One solution to avoid causeofdeath misclassification is to use populationbased methods. Indeed, such methods account for the competing causes of death by adjusting for the population survival rate obtained from population life tables. They rely on the assumption that mortality due to other causes than cancer of interest is the same as the expected mortality in the general population with same demographic characteristics derived from general population life tables. The observed hazard (λ_{O}) is then considered as the sum of the hazard due to the cancer of interest, the excess hazard (λ_{E}), plus the hazard derived from the othercause of death than cancer of interest ( λ_{P}); this latter quantity is not estimated but drawn from the general population life tables. The net survival is then the survival function that derives from the excess hazard function and this relation may be expressed as: \( {S}_E(t)=\exp \left[{\int}_0^t{\lambda}_E(u) du\right] \). It can be estimated nonparametrically (e.g., PoharPerme estimator [12] and doubly robust estimator [13]) or parametrically (e.g., Estève et al. [14], Giorgi et al. [15], or Remontet et al. [16] regression models). Simulation studies have shown that PoharPerme estimator and regression model adjusted on demographic covariates have good performances in estimating net survival [17]. The main assumption of populationbased methods is that othercause mortality in the studied group is comparable to that of the general population [9, 18]. However, patients included in a cancer trial are rather selected. Therefore, they are neither representative of all patients diagnosed with the same cancer nor representative of the general population (different characteristics and different othercause mortality), even within the same area of residence, and their othercause mortality as derived from the general population life tables will be surely biased (underestimated or overestimated). To our knowledge, only Cheuvart and Ryan [19] have proposed a populationbased regression model to analyze longterm cancer clinical trial that accounts for this type of bias. They introduced a rescaling parameter that allows the mortality from other causes of the studied group to differ from the mortality of the general population. However, i) the baseline hazard is a piecewise exponential, which is not very flexible and requires potentially a high number of parameters to estimate in case of longterm followup; and ii) the approach relies on groupeddata, which is source of loss of information.
The present article proposes a new flexible populationbased model to estimate longterm net survival in clinical trials and that allows for causeofdeath misclassification and for effects of selection. Comparisons with other causespecific and populationbased methods are carried out to examine the newmodel limits.
The article is organized as follows. The Methods section describes parametric and nonparametric estimators of net survival as well as the new model and the simulation study. The Results section presents the results of simulations performed with different combinations of proportion of misclassification / degree of selection effect. This section also shows an application to data from a clinical trial on prostate cancer patients. The article ends with a discussion of the findings.
Methods
Models and estimators of net survival
Data settings, assumptions and notations
To estimate net survival, two setting are defined according to cause of death information. When one considers that this information is available for each patient, net survival is estimated in “causespecific setting”. But if cause of death information is unavailable, or if one wants to get rid of cause of death, net survival is estimated in “populationbased setting”. In this context, cause of death information is indirectly obtained by matching the observed data with othercause hazard drawn from general population life tables. Indeed, the tables contain daily hazard rate \( {\lambda}_{P_i} \) for each matched individual i from the general population of interest. The main assumption to consider that, is the fact that the cancer part in the whole mortality is negligible. In consequence, the othercause hazard of the studied sample is equal to that of general population of interest. In these two settings the common assumption is that excess hazard and othercause hazard are independent.
Furthermore, in the two settings, net survival may be estimated using nonparametric estimators or parametric models; the latter allow estimating and testing effects of covariates on the excess mortality.
Some wellknown estimators presented in the next subsection (KaplanMeier, NelsonAalen, Cox) were firstly developed in the overall cause of death setting. For simplicity reason, we present their adaptation directly in the causespecific setting, where the main change concerns the event indicator.
Nonparametric estimators of net survival
Here, we present briefly the properties of 1) one causespecific method, 2) one mixed populationbased and causespecific method, and 3) one populationbased method. The first is the popular KaplanMeier (KM) estimator in the causespecific setting with rightcensoring of the times to death from noncancer causes. The second uses an adaptation of the NelsonAalen estimator to account for the informative censoring problem and uses othercause mortality information from population life tables. The third is the PoharPerme (PP) estimator, a reliable estimator of net survival in populationbased studies.
The KaplanMeier estimator
Estimation of net survival in the causespecific setting leads to consider deaths from cancer as events and to right censor deaths from other causes and live patients. The KM estimator of net survival is then:
In this equation, n is the number of patients, \( {N}_E(t)=\sum \limits_{i=1}^n{N}_{E,i}(t) \) is the number of cancerrelated deaths up to the time t obtained by summing up the individual counting processes N_{E, i}(t), and \( Y(t)=\sum \limits_{i=1}^n\mathbb{l}\left[{t}_i\ge t\right] \) is the atrisk process just before time t (i.e., alive or not censored patients; the at risk process counts the subjects who did not experience the event by time t and, thus, who are still “at risk” of experiencing the event).
A Weighted NelsonAalen estimator
In the causespecific setting, PoharPerme et al. proposed an adaptation of the NelsonAalen estimator [12]. When the assumption of independence between the censoring process and the cancer death process is violated, mainly due to age, the censoring becomes informative. Using the inverse probability of censoring weighting approach on the NelsonAalen estimator [12], PoharPerme et al. derived an asymptotically unbiased estimator of the net survival:
In this equation, \( {N}_E^w(t)={\sum}_{i=1}^n{N}_{E,i}^w(t) \) and \( {Y}^w(t)={\sum}_{i=1}^n{Y}_i^w(t) \) are, respectively, the weighted aggregated counting process and the atrisk process. More precisely, \( {dN}_{E,i}^w(t)=\frac{dN_{E,i}(t)}{S_{P,i}\left({t}_i\right)} \) and \( {Y}_i^w(t)=\frac{Y_i(t)}{S_{P,i}\left(t\right)} \), where each of components are weighted, respectively, by S_{P, i}(t_{i}−) and S_{P, i}(t−) the inverse of the individual expected survival derived from population life tables obtained respectively at times t_{i}− and t−. Thereafter, we called this estimator, with these types of weights, the weighted NelsonAalen (wNA) estimator.
The PoharPerme estimator
The PP estimator [12] is a reliable nonparametric estimator of net survival developed to overcome some assumptions of excess hazard modeling. It corresponds to the difference between the NelsonAalen estimate and the cumulative population of the patients still at risk at each death, where the atrisk process and the counting process are weighted to give greater weight to subjects with high risk of othercause mortality. The PP estimator of net survival is:
where \( {N}^w\left(\mathrm{t}\right)=\sum {N}_i^w(t) \) is the sum of the individual allcause counting process \( {N}_i^w(t) \) and with \( d{N}_i^w(t)=\frac{d{N}_i(t)}{S_{pi}(t)} \), which, as \( {Y}_i^w(t) \), is weighted by the inverse of the individual expected survival. This latter quantity and the general population othercause mortality λ_{P} are derived from population life tables.
Among these nonparametric estimators, only the KM estimator is a purely causespecific estimator because, in our case of causespecific setting, it uses only cancer specific death information. Though used in the causespecific setting, the wNA estimator uses also the population othercause mortality to correct the estimation of net survival. The PP estimator is used only in populationbased settings; it needs the othercause mortality and the vital status to provide an estimate of net survival.
Parametric and semiparametric models
Cox model
In the causespecific setting, the semiparametric Cox proportional hazards model expresses the excess hazard at time t as: λ_{E}(t, X) = λ_{E, 0}(t) exp(β^{T}X) where λ_{E, 0}(t), the baseline excess hazard at time t, and β corresponds to the proportional linear effect of covariate X on the baseline excess hazard estimated separately through the semiparametric approach.
One solution to derive the baseline cumulative excess hazard function from the Cox model was given by Breslow [20]. Using Breslow estimator with cancer death as status indicator δ_{i}, the baseline cumulative excess hazard function Λ_{E, 0}(t) may be estimated with the expression applied to times t_{i} at which the events take place:
In this equation, \( \mathcal{R}(t) \) denotes the risk at time t of all individuals still at risk of death from cancer at time t, \( \widehat{\upbeta} \) corresponds to the effect of covariates and Δ_{i} corresponds the indicator of death due to the cancer of interest. In this formula, we estimate unadjusted cumulative excess hazard by setting \( \widehat{\beta}=0 \) [21].
The corresponding net survival from Breslow’s estimator is therefore:
where \( {\widehat{\varLambda}}_{E,0} \) corresponds to the baseline cumulative excess hazard function estimated separately from \( \widehat{\beta} \).
The new flexible model
The new model is an extension of the flexible parametric excess hazard model proposed by Giorgi et al. [15]. It is based on the seminal excess hazard model of Estève et al. [14] where the observed hazard of a patient i at time t_{i} is:
and where the baseline excess hazard (λ_{E, 0}) is modelled by a piecewise constant function; β represents the effects of the vector of covariates X including demographic variables Z (such as age at diagnosis, year of diagnosis, sex, of the individual i).
In the model of Giorgi et al. [15], the baseline excess hazard and the timedependent covariates are both modelled using specific Bspline functions. More precisely, for a higher degree of flexibility, Giorgi et al. used quadratic Bsplines (order 3) and two interior knots.
For simplicity, only the case of proportional hazard effects of prognostic covariates is considered. The simplified version of Giorgi’s model is then:
where v_{j} are the spline coefficients, B_{j, 3}(t_{i}) the value at time t_{i} of the j ^{th} Bspline of order 3 and degree 2, X the vector of covariates with proportional hazard effects β, and λ_{Pi} the othercause hazard of individual i, at age a_{i} + t_{i} in year y_{i} + t_{i}.
In agreement with Cheuvart and Ryan [19], we considered that the othercause mortality of a participant in a clinical trial may be corrected by multiplying the population hazard obtained from the life table by a scale parameter α. This parameter is the average effect of selection on the othercause mortality in the trial participants. This effect in the general population with same demographic characteristics equals 1 (α = 1). The new flexible model we call the “rescaled Bspline” model (RBS) can be written as follows:
To estimate the parameters of the RBS model, we used the maximum likelihood procedure. The loglikelihood of the RBS model can be written:
where, given an individual i, observation δ_{i} = 1 is the indicator of death from any cause, α the scale parameter of the instantaneous othercause mortality λ_{P}, and Λ_{P} the cumulated value of λ_{P} over all the followup duration. The rescaled cumulative population hazard αΛ_{P} may not be a constant as in the classical additive excess hazard model. In addition, the scale parameter α will be considered in the estimation process. For mathematical convenience and because all patients may die from another cause than the cancer under study, it is assumed that α > 0 and constant over time. The maximization of the loglikelihood was performed using optim function in R based on Byrd method for nonlinear optimization problems with box constraints [22]. The estimates of net survival were derived from the cumulative excess hazard calculated by derivation of the corresponding estimate of the excess hazard function. The confidence interval of the net survivals was obtained with a MonteCarlo method [17]. The R code that implements these estimation procedures is available on request from the authors.
The simulation study
We carried out a simulation study to assess the performance of the RBS model in estimating the net survival in clinical trials and compare this performance with those of previous models and estimators used in clinical trials.
The simulation design
The simulation considered a randomized clinical trial that would compare treatment vs. placebo effects. The French generalpopulation lifetable was used to construct a “corrected” life table that would correspond to othercause mortality of trial participants. The “corrected” life table mortality rates were the mortality rates of the initial life table multiplied by the scale parameter α. This “corrected” life table was then used to generate T_{P}, the individual timetodeath from another cause than cancer in a trial. For each patient, we generated also T_{E}, the timetodeath from cancer and T_{C}, the time to rightcensoring (see the Data generation section). All the times T_{E}, T_{P} and T_{C} were generated independently from each other. An individual observed timetodeath T_{O} was then the smallest of T_{P}, T_{E}, and T_{C}. In addition, as in Grafféo et al. [23], from these generated times, we inferred a timetodeath T_{N} in the net survival setting where the patients would only die from cancer. Thus, T_{N} is the smallest of T_{E} and T_{C}.
In this simulation, the causes of death were considered as known; thus, T_{N} and wellclassified cancerspecific causes of death can be used with the KM estimator to obtain a gold standard (GS) estimator of the net survival; that is, the survival that would be obtained if cancer were the only possible cause of death (Table 1).
Various scenarios were built by combining various values of α with various proportions of causeofdeath misclassification. Within each scenario, 1000 datasets of 1000 patients each were simulated.
Data generation
Adjusted excess mortality
In this simulation, for simplicity, the patients were considered to be of same sex. Trial group (placebo or treatment) covariates and age at diagnosis were independently generated. The treatment group was generated so as to obtain 50% of patients in each trial group. The age at diagnosis was obtained from a uniform distribution so as to obtain 25% of patients aged 24 to 45 years old, 50% aged 46 to 64 years, and 25% aged 65 to 70 years. We assumed a linear effect β_{age} = 0.05 for the effect of centered age on the excess hazard. The beneficial effect of the treatment on the excess hazard of death was β_{treatment} = − 0.5.
In all scenarios, a generalized Weibull distribution [23] was assumed for the distribution of the baseline excess hazards and the individual timestodeath from cancer T_{E} were generated using the inverse transformation method to account for covariate effects [23]. The time to rightcensoring T_{C} was generated from a uniform distribution \( \mathcal{U}\sim \left[0;b\right] \) with b chosen so as to obtain a censoring rate of 50%.
Rescaled othercause mortality
T_{P}, the timetodeath from another cause (i.e., the othercause mortality in the general population multiplied by a scale parameter that depends on the characteristics of the trial participants) was generated using the French life table survexp.fr available in the eponymous R package [24]. This life table is stratified by age, sex, and year of cancer diagnosis. For each patient i, the “rescaled” othercause mortality rate was considered as αλ_{P}(a_{i} + t_{i}, Z_{i}), where α is the selection effect and λ_{P}(a_{i} + t_{i}) the general population othercause mortality at age a_{i} + t_{i} adjusted on demographic covariates Z as derived from the life table.
Four scenarios were considered for the othercause mortality of cancer patients in clinical trials: (1) patients comparable to the general population in terms of othercause mortality (α = 1); (2) patients more robust than the general population (α = 0.5); (3) patients frailer than the general population (α = 2); and (4) patients much more frail than the general population (α = 4).
Misclassification of the cause of death
Three conditions were considered regarding the proportion of errors in identifying the causes of death. These conditions are useful to compare causespecific with populationbased methods in realistic settings. Indeed, in many clinical trials with medium to longterm followups (10 to 15 years), it is difficult to obtain accurate causes of death. The simulation considered three causeofdeath misclassification rates: (i) 0%, a rare condition where all information on cancerrelated death is true; (ii) 20% of misclassification beyond 5 years of followup, which means that 20% of deaths from cancer are wrongly attributed to another cause; and (iii) 30% of misclassification beyond 5 years of followup. Actually, in active followups in clinical trials, the causes of death over shortterm followups (e.g., 5 years) are rather reliable.
To sum up, Scenarios 1 to 4 and conditions i to iii were designed to: a) assess the performance of the estimators in case of no misclassification and no selection effect; b) assess the bias due to misclassification with causespecific approaches and examine then the interest of populationbased estimators; c) assess the performance of the new RBS model in correcting for selection bias alone; and, d) assess the performance of the causespecific methods in presence of selection effect and misclassification.
Performance criteria
The theoretical net survival in each of the treatment and the placebo group is the average of the individual net survivals. Thus, the theoretical net survival can be written:
n_{j} is the number of patients in each group j and Λ_{0}(t) is the excess cumulative baseline hazard from the generalized Weibull distribution.
The performance in estimating the net survival is established on: (1) the bias \( \frac{1}{m}\sum \limits_{j=1}^m{\widehat{S}}_{M,E,j}(t){S}_{\mathrm{E},\mathrm{j}}(t) \), where \( {\widehat{S}}_{M,E,j}(t) \) is the mean of net survival estimates by model or estimator M at time t in group j, S_{E, j}(t) is the theoretical net survival at time t in group j, and m is the number of simulations; (2) the relative bias (R. Bias) \( \left(\frac{\frac{1}{m}\sum \limits_{j=1}^m\ {\widehat{S}}_{M,E,j}(t){S}_{\mathrm{E},\mathrm{j}}(t)}{S_{\mathrm{E},\mathrm{j}}(t)}\right)\ast 100 \); (3) the root mean square error \( \sqrt{\frac{1}{m}\sum \limits_{j=1}^m{\left({\widehat{S}}_{M,E,j}(t){S}_{\mathrm{E},\mathrm{j}}(t)\ \right)}^2} \); (4) the empirical coverage rate (ECR); i.e., the proportion of samples in which the 95% confidence interval of the estimated net survival at time t in group j contains S_{E, j}(t). These statistical indicators were calculated at 5, 10, and 15 years of followup. We also calculated the performances of covariates (centered age and treatment) effects estimated by Cox and RBS models using bias, root mean square error and ECR.
Results
Simulation results
In this section, we present the simulation results. We provide only the results relative to the placebo group (because the effects of causeofdeath misclassification were not different between the treatment and the placebo group) and at 5, 10, and 15 years of followup. The performance criteria of each method by scenario are shown in Table 2. The corresponding net survival curves are presented in Fig. 1. The results relative to the estimation of Cox and RBS models’ parameters are shown in Table 3. The boxplots of Cox and RBS models’ parameters estimated in all scenarios are presented in Additional files 1, 2 and 3.
No selection effect and no causeofdeath misclassification (scenario 1a)
Scenario 1a allowed evaluating the performance of causespecific vs. populationbased methods in a theoretical setting. As expected, the estimates of net survival obtained with all methods had very small bias. Whatever the estimator, the ECR was close to 95%. In comparison with the root mean square error (RMSE) obtained with the GS (KM estimator applied to data where patients would only die from cancer, i.e. net survival setting), the RMSEs of the KM, Cox, and wNA estimators were similar, whereas the RMSEs obtained with populationbased methods were slightly higher. For example, at 15 years followup, the RMSEs were: 0.016 with GS, 0.018 with KM, 0.016 with Cox, 0.016 with wNA, 0.020 with PP, and 0.036 with RBS). Otherwise, Cox and RBS parameters estimates had good performances and globally better in Cox than RBS model (Table 3).
No selection effect but presence of causeofdeath misclassification (scenario 1b)
As expected, increasing the proportion of causeofdeath misclassification resulted in increased biases with KM and wNA estimators. With 30% misclassification, the differences in terms of bias between the GS and each of KM, Cox, and wNA were, respectively, 0.072, 0.063 and 0.067. The ECRs with KM, Cox, and wNA estimators were close to 0 when the misclassification was 30%. With 30% misclassification, the differences in RMSE between GS and each of KM, Cox, and wNA estimates at 15 years followup were, respectively, 0.058, 0.049, and 0.053. As the populationbased methods did not use the cause of death information, we can compare directly their results with those of causespecific methods. In this scenario, the performances of the PP and the RBS estimator were better than those obtained with the causespecific methods in case of no selection effect (α = 1) and 20 to 30% misclassification.
In summary, in Scenario 1 and with α= 1, the performance of the RBS regression model in net survival estimation was better than that of another model. However, Cox model parameters (effect of age and treatment) estimates were very slightly impacted in terms of relative bias, RMSE and ECR (Table 3).
Presence of selection effect but no causeofdeath misclassification (scenarios 2a, 3a, 4a)
With causespecific methods KM and wNA, the bias in the estimate of the net survival increased with the increase of α (the effect of selection). The bias with the RBS was much more important than with the GS. The bias with PP was much more important than with the GS. The estimates of net survival obtained with the RBS model had higher RMSEs than those obtained with wNA or KM and the estimates obtained with PP had higher values than those obtained with the other methods. The ECRs obtained with KM and wNA were far from 95%, the nominal value; they approached zero with PP with the increase of the selection effect; i.e., with the increase of α values over 1. The ECRs obtained with Cox and RBS models remained stable and close to their nominal values (Table 2). Besides, Cox model parameters (effect of age and treatment) estimates were not impacted in terms of relative bias, RMSE, ECRs than that of RBS (Table 3).
Presence of selection effect and causeofdeath misclassification (scenarios 2b, 3b, 4b)
In Scenarios 2 to 4 with 20% or 30% causeofdeath misclassification, the biases in the estimates of net survival obtained with each of KM, Cox, and wNA were 3 to 4 times more important than those obtained with the GS. The biases obtained with the RBS model did not change much. The ECRs of the estimate of net survival obtained with KM, Cox, or wNA was close to 0 when the causeofdeath misclassification was 30%. With 30% misclassification at 15 years, the differences between the RSME obtained with GS and each of the tree causespecific methods (KM, Cox, wNA) were 0.073, 0.048 and 0.069 (Scenario 4b). However, Cox model parameters (effect of age and treatment) estimates were slightly impacted in terms of relative bias, RMSE, ECRs than that of RBS.
Application to clinical trial data
To illustrate the interest of the RBS model in estimating net survival in clinical trials and be able to compare it with other approaches, we used data from a clinical trial that included 506 prostate cancer patients [25] (USA, 1967–1969, data published by Andrews and Herzberg [26]). The patients were randomly allocated to one of four treatment regimens: placebo or 0 mg/d, 0.2 mg/d, 1 mg/d, and 5 mg/d of per os diethylstilbestrol. As in previous works on these data, we gave indicator 0 to the lowdoses (0 and 0.2 mg/d) and 1 to highdoses (1 and 5 mg/d) and considered seven other covariates: age, weight index, performance rating, history of cardiovascular disease, serum hemoglobin, size of primary lesion, and Gleason stage/grade category [25]. However, in the parametric model as in the simulation study, age was centered on the median age (73 years). Because 23 patients have missing information, 483 patients were kept for analysis: 241 in the lowdose and 242 in the highdose treatment group. Concerning the stage variable, 278 patients were in stage 3 and 205 in stage 4. The median followup was about 65 months and the whole followup was 76 months.
In estimating the net survival, populationbased methods used the USA life tables from 1967 to 1973 that included covariates age, sex, and year of diagnosis. The distributions of the two treatment groups were compared using a specific logranktype test for net survival comparisons [23] and considering p < 0.05 as significance level.
The parametric causespecific Cox and the proposed RBS models were used to estimate the effect of treatment on the excess hazard of death from prostate cancer. The net survival estimates over time by KM, wNA, PP and RBS methods in placebo group and in treatment group for all the patients are shown respectively in Fig. 2.a and Fig. 2.b. The logranktype test did not find a statistically significant difference in net survival estimates by PP method between the lowdose and the high dose group (\( {\chi}_1^2=3.61,\kern0.5em \mathrm{p}=0.057 \)). At the end of the 76month followup period, the estimates of net survival in the highdose group were: S_{KM} = 0.682, S_{Cox} = 0.662, S_{wNA} = 0.684, S_{PP} = 0.478, and S_{RBS} = 0.565. The estimates in the lowdose group were: S_{KM} = 0.475, S_{Cox} = 0.498, S_{wNA} = 0.491, S_{PP} = 0.293, and S_{RBS} = 0.396. Table 4. shows results obtained with the two parametric Cox and RBS model adjusted for the covariates. Both models concluded to a significant reductive effect of highdose treatment on the excess hazard of death from prostate cancer and to a higher estimated effect with the RBS model vs. Cox model (Excess hazard ratio, \( {EHR}_{RB{S}_{high} dose}=0.62\ \left[0.40;0.96\right] \)) vs. \( {EHR}_{CO{X}_{high} dose}=0.43\ \left[0.30;0.64\right] \)) . The RBS model showed that othercause mortality in the trial participants was 1.51 times that of the US general population. Thus, on average, the expected mortality provided by the US life table is 1.51 times lower compared to the appropriate othercause mortality in the trial participants. Correcting for both possible causeofdeath misclassification and effect of selection indicated that treatment effect (excess risk reduction = 38%) was lower than using a causespecific Cox (excess risk reduction = 57%).
Discussion
The present work proposes a new flexible populationbased model to estimate longterm net survival in clinical trials. To account for biases due to causeofdeath misclassification and patient selection, the model extends the excess hazard models developed by Estève et al. [14, 15] or Giorgi et al. [14, 15] and considers settings where the othercause mortality is different from that of the general population of same general characteristics.
One advantage of the new RBS model is that it corrects the othercause mortality of trial participants and provides more precise estimates of the excess hazard and the net survival in presence of selection. Simulation study has shown that the RBS model provides accurate estimates of the net survival. In the application, the RBS model showed that, due to selection, the othercause mortality of trial participants was, on average, 1.51 times that of the general population, and the treatment effect was lower, compare to estimation obtained using causespecific approach. This selection problem was already mentioned by Augustin et al. [27] who found that the effects of recent therapeutics on net survival in mantlecell lymphoma patients included in a clinical trial were different from those seen in a larger group of patients found in cancer registries and concluded that trial patients are highly selected and may not be representative of the patients encountered in everyday practice. However, in accordance with previous comments on Augustin’s study [28], the present simulation study shows that some standard populationbased methods, such as the PP estimator used by Augustin, provide biased estimates of net survival in the presence of a selection effect. Within this context, RBS model may allow for selection and provide more accurate comparisons between trial and other patients’ survivals.
Another advantage of the RBS model is that it is more flexible than that of Cheuvart and Ryan [19] in estimating the baseline excess hazard function. Estimating cancer excess hazard with the RBS model by a flexible function instead of a step function (that represents only discontinuous constant excess mortality rates) is more realistic and reliable in an epidemiological or clinical setting. Contrarily to causespecific methods, the RBS, as other populationbased methods, does not require knowing the cause of death and is thus insensitive to the causeofdeath misclassification. In the application, the cause of death information was used with cause specific methods (KM, Cox, and wNA) without any indication about its accuracy and the vital status was used with the PP estimator and the RBS model. Also, our results from the application have shown that net survival estimates with causespecific methods (KM, Cox and wNA) were higher compared to that PP and RBS. Due to the robustness of the RBS model to estimate net survival in presence of selection effect, we are more confident with the RBS model estimates of net survival although the followup in the trial was less than 10 years. Indeed, net survival is probably overestimated using causespecific methods because of an underestimation of the prostate cancerrelated death. Besides, the misclassification phenomenon is not rare in prostate cancer data because of treatment impact [29]. The RBS model allowed rescaling the estimate of the net survival in treatment and placebo groups.
Causespecific methods performed better than populationbased methods in the absence of causeofdeath misclassification and/or selection effect. However, 0% misclassification is highly improbable; even in shortterm clinical trials, there is always some proportion of causeofdeath misclassification because of information unreliability [7] or because of competing causes of death [5].
KM, wNA, and Cox model showed similar performances in estimating net survival in the absence of causeofdeath misclassification. In fact, KM and Cox model should be avoided because the assumption of independence between the othercause and cancer mortality is not met; actually, some variables, such as age, may affect both mortalities. Thus, censoring death from other cause than cancer could be informative. Likewise, PoharPerme et al. [12] have shown that only the NelsonAalen estimator is consistent (asymptotically unbiased) in estimating the net survival using the inverse probability of censoring weighting. Therefore, the selection bias impacts obviously the causespecific survival estimates with wNA because incorrect life tables are used to calculate the weights. In a trial, when the othercause mortality of the participants is higher than that of the general population, the net survival estimate that uses a general population life table is underestimated, and inversely [28]. These results are in agreement with those of Baili et al. [30] and Stroup et al. [31]. For prostate cancer, Stroup et al. showed that prostate cancer patients with early stage have better health status than US general population. In addition, they also found that causespecific methods are preferable to estimate net survival compared to populationbased survival for prostate cancer patients with early diagnosis. These same conclusions were also found by Skyrud et al. [8] in Norway cancer registry among prostate cancer patients. However, in our application, the proposed model found that the selected patients were more frail than US general population. This can be explained by the fact that patients included in this trial had an advanced prostate tumor (stage 3 and 4) and were also more exposed to comorbidities due to their high median age.
In this work, we considered identical degrees of causeofdeath misclassification in the treatment and the placebo group, which is a plausible assumption in real clinical trials with long followup durations. This may explain the similar performances of causespecific methods in the treatment and the placebo group in case of 20 or 30% causeofdeath misclassification after a fiveyear followup. However, as in populationbased studies, and despite longterm followups, distinct degrees of misclassification between the treatment and the placebo group are also possible. Some authors have shown that identical degrees of misclassification had less impact than distinct misclassifications on net survival estimation [9, 12].
Furthermore, Morisot et al [32] investigated the interest of multiple imputation approach in the estimation of causespecific survival notably when a subset of cause of death was available. In their case there is a confidence in cause of death classification of some patients contrary to our case where we assume it exists an overall uncertainty on cause of death classification. Up to 50% missing values in the “cause of death” variable, Morisot et al have recommended multiple imputation method to obtain accurate estimates of causespecific survival, notably in not large database. Indeed, this approach may be timeconsuming and not satisfactory if a representative percentage of causes of death is not validated by experts. However, it is wellknow that cause of death information may be difficult to be validate in longterm followup without autopsy. The consequence could be finally a high risk for misclassification of cause of death up to 68.2% [33], resulting in bias on net survival estimates using causespecific method even after multiple imputation approach.
Despite its advantages, the RBS model has the limitations of most parametric excesshazard models because of the assumptions regarding the baseline excess hazard and the effects of the covariates. For example, the effect of selection on the general population mortality was assumed to be multiplicative; this assumption is reasonable from an epidemiological or clinical point of view but may not be always met [19]. Another clinically plausible assumption would be to consider a nonproportional effect. Also, one may consider a heterogeneous selection effect between trial centers or individuals. For example, in the application, the effect of selection may be considered different between hospitals of the Veteran’s Administration, and the use of a frailty model for the risk of othercause death could improve the RBS model. Within this context, the works of Zahl that account for heterogeneity in the competing risk model may improve the RBS model [34].
Besides, one assumption for net survival estimation is that T_{P} and T_{E} are conditionally independent on a set of explanatory covariates [14]. This assumption may not be verified and that raised some issues as well as in the classical framework of competing risks [35] and in the net survival setting resulting in informative censoring bias [17, 36]. In latter setting we have also to consider that general population life tables exist for each combination of demographic covariables, and observed data contain these demographic covariables. Indeed, the demographic covariables acts on both the excess hazard and the population hazard. However, these covariables are not always available on data or on general population life tables. The impact of their absence has been showed in Grafféo et al. [37], Danieli et al. [17] and studied by Pavlik and PoharPerme [38]. As shown by Danieli et al. [17] a regression model for excess hazard modelling adjusted on demographic covariables when there are present can deal with the informative censoring problem. In the absence of some important covariables the proposed model can be used to rescale the population hazard and offers potential research opportunities.
Conclusion
In conclusion, the new RBS model allows estimating net survival in clinical trials. It corrects the biases of causeofdeath misclassification and of selection effect on the expected mortality in the general population. This makes it particularly useful in clinical trials with long followups. With the RBS model, the researcher obtains accurate estimates of the excess hazard and, therefore, of net survival; however, he/she should check the strong assumption of homogeneous selection. Finally, the RBS model paves the way for new methodological developments in the field of net survival methods in multicenter clinical trials.
Abbreviations
 ECR:

Empirical coverage rate
 EHR:

Excess hazard rate
 GS:

Net survival setting or Gold Standard net survival
 KM:

KaplanMeier estimator
 PP:

PoharPerme estimator
 RBS:

Rescaled Bspline model
 RMSE:

Root Mean Square Error
 wNA:

Weighted NelsonAalen estimator
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Acknowledgements
The authors thank Jean Iwaz (Hospices Civils de Lyon, France) for the revision of the final version of this manuscript.
Funding
The project leading to this publication has received funding from Excellence Initiative of AixMarseille University  A*MIDEX, a French “Investissements d’Avenir” programme". JAG was funded by this grant. This work has been also carried out with supports from the French National Research Agency (CENSUR project, grant ANR12BSV1–0028).
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Data obtained from http://biostat.mc.vanderbilt.edu/DataSets. The R code used in this paper is available on request from the authors.
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RG and JAG designed the study and drafted the manuscript. JAG carried out the programming for the simulation study. CT and NG contributed to methods, analysis, interpretations of the findings and drafting the manuscript. CT and NG critically revised the article. All authors read and approved the final manuscript.
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Additional files
Additional file 1:
Boxplots of the effect of centered age estimated with the RBS and Cox models’ in the simulation study. (PDF 27 kb)
Additional file 2:
Boxplots of the effect of treatment estimated with the RBS and Cox models’ in the simulation study. (PDF 28 kb)
Additional file 3:
Boxplots of the selected effect estimated with the RBS model in the simulation study. (PDF 17 kb)
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Goungounga, J.A., Touraine, C., Grafféo, N. et al. Correcting for misclassification and selection effects in estimating net survival in clinical trials. BMC Med Res Methodol 19, 104 (2019). https://doi.org/10.1186/s1287401907473
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DOI: https://doi.org/10.1186/s1287401907473
Keywords
 Cause of death
 Cancer clinical trials
 Life tables
 Excess hazard model
 Net survival
 Selection bias