Comparison of methods for estimating the attributable risk in the context of survival analysis
 Malamine Gassama^{1},
 Jacques Bénichou^{2, 3},
 Laureen Dartois^{4, 5} and
 Anne C. M. Thiébaut^{1}Email author
DOI: 10.1186/s1287401602851
© The Author(s) 2017
Received: 11 July 2016
Accepted: 22 December 2016
Published: 23 January 2017
Abstract
Background
The attributable risk (AR) measures the proportion of disease cases that can be attributed to an exposure in the population. Several definitions and estimation methods have been proposed for survival data.
Methods
Using simulations, we compared four methods for estimating AR defined in terms of survival functions: two nonparametric methods based on KaplanMeier’s estimator, one semiparametric based on Cox’s model, and one parametric based on the piecewise constant hazards model, as well as one simpler method based on estimated exposure prevalence at baseline and Cox’s model hazard ratio. We considered a fixed binary exposure with varying exposure probabilities and strengths of association, and generated event times from a proportional hazards model with constant or monotonic (decreasing or increasing) Weibull baseline hazard, as well as from a nonproportional hazards model. We simulated 1,000 independent samples of size 1,000 or 10,000. The methods were compared in terms of mean bias, mean estimated standard error, empirical standard deviation and 95% confidence interval coverage probability at four equally spaced time points.
Results
Under proportional hazards, all five methods yielded unbiased results regardless of sample size. Nonparametric methods displayed greater variability than other approaches. All methods showed satisfactory coverage except for nonparametric methods at the end of followup for a sample size of 1,000 especially. With nonproportional hazards, nonparametric methods yielded similar results to those under proportional hazards, whereas semiparametric and parametric approaches that both relied on the proportional hazards assumption performed poorly. These methods were applied to estimate the AR of breast cancer due to menopausal hormone therapy in 38,359 women of the E3N cohort.
Conclusion
In practice, our study suggests to use the semiparametric or parametric approaches to estimate AR as a function of time in cohort studies if the proportional hazards assumption appears appropriate.
Keywords
Attributable risk Weighted KaplanMeier estimator Piecewise constant hazards model Cox model Cohort studies Breast cancerBackground
where P(D) is the probability of disease (incidence) in the population, which includes exposed E and unexposed \(\bar {E}\) subjects, and \(\mathbf {P}(D  \bar {E})\) is the hypothetical probability of disease in the same population but with all exposure eliminated.
The AR can be estimated from different types of studies including casecontrol studies for which many estimation methods exist (as reviewed in [3]), but it is rarely estimated from cohort studies. In the context of cohort studies and timetoevent outcomes, AR measures can be defined as functions of time [49] although a single AR estimate has been proposed alternatively [10].
Recent developments for estimating AR as a function of time from cohort studies in the survival analysis context have not so far led to a consensus definition. Several definitions have been proposed depending on whether authors interpret disease incidences P(D) and \(\mathbf {P}(D\bar {E})\) in Eq. (1) as cumulative distribution functions (CDFs) [69] or as instantaneous hazard functions [4, 5]. The two definitions converge only for rare diseases or low exposure prevalence [4]. Here we focus on the first definition of AR based on CDFs which looks more consistent with the standard AR definition and appears to be the most used in the literature. Several methods of estimation have been proposed for the AR defined in this case, including nonparametric approaches based on KaplanMeier’s estimator of the survival function [7], a semiparametric approach based on Cox’s proportional hazards model [7] and a fully parametric approach assuming a piecewise constant hazards model [8]. Some evaluations were made for the nonparametric and semiparametric approaches [7] but, to the best of our knowledge, the performances of these various approaches have not been systematically compared.
The aim of this paper was to compare available methods for estimating AR when defined using CDFs. In the sections to follow, we first review the corresponding estimation methods so far published in the statistical literature. Simulations were conducted to assess the performance of the proposed AR estimators. The methods were then applied to data on menopausal hormone therapy (MHT) and breast cancer from the E3N women cohort (Étude Épidémiologique auprès de Femmes de la Mutuelle Générale de l’Éducation Nationale) [11]. For the purpose of our illustration, we considered 38,359 participants who were postmenopausal and free of cancer when they completed a selfadministered questionnaire on their past use of any MHT in January 1992. In total, 17,185 (44.8%) women had ever used MHT at baseline and were considered exposed thereafter. By June 2008 (for a maximal 16.4 years and mean 14.0 years of followup), 2,228 invasive breast cancers had been diagnosed (1, 106 in unexposed women). A recent work on the E3N cohort estimated a 14.5% postmenopausal breast cancer risk attributable to MHT use after 15 years of followup [12]. We estimated AR as a CDFbased function of time at four time points using nonparametric, semiparametric and parametric approaches, as well as the single overall AR measure proposed by Spiegelman et al. [10].
Methods
Review of estimation methods
A natural estimate of A(t) is obtained by replacing the survival functions S _{0}(.) and S(.) by their respective estimators \(\hat {S}_{0}(.)\) and \(\hat {S}(.)\). Various estimators \(\hat {S}_{0}(.)\) and \(\hat {S}(.)\) have been proposed, as detailed in the following subsubsections.
Nonparametric approaches
Chen et al. [7] considered several approaches for estimating survival functions S _{0}(.) and S(.) depending on covariate type: nonparametric when all p covariates are categorical and independent of time, otherwise semiparametric. The former case applies to a single categorical covariate or several covariates forming K+1 categories.
When all p covariates are categorical and independent of time and under the assumption that censoring is independent of the covariates, Chen et al. [7] suggested estimating both S _{0}(.) and S(.) by the KaplanMeier method [13].
In all cases, the estimation of the variance of \(\hat {A}(t)\) is based on the expression of \(\{\hat {A}(t)  A(t)\}\) as a linear combination of \(\{\hat {S}_{0}(t)  S_{0}(t)\}\) and \(\{\hat {S}(t)  S(t)\}\) and relies on counting process results [7].
Semiparametric approach
The expression of the variance of \(\hat {A}(t)\) follows the same general principles as for the nonparametric approaches above [7].
Parametric approach
The model parameter estimates \(\hat {\alpha }=(\hat {\alpha }_{1}, \ldots,\hat {\alpha }_{J})\) and \(\hat {\beta }\) are obtained by maximum likelihood estimation. The variance of \(\hat {A}(t)\) is estimated using the delta method [8].
Global approaches over the whole followup period
Upon using Cox’s proportional hazards model, the overall AR can be estimated using estimated hazard ratio (HR) for relative risk and personyears for exposure prevalence in the cohort. The asymptotic variance is estimated using the multivariate delta method [10].
where q denotes the exposure prevalence and RR the relative risk of exposed relative to nonexposed subjects. This formula resembles the wellknown formula used by epidemiologists [1, 2] where q is estimated by the proportion of exposed subjects at baseline (instead of exposed personyears over the whole followup).
Simulations
In this work, we considered a single, binary covariate Z representing exposure with Z=0 and 1 for unexposed and exposed subjects respectively, simulated as a Bernoulli random variable with probability of exposure (q) set to 0.25, 0.50 and 0.75. To compare the different approaches for estimating AR, we considered either proportional or nonproportional hazards between the exposed and the unexposed.
For proportional hazards, we used instantaneous hazard functions of the form λ(tZ)=λ _{0}(t) exp(β Z) where β denotes the regression parameter set to ln(2) or 0, and λ _{0}(t) the baseline hazard function taken from a Weibull distribution with shape parameter γ and scale parameter θ: λ _{0}(t)=γ θ ^{−γ } t ^{ γ−1}, and generated event times from (1/θ)[− ln(U)/ exp(β Z)]^{1/γ } with U uniform on (0,1). We explored situations where the baseline hazard was constant (γ=1) or dependent on time, either increasing (γ=4/3) or decreasing (γ=3/4) with time. The scale parameter θ was chosen as a function of the shape parameter γ so as to obtain median survival time equal to 15 years for unexposed subjects in all scenarios. We calculated survival functions S _{0}(.) and S(.) as exp{−(t/θ)^{ γ }} and (1−q) exp[−(t/θ)^{ γ }]+q exp[−(t/θ)^{ γ } exp(β)] respectively and derived theoretical values of AR as a function of time from Eq. (2). For the global AR derived from the simpler approach, theoretical values were obtained as q[ exp(β)−1]/{1+q[ exp(β)−1]}.
For nonproportional hazards, we generated event times from G ^{−1}[− ln(U)]/[λ _{0} exp(β Z)] assuming a cumulative hazard function of the form Λ(tZ)=G[λ _{0} t exp(β Z)] where G is the logarithmic transformation G(t)= ln(1+2t)/2 [7]. Setting λ _{0}=0.1 yielded a median survival time for unexposed subjects equal to 15 years as in the proportional hazards case. Setting the regression coefficient β to ln(2), the HR between the exposed and the unexposed decreased from 2 toward 1 over time. We calculated survival functions S _{0}(.) and S(.) as exp{− ln(1+2λ _{0} t)/2} and (1−q) exp{− ln(1+2λ _{0} t)/2}+q exp{− ln(1+2λ _{0} t exp(β))/2} respectively and derived theoretical values of AR as a function of time from Eq. (2).
We generated censoring times independent of the covariate Z and event times T from a uniform distribution on [0,τ], with τ the maximal followup time of the study set at 20 years. Depending on scenarios, we obtained censoring percentages around 4768% (ranges across simulations from 42% to 73%).
We generated 1, 000 data sets of n=1, 000 or 10, 000 independent observations and calculated estimators \(\hat {A}(.)\) of the AR as a function of time and their associated variances using the four approaches: two nonparametric approaches corresponding to the case where S _{0}(.) and S(.) are both estimated by the KaplanMeier method (KM) and to the case where S _{0}(.) and S(.) are estimated by the KaplanMeier and the weighted KaplanMeier methods, respectively (WKM) [7], one semiparametric approach using Cox’s proportional hazards model (COX) [7], and one parametric approach corresponding to the case where survival functions are estimated assuming piecewise constant hazards (PCH) [8] considering four intervals of 5year width. In the case where no event was generated in any fiveyear interval, the simulated dataset was discarded and replaced by a new one. We also considered the simpler approach based on Eq. (3) to estimate a global AR.
Results of the timedependent approaches are presented at times t=τ/4, τ/2, 3τ/4 and τ (respectively, 5, 10, 15 and 20 years). For the nonparametric and semiparametric approaches, estimates were obtained at times actually observed in the dataset so we considered values taken at the closest preceding time point. While nonparametric estimations are based on data available until the time of interest, semiparametric and parametric methods use data available over the whole followup period. To allow for a fairer comparison under the proportional hazards assumption, we also computed semiparametric and parametric estimators after censoring observation times at either τ/4 or τ/2. The parametric approach was then based on one or two interval(s) of 5year width respectively.
For all five approaches, results displayed are the average absolute bias relative to the theoretical value A(.), the Sampling Standard Deviation of \(\hat {A}(.)\) (SSD), the average Standard Error Estimator of A(.) (SEE) and the coverage probability (CP) of the 95% confidence interval (CI) of A(.). Although authors [7, 8] have suggested to use the complementary logarithmic transformation ln {1−A(.)black} to improve coverage probabilities in case of small sample size, this did not notably improve coverage probabilities in our results (data not shown) so results presented are for the untransformed A(.).
Simulations were performed using R release 3.0.1. We coded the nonparametric methods using R software and tested the validity of our code by comparing our simulation results with those of the authors using the same parameters [7]. For the semiparametric method [7], we used the R package paf developed by Chen [17]. For the parametric method [8], we used SAS release 9.3 and a set of macros developed by Laaksonen et al. [18]. For the global approach by Spiegelman et al. [10], we used the %par SAS macro developed by the authors.
Results
Simulations
Simulation results for the estimation of attributable risk A(.) under proportional hazards, constant baseline hazard (γ=1) with regression parameter β= ln(2) and probability of exposure q=0.5
Estimation method  n=1,000  n=10,000  

Time  A(t)  Bias  SEE  SSD  CP  Bias  SEE  SSD  CP  
KM  τ/4  0.284  0.001584  0.052440  0.052591  0.949  −0.000011  0.016622  0.016349  0.944 
τ/2  0.240  0.001496  0.039210  0.039099  0.948  0.000235  0.012434  0.012420  0.944  
3 τ/4  0.200  0.001100  0.035666  0.035948  0.946  −0.000333  0.011353  0.011354  0.949  
τ  0.166  0.004047  0.043238  0.053015  0.912  0.001025  0.017251  0.019598  0.943  
WKM  τ/4  0.284  0.001594  0.052516  0.052483  0.949  0.000003  0.016613  0.016357  0.946 
τ/2  0.240  0.001541  0.039144  0.038926  0.950  0.000285  0.012401  0.012398  0.946  
3 τ/4  0.200  0.001093  0.035402  0.035479  0.953  −0.000286  0.011283  0.011297  0.952  
τ  0.166  0.002922  0.040635  0.048602  0.902  0.000497  0.016646  0.018245  0.942  
COX  τ/4  0.284  0.000977  0.038843  0.038208  0.958  −0.000136  0.012292  0.012206  0.956 
τ/2  0.240  0.001108  0.033847  0.033524  0.951  0.000006  0.010700  0.010616  0.958  
3 τ/4  0.200  0.001031  0.029264  0.028893  0.958  −0.000081  0.009237  0.009253  0.954  
τ  0.166  0.002577  0.027146  0.027753  0.946  0.000148  0.008965  0.009087  0.950  
PCH  τ/4  0.284  0.001356  0.038338  0.038248  0.952  −0.000086  0.012120  0.012209  0.953 
τ/2  0.240  0.001372  0.033380  0.033529  0.948  0.000034  0.010543  0.010608  0.952  
3 τ/4  0.200  0.001113  0.028804  0.028870  0.957  −0.000081  0.009088  0.009263  0.952  
τ  0.166  0.001564  0.025811  0.025420  0.961  −0.000154  0.008105  0.008153  0.952  
Simpler  –  0.333  0.000826  0.043356  0.043147  0.952  −0.000209  0.013715  0.013776  0.955 
When considering samples of size 10, 000 (Table 1, righthand side), bias decreased in magnitude compared to a sample size of 1, 000 observations (relative bias < 0.7% for AR in all methods and <0.04% for β in the semiparametric and parametric approaches). As expected, precision increased markedly for all methods, by a factor of about \(\sqrt {10}\). Moreover, SEEs and SSDs were in closer agreement even at time τ with nonparametric methods and all coverage probabilities fell within the 0.940 to 0.960 range.
Simulation results for the estimation of attributable risk A(.) under proportional hazards, decreasing baseline hazard (γ=3/4) with regression parameter β= ln(2) and probability of exposure q=0.5
Estimation method  n=1, 000  n=10, 000  

Time  A(t)  Bias  SEE  SSD  CP  Bias  SEE  SSD  CP  
KM  τ/4  0.269  0.001799  0.044659  0.045486  0.940  0.000129  0.014162  0.014200  0.946 
τ/2  0.231  0.001217  0.036054  0.036037  0.943  0.000351  0.011437  0.011547  0.946  
3 τ/4  0.200  0.001164  0.034218  0.034637  0.948  −0.000204  0.010895  0.010746  0.956  
τ  0.176  0.003532  0.041550  0.047835  0.915  0.000299  0.016351  0.019086  0.948  
WKM  τ/4  0.269  0.001832  0.044713  0.045359  0.942  0.000131  0.014153  0.014197  0.946 
τ/2  0.231  0.001283  0.035999  0.035858  0.947  0.000368  0.011408  0.011509  0.947  
3 τ/4  0.200  0.001132  0.034004  0.034272  0.950  −0.000193  0.010838  0.010716  0.956  
τ  0.176  0.002628  0.039647  0.045615  0.906  0.000116  0.015851  0.017720  0.947  
COX  τ/4  0.269  0.000957  0.036029  0.035611  0.955  0.000107  0.011401  0.011229  0.955 
τ/2  0.231  0.001067  0.031741  0.031499  0.954  0.000129  0.010031  0.009949  0.953  
3 τ/4  0.200  0.000972  0.028300  0.028071  0.962  0.000060  0.008937  0.008899  0.949  
τ  0.176  0.002177  0.026818  0.027274  0.955  0.000168  0.008790  0.008771  0.956  
PCH  τ/4  0.269  0.003717  0.035027  0.035896  0.940  0.002630  0.011076  0.011300  0.939 
τ/2  0.231  0.002926  0.030819  0.031734  0.945  0.001853  0.009736  0.009995  0.936  
3 τ/4  0.200  0.002124  0.027440  0.028260  0.949  0.001247  0.008666  0.008949  0.940  
τ  0.176  0.001883  0.025457  0.025679  0.958  0.000621  0.008014  0.008240  0.946  
Simpler  –  0.333  0.000814  0.041900  0.041749  0.952  0.000050  0.013257  0.013257  0.947 
Simulation results for the estimation of attributable risk A(.) under proportional hazards, increasing baseline hazard (γ=4/3) with regression parameter β= ln(2) and probability of exposure q=0.5
Estimation method  n=1, 000  n=10, 000  

Time  A(t)  Bias  SEE  SSD  CP  Bias  SEE  SSD  CP  
KM  τ/4  0.299  0.000814  0.064311  0.064377  0.947  −0.000024  0.020388  0.020204  0.956 
τ/2  0.250  0.002020  0.043388  0.043169  0.952  0.000210  0.013761  0.013651  0.944  
3 τ/4  0.200  0.001174  0.037152  0.037027  0.955  −0.000469  0.011824  0.011798  0.960  
τ  0.153  0.007382  0.043968  0.054032  0.891  0.000554  0.018140  0.021081  0.939  
WKM  τ/4  0.299  0.000805  0.064427  0.064296  0.950  −0.000010  0.020380  0.020196  0.954 
τ/2  0.250  0.002055  0.043322  0.042973  0.949  0.000272  0.013722  0.013643  0.947  
3 τ/4  0.200  0.001193  0.036838  0.036463  0.962  −0.000410  0.011739  0.011741  0.958  
τ  0.153  0.005596  0.040652  0.048586  0.898  0.000055  0.017280  0.019095  0.935  
COX  τ/4  0.299  0.001207  0.041863  0.040891  0.960  −0.000209  0.013250  0.013076  0.962 
τ/2  0.250  0.001321  0.036377  0.035580  0.954  −0.000062  0.011499  0.011341  0.958  
3 τ/4  0.200  0.001300  0.030350  0.029672  0.956  −0.000121  0.009572  0.009502  0.965  
τ  0.153  0.002791  0.027165  0.028199  0.945  −0.000309  0.009206  0.010402  0.945  
PCH  τ/4  0.299  −0.000084  0.041594  0.040674  0.961  −0.001831  0.013151  0.013022  0.957 
τ/2  0.250  0.000876  0.036176  0.035464  0.956  −0.000759  0.011424  0.011313  0.958  
3 τ/4  0.200  0.001462  0.030163  0.029655  0.959  −0.000051  0.009509  0.009485  0.961  
τ  0.153  0.002572  0.025716  0.024704  0.961  0.000622  0.008058  0.007962  0.945  
Simpler  –  0.333  0.001129  0.044983  0.044481  0.955  −0.000242  0.014226  0.014195  0.957 
With a lower or greater prevalence of exposure (25% or 75% exposed), coverage probabilities in τ for the nonparametric approaches improved but sometimes remained lower than the nominal value despite a sample size of 10, 000 (Additional file 1: Table S1 and Additional file 2: Table S2 for γ=1 and β= ln(2)). The same general picture held with other values of γ (data not shown), except for the parametric approach which showed slightly insufficient (93%) coverage at times < τ for γ=3/4 and both exposure probabilities 0.25 and 0.75.
Under the same parameters but β=0 (Additional file 3: Table S3 for γ=1 and 50% exposed), results were similar to those with β= ln(2) except for slightly improved coverage probabilities in τ for the nonparametric approaches and a sample size of 1, 000.
Under all scenarios with proportional hazards (Tables 1, 2 and 3, Additional file 1: Table S1, Additional file 2: Table S2 and Additional file 3: Table S3), estimators of global AR from the simpler approach were virtually unbiased with satisfactory coverage probabilities. The estimated single values were generally greater than those of timedependent approaches at any point in time.
When followup was stopped at τ/4 or τ/2, under proportional hazards (data not shown), estimates for the two nonparametric methods were of course identical to those obtained at the same time points with a complete followup. SEEs for the semiparametric and parametric methods increased, getting closer to those of nonparametric methods with censoring at τ/2 and even closer with censoring at τ/4. Coverage probabilities remained satisfactory except for the parametric method under decreasing baseline hazard (γ=3/4) where they tended to be lower than the nominal value e.g., 0.935 and 0.918 at τ/4 for censoring at τ/4, β= ln(2) and 50% exposed, and for n=1, 000 and n=10, 000 respectively.
Simulation results for the estimation of attributable risk A(.) under nonproportional hazards with regression parameter β= ln(2) and probability of exposure q=0.5
Estimation method  n=1, 000  n=10, 000  

Time  A(t)  Bias  SEE  SSD  CP  Bias  SEE  SSD  CP  
KM  τ/4  0.181  0.001124  0.045053  0.045787  0.954  0.000289  0.014277  0.014126  0.949 
τ/2  0.133  0.001330  0.037581  0.037647  0.953  −0.000029  0.011915  0.012154  0.935  
3 τ/4  0.109  0.001211  0.036543  0.036593  0.953  −0.000301  0.011618  0.011608  0.952  
τ  0.093  0.002743  0.043713  0.051764  0.933  −0.000888  0.016362  0.019957  0.950  
WKM  τ/4  0.181  0.001138  0.045090  0.045739  0.954  0.000291  0.014274  0.014130  0.949 
τ/2  0.133  0.001347  0.037587  0.037593  0.956  −0.000024  0.011911  0.012151  0.938  
3 τ/4  0.109  0.001165  0.036511  0.036518  0.952  −0.000293  0.011612  0.011607  0.956  
τ  0.093  0.001685  0.042617  0.049261  0.920  −0.000708  0.016157  0.019107  0.946  
COX  τ/4  0.181  −0.018761  0.037521  0.037543  0.933  −0.019843  0.011869  0.011939  0.621 
τ/2  0.133  0.010548  0.033500  0.033580  0.941  0.009504  0.010588  0.010676  0.847  
3 τ/4  0.109  0.023376  0.030960  0.031017  0.879  0.022314  0.009775  0.009879  0.368  
τ  0.093  0.030360  0.029427  0.029588  0.830  0.029168  0.009323  0.009456  0.127  
PCH  τ/4  0.181  0.026479  0.048525  0.049191  0.908  −0.017516  0.011688  0.012080  0.672 
τ/2  0.133  0.057418  0.044915  0.045594  0.738  0.011082  0.010391  0.010768  0.806  
3 τ/4  0.109  0.070045  0.042342  0.043042  0.607  0.023478  0.009571  0.009936  0.313  
τ  0.093  0.075924  0.040403  0.041050  0.525  0.029848  0.009011  0.009360  0.098 
Data example
As in our simulations, we used timeonstudy rather than attained age as the timescale after checking that both yielded similar results. Fitting a Weibull distribution to the observed survival data and considering incident invasive breast cancer as the event of interest (i.e., considering time to breast cancer occurrence), the shape (γ) and scale (θ) parameters were estimated as 1.2 and 178.2 respectively and the corresponding estimated Weibull survival function almost coincided with nonparametric KaplanMeier estimate (data not shown). The assumption of proportional hazards between women everexposed and those neverexposed to any MHT at baseline seemed appropriate (Schoenfeld residual test, p=0.7), with an estimated HR at 1.22 (95% CI, 1.13 to 1.33) for MHT exposure from the Cox model.
Finally, using the method proposed by Spiegelman et al. [10], we found that 9.2% (95% CI, 5.4 to 13.0%) of cases who developed invasive breast cancer at various times in the cohort followup were attributable to MHT exposure. Using the simpler approach with the proportion of exposed subjects at inclusion, we obtained a close, slightly smaller estimate at 9.1% (95% CI, 5.3 to 12.8%).
Discussion
Comparing different methods of AR estimation when disease incidence is interpreted as a CDF [7, 8], we observed that AR estimators were essentially unbiased for all approaches when we generated event times from a proportional hazards model. Empirical and estimated variances were close, with proper coverage probabilities except at the end of followup for the nonparametric methods and a smaller sample size. When considering a nonconstant baseline hazard, estimates using the parametric approach were robust despite misspecification of the baseline hazard. For nonparametric approaches, biases tended to increase at the end of followup (time τ) when the baseline hazard increased with time. With the simpler approach, results were satisfactory. However, under nonproportional hazards, estimates using the semiparametric and parametric approaches were biased with poor coverage probabilities.
To our knowledge, this is the first simulation study comparing nonparametric, semiparametric and parametric methods of AR estimation as a function of time as well as a simpler, global approach for a diversity of scenarios (proportional or nonproportional hazards, constant or nonconstant baseline hazard, varying exposure probabilities, strengths of association and sample sizes) in the survival analysis context. Chen et al. [7] reported simulations for the KaplanMeier, weighted KaplanMeier and transformation models when event times were generated from proportional or nonproportional hazards models with regression parameter β=1, 40% probability of exposure and a sample size of 1, 000 observations. Like them, we found that, under the assumption of independent censoring, results with KM and WKM approaches were very close. Differences between the two nonparametric approaches were apparent when censoring was dependent on covariates [7], which we did not evaluate in this study.
Also in line with Chen et al. [7], when we generated event times from a proportional hazards model, we found that nonparametric and semiparametric estimates were all unbiased, nonparametric estimates had larger variances than semiparametric estimates and estimated variances accurately reflected the true variance except in τ for the nonparametric approaches and a sample size of 1, 000 observations. Nonparametric approaches tended to perform better (respectively worse) when exposure prevalence was lower (respectively higher) which could be expected from the possibly unstable and inefficient KaplanMeier estimator of survival among the unexposed when the proportion of those is small [7]. This general picture held in our simulations whether event times were generated with constant, decreasing or increasing baseline hazard. We note, however, that, when we considered a larger sample size, the discrepancies between estimated and empirical variances tended to diminish, with most often satisfactory coverage probabilities in τ.
For nonproportional hazards, we generated event times using a transformation model considered by Chen et al. [7] and found consistent results for the nonparametric approaches, similar to those in the case of proportional hazards. However, while Chen et al. [7] applied the same nonproportional hazards model for both data generation and analysis (AR estimation), we generated data under nonproportional hazards and estimated AR from (misspecified) Cox’s proportional hazards model. This explains the impaired performance we observed when the proportional hazards assumption was violated in contrast with the satisfactory results obtained by Chen et al. [7]. Sjölander and Vansteelandt [9] recently proposed an alternative semiparametric estimator of AR also based on Cox’s proportional hazards model that proved robust to various model misspecifications. However these authors did not evaluate deviations from the proportional hazards assumption.
Like Chen et al. [7] in their simulation and example analysis, we observed greater imprecision of the nonparametric estimators at the start of followup, which could explain possible early negative AR values. This imprecision could be expected because the estimation of the survival function relies upon the information available until the time of interest and not many events have yet occurred by then. This differs from the semiparametric and parametric methods which take advantage of the estimation of parameter β being performed over the entire followup. Consistently, we found larger variances for the semiparametric and parametric approaches with shorter lengths of followup.
Another novelty of this work was the evaluation of the parametric approach to AR estimation proposed by Laaksonen et al. [8] using simulations and its comparison with nonparametric and semiparametric approaches. Generally under proportional hazards, we found close agreement between the semiparametric and parametric approaches, in our simulations as well as in the example. Of note, the parametric approach seemed robust despite misspecification of baseline hazard, i.e., when we considered decreasing or increasing (instead of constant) baseline hazard and proportional hazards. However, like the semiparametric approach based on Cox’s model, the parametric approach was sensitive to the proportional hazards assumption and performed poorly in our simulations when this assumption was violated. We also evaluated the simpler, global approach and our results were satisfactory under proportional hazards.
As noted by several authors [4, 7], simpler approaches based on Eq. (1) or equivalent formulas [1, 2] are generally defined for binary outcomes with timeindependent risk factors. Consequently, they prove to be inadequate for cohort studies with censored timetoevent outcomes and possibly timedependent covariates. In contrast, the nonparametric, semiparametric and parametric approaches we considered here have been specifically developped for censored timetoevent outcomes and produce AR estimate as a function of time, thus allowing the AR to be timevarying. A major limitation of the simpler approach in the context of cohort studies is that it only takes account of the proportion of exposed subjects at the beginning of followup. The proportion of exposed subjects indeed decreases as followup time increases (because exposed subjects fail earlier than nonexposed subjects) [6]. This explains why our AR estimates from the simpler approach were generally greater than those from timedependent approaches and further underlines why approaches estimating AR as a function of time are an improvement on the simpler approach in the context of survival analysis.
In our study, we used the definition of AR based on CDFs because it is a natural extension of the standard AR definition (Eq. (1)) for timetoevent outcomes [6–9] and it is equivalent to the standard definition when time t is the end of followup in cohort studies [4]. In addition several estimation methods have been proposed for the CDFbased AR definition in cohort studies and the survival analysis context in contrast to the alternative definition based on instantaneous hazard functions [4, 5] for which only one method of estimation based on Cox’s proportional hazards model has been published [4].
In cohort studies where exposed individuals are oversampled relative to the exposure prevalence in the population, AR will correctly reflect the impact of exposure in the cohort, but the impact at the population level will be overestimated. The marginal survival function S(t) should be corrected in order to alleviate this upward bias on AR estimates. The AR (and its estimates) being a function of time, various representations of AR estimates can be used. We used a graphical representation of the whole time function in our example and produced estimates at four equally spaced times in our simulations. Alternatively, a single overall estimate could be obtained by averaging out the time function of AR estimates or by using the alternative approach by Spiegelman et al. [10] as in our example.
We chose our simulation parameters to resemble real epidemiologic cohorts. These often include a few thousands participants followed for several years. For a smaller sample size (n=500), whether we used the logarithmic transformation or not, we observed findings generally similar to those presented with a sample size of 1,000 observations. This was true with the notable exception of the less than nominal coverage probability for the semiparametric approach at time τ for constant (γ=1) and decreasing (γ=3/4) baseline hazards, and at times τ/4 and τ for increasing (γ=4/3) baseline hazard (data not shown).
In our application, the proportional hazards assumption seemed appropriate, as well as a Weibull distribution for event times with an increasing baseline hazard and shape parameter halfway between the values γ=1 and 4/3 considered in our simulation study. Exposure frequency was also close to our simulated 0.5 probability of exposure. However, as in many epidemiologic cohorts, the censoring rate was much greater in our example (94.2%) than in our simulations. The resulting imprecision may explain the nonparametric AR estimates apparently increasing until three quarters of total followup but compatible with the more expected decreasing trend. Chen et al. [7] observed the same finding in their application on a shorter length of followup.
Using the approach described by Spiegelman et al. [10], the overall AR estimate for ever use of MHT at baseline was 9.2% in the E3N cohort. This estimate was close to those obtained at the end of followup with the nonparametric methods and at the start of followup with the parametric and semiparametric approaches. In a recent publication, Dartois et al. [12] reported a higher AR estimate of 14.5% (95% CI, 9.2 to 19.6%) for recent MHT use and postmenopausal invasive breast cancer from the E3N cohort data, using the approach proposed by Spiegelman et al. [10] and a more refined, adjusted analysis with MHT exposure as a timedependent covariate.
This study has some limitations. First, we did not evaluate AR estimates adjusted for covariates. Adjustment for multiple variables is common practice in epidemiology, especially age which can also be used as the underlying timevariable [19]. In our example, using analyses unadjusted or parametrically adjusted for age, there was a statistically significant association between baseline MHT ever use and breast cancer risk, in line with findings from more complex models with age as the timescale and adjustment for other covariates in the original study [11]. Although adjustment for covariates is available in packages for semiparametric and parametric approaches [7, 8], there are constraints in nonparametric approaches as the number of covariates must be limited and adjustment for continuous variables is not possible. Moreover, available packages for estimating the AR would need to be adapted to allow left truncation resulting from using age as the timescale. Second, in our example, we only considered women who had ever received MHT at baseline as exposed whereas exposure can vary during followup. Other methodological studies are needed to take into account the exposure time dependency for estimating AR as a function of time [9]. Finally, we have ignored the competing risk of death and cancers of other sites (11.2% of our 94.2% censored observations) which might also bias our estimate of breast cancer risk attributable to MHT [20].
Conclusions
The AR estimators from the four timedependent methods had satisfactory performance under the proportional hazards assumption. Estimators using semiparametric and parametric approaches were not robust in case of nonproportional hazards. Lack of precision could be an issue for nonparametric methods at the beginning of the followup time in cohorts of relatively low sample size. In practice, if the proportional hazards assumption seems appropriate, the semiparametric or parametric approaches should be used.
Abbreviations
 AR:

Attributable risk
 CDF:

Cumulative Distribution Function
 CI:

Confidence Interval
 COX:

Cox’s proportional hazards model
 CP:

Coverage Probability
 E3N:

Étude Épidémiologique auprès de Femmes de la Mutuelle Générale de l’Éducation Nationale
 HR:

Hazard Ratio
 KM:

KaplanMeier
 MHT:

Menopausal Hormone Therapy
 PCH:

Piecewise Constant Hazards
 RR:

Relative Risk
 SEE:

Standard Error Estimator
 SSD:

Sampling Standard Deviation
 WKM:

Weighted KaplanMeier
Declarations
Acknowledgements
The authors wish to thank Pascale TubertBitter for her constructive comments, Mohammed Sedki for statistical advice and Agnès Fournier for kindly sharing the E3N data. The authors are also grateful to all participants, practitioners and study staff of the E3N study. The E3N cohort is conducted with the financial support of ‘Mutuelle Générale de l’Éducation Nationale’ (MGEN); the European Community; ‘Ligue nationale contre le Cancer’; ‘Institut GustaveRoussy’; ‘Institut National de la Santé et de la Recherche Médicale’ (Inserm); and ‘Fondation de France’.
Funding
This research was supported by the French Medicines Agency (‘Agence Nationale de Sécurité du Médicament et des produits de santé’, ANSM) and French Government’s Investissement d’Avenir program, Laboratoire d’Excellence “Integrative Biology of Emerging Infectious Diseases” (grant number ANR10LABX62IBEID). MG was a recipient of PhD grant from the French Ministry of Research.
Availability of data and materials
The E3N dataset analyzed during the current study was available from the E3N study team but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the E3N study team upon reasonable request and permission of E3N principal investigator.
Authors’ contributions
MG conducted the simulation study and prepared the first draft of the manuscript. JB and ACMT devised the analytical strategy and codrafted the manuscript. LD contributed to cohort data analysis. All authors contributed to the preparation of the final manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
The E3N cohort received ethical approval from the French National Commission for Computed Data and Individual Freedom (‘Commission Nationale de l’Informatique et des Libertés’, CNIL) under the reference CNIL 186 and all participants in the study provided informed consent.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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