Abstract
Background
Analyzing timetoonset of adverse drug reactions from treatment exposure contributes to meeting pharmacovigilance objectives, i.e. identification and prevention. Postmarketing data are available from reporting systems. Timestoonset from such databases are righttruncated because some patients who were exposed to the drug and who will eventually develop the adverse drug reaction may do it after the time of analysis and thus are not included in the data. Acknowledgment of the developments adapted to righttruncated data is not widespread and these methods have never been used in pharmacovigilance. We assess the use of appropriate methods as well as the consequences of not taking right truncation into account (naive approach) on parametric maximum likelihood estimation of timetoonset distribution.
Methods
Both approaches, naive or taking right truncation into account, were compared with a simulation study. We used twelve scenarios for the exponential distribution and twentyfour for the Weibull and loglogistic distributions. These scenarios are defined by a set of parameters: the parameters of the timetoonset distribution, the probability of this distribution falling within an observable values interval and the sample size. An application to reported lymphoma after anti TNF α treatment from the French pharmacovigilance is presented.
Results
The simulation study shows that the bias and the mean squared error might in some instances be unacceptably large when right truncation is not considered while the truncationbased estimator shows always better and often satisfactory performances and the gap may be large. For the real dataset, the estimated expected timetoonset leads to a minimum difference of 58 weeks between both approaches, which is not negligible. This difference is obtained for the Weibull model, under which the estimated probability of this distribution falling within an observable values interval is not far from 1.
Conclusions
It is necessary to take right truncation into account for estimating timetoonset of adverse drug reactions from spontaneous reporting databases.
Keywords:
Pharmacovigilance; Reporting databases; Right truncation; Parametric estimation; Maximum likelihood estimation; Bias; Simulation studyBackground
Identifying and preventing adverse drug reactions are major objectives of pharmacovigilance. Owing to design constraints, premarketing clinical trials fail to identify rare events, which lead in the last decades to an increased focus placed on the development of postmarketing surveillance methods [111]. Postmarketing spontaneous reporting of suspected adverse drug reactions has proved a valuable resource for signal detection [1217]. It has recently been suggested that the modeling of the timetoonset of adverse drug reactions could be a useful adjunct to signal detection methods, either from spontaneous reports [18,19] or longitudinal observational data [20]. Timely acquiring knowledge with respect to the timetoonset distribution of adverse drug reactions contributes to meeting pharmacovigilance objectives. Early estimation procedures tailored to available pharmacovigilance data, i.e. spontaneous reporting data, should be sought.
The data consisting of the timetoonset among patients who were reported to have potentially developed an adverse drug reaction are righttruncated. Truncation arises because some patients who were exposed to the drug and who will eventually develop the adverse drug reaction may do it after the time of analysis (Figure 1). Among patients exposed to the drug, only those who experienced adverse reactions before time of analysis are included in the database. No information is available for the other patients. If all the patients begin their treatment at the same time, the data are righttruncated with a single truncation time. If they do not all begin their treatment at the same time, the data are righttruncated with different truncation times. In spontaneous reporting, data are righttruncated with different truncation times and they require appropriate statistical methods.
Figure 1. Right truncation and data on timetoonset of adverse drug reactions from spontaneous reporting databases. Some patients who were exposed to the drug and who will eventually develop the adverse drug reaction may do it after the time of analysis. Here, in these hypothetical examples, the patient on the top line is included in the database because he experienced the adverse drug reaction before the time of analysis, i.e.x_{1 }≤ t_{1}. The patient on the bottom line is not included in the database because he has not yet experienced the adverse drug reaction, i.e. , when data are analyzed.
This paper investigates parametric maximum likelihood estimation of the timetoonset distribution of adverse drug reactions from spontaneous reporting data for different types of hazard functions likely to be encountered in pharmacovigilance. Acknowledgment of the developments adapted to righttruncated data is not widespread and these methods have never been used in pharmacovigilance. No simulation studies are available on the accuracy of their estimates. Furthermore, a naive approach that does not take into account right truncation features of spontaneous reports and uses classical parametric methods instead of appropriate methods may lead to misleading estimates. We consider the two approaches, i.e. taking or not taking right truncation into account, and the corresponding parametric maximum likelihood estimators. Both approaches are compared with a simulation study conducted to evaluate the consequences, notably in terms of bias, of not considering right truncation on the maximum likelihood estimates, as well as assessing the performances of the right truncationbased estimation. We also apply these methods to a set of 64 cases of lymphoma occurring after anti TNF α treatment from the French pharmacovigilance.
Methods
Proper estimation of the timetoonset distribution
We consider a given time of analysis and the population of exposed patients who will eventually experience the adverse drug reaction before they die. Let X be the timetoonset of the adverse drug reaction of interest in that population and F its cumulative distribution function one is willing to estimate. Observations arising from n reported cases are (x_{1},t_{1}),(x_{2},t_{2}),…,(x_{n},t_{n}), where x_{i }is the timetoonset calculated as the lag between the time of the occurrence of the reaction and the time of initiation of treatment, and t_{i }is the truncation time calculated as the lag between the time of analysis and the time of initiation of treatment. Let t^{∗} be the maximum of the observed truncation times. All observed data meet the condition x_{i }≤ t_{i}.
We consider a parametric model for the timetoonset X, with cumulative distribution function F (x; θ) and density f(x; θ), and derive the following maximum likelihood estimations of θ.
When right truncation, i.e. the condition x_{i }≤ t_{i}, is ignored, the likelihood of the sample is written as:
maximizing this likelihood yields the naive estimator of θ.
When right truncation is considered, the likelihood is modified. Observed timestoonset consist of n independent realizations of random variables with respective distribution the conditional distribution of X_{i }given {X_{i }≤ t_{i}}, that is with cumulative distribution function and density . The likelihood is now written as:
the maximum likelihood estimator from this likelihood, , is the proper estimation of θ and is called the truncationbased estimator (TBE).
The nonparametric maximum likelihood estimation for righttruncated data was developed and used to estimate the incubation period distribution for AIDS [21,22]. However, in a nonparametric setting, one can only estimate the distribution function conditional on the time to event as being less than t^{∗} :
where the v_{j}’s are the m distinct values of the x_{i}’s, i =1,…,n, taken by patients and for 1 ≤ j ≤ m, I denoting the indicator function. The unconditional distribution function is not identifiable, as F (t^{∗}) is not known and cannot be estimated from the data.
In a parametric framework, the unconditional distribution is completely specified by a parameter θ of finite dimension. Maximum likelihood estimation of the parameter of interest can be conducted with the conditional distributions that describe the observations and the unconditional distribution can be estimated secondarily by . Hence parametric maximum likelihood estimation is potentially more useful than nonparametric estimation since the unconditional distribution is of interest for pharmacovigilance purposes [18,20].
Simulation study
Some adverse reactions have a very short timetoonset, from several minutes to several hours after the beginning of treatment. Others occur only after several days, weeks, months or even years of exposure. This variation depends on numerous factors such as the pharmacokinetics of the drug and its metabolites, or the pathophysiological mechanism of the effect. The multiplicity of the underlying mechanisms results in a range of possible hazard functions that can be observed in pharmacovigilance [23]. The simplest model is given by a constant hazard function of time; the corresponding distribution is the exponential distribution with a rate parameter λ. Effects may also have an early or a late onset, the latter being the case for instance, when the rate of occurrence of the adverse reaction depends on the duration of exposure. Two distribution families among others make it possible to handle a wide range of hazard functions: the Weibull distributions and the loglogistic distributions (Table 1). Both are defined with two scalar parameters (λ,β); λ is the scale parameter and β is the shape parameter. The hazard function for the Weibull model is increasing if β > 1, decreasing if β < 1 and constant if β = 1 where it reduces to the exponential distribution. The hazard function for the loglogistic model is decreasing if β < 1 and has a single maximum if β > 1. We therefore consider the families of the exponential, Weibull and loglogistic distributions.
Table 1. Exponential, Weibull and loglogistic distributions
The timestoonset were generated from these three distributions. Two values of λ were considered for the exponential distribution: 0.05 and 1. The same values were used for the scale parameter λ of the Weibull and loglogistic distributions. For the shape parameter β, the values 0.5 and 2 were chosen. The truncation times were uniformly distributed in [0,τ]. Survival and truncation times were independently generated. For a chosen value of p, with p representing the probability of X falling within the observable values interval [0, τ], the parameter τ was determined as P (X< τ) = p. The probability 1  p is also a lower bound of the actual proportion of truncated data P (X> T), the truncation time T being randomly generated. The probability p was chosen in {0.25, 0.50, 0.80}. The sample size n was chosen in {100, 500}. For each drawn pair (X,T), if the timetoonset was shorter than the truncation time, then the pair was included in the data. If not, another pair (X,T) was generated. Pairs were generated until the sample size of observations included was equal to n.
Parametric likelihood maximization with and without considering right truncation were performed for each generated sample. An iterative algorithm is necessary to solve this optimization problem except for the naive exponential estimation. Calculations were made with the R [24] function maxLik from the package maxLik. For each set of simulation parameters, 1000 replications were run.
Application study
We analyzed 64 French cases of lymphoma that occurred after anti TNF α treatment using the national pharmacovigilance database at the date of February 1, 2010 [25]. The population included patients suffering from rheumatoid arthritis, Crohn’s disease, ankylosing spondylitis, psoriatic arthritis, psoriasis, Sjögren’s syndrome, dermatomyositis, polymyositis or polyarthropathy and exposed to one or (successively) more of the three anti TNF α available at the study date: etanercept, adalimumab and infliximab. The occurrence of a malignant lymphoma was confirmed by histopathological analysis. Marketing authorization was obtained in August 1999 for infliximab, in September 2002 for etanercept and in September 2003 for adalimumab. These 64 adverse effects occurred between July 2001 and October 2009. None of the survival or truncation times was missing in the database. The observed maximum truncation time was 529 weeks.
All anti TNFagents taken together, we derived the parametric maximum likelihood estimates and secondarily corresponding estimated mean times, with and without considering right truncation, for the exponential, Weibull and loglogistic distributions. For completeness, we also derived the nonparametric maximum likelihood estimation.
The French pharmacovigilance database is developed by the French drug agency (Agence Nationale de Sécurité du Médicament et des produits de santé, ANSM) and is not publicly available. It is build up and used on an ongoing basis by the network of regional pharmacovigilance centres, which have a direct access to the data. This set of data has already been extracted for another study [25] with the authorization of the ANSM and the network of regional centres, according to the internal rule.
Results
Simulation study
For each set of simulations parameters, for both approaches and for both parameters, the bias and the mean squared error of the parametric maximum likelihood estimator, based on the 1000 replications, were calculated as well as the proportion of replications where the estimate is larger than the true value. As the iterative algorithm may fail to find a maximum, those three quantities were actually calculated on the replications where there was no problem of maximization. The mean squared error is a measure of the dispersion of the estimator around the true value of the parameter  the smaller the better  and is used for global comparative purposes between two estimation procedures, as it incorporates both the variance of the estimator and its bias. The proportion of replications where the estimate is larger than the true value makes it possible to know if the estimators tend to overestimate or underestimate systematically the true value of the parameter.
Bias and mean squared error
For both approaches, for all distributions and for both parameters, the smaller is p, the larger are the bias and the mean squared error (Tables 2, 3 and 4). This increase with p is smaller for the parameter β than for the parameter λ. These estimators tend to be positively biased. However, the bias might be almost naught for the TBE. The bias and the mean squared error of the naive estimator are always larger than the bias and the mean squared error of the TBE, but to a lesser extent for the parameter β. When the sample size n increases, the bias and the mean squared error are almost constant for the naive estimator, while for the TBE, they decrease clearly (Tables 2, 3 and 4). The naive estimator might be unacceptably large whatever the value of p, whereas the TBE shows good performances when p is equal to 0.8, and often even less according to the distribution.
Table 2. Simulation results: estimations of bias and mean squared error for the exponential model
Table 3. Simulation results: estimations of bias and mean squared error for the Weibull model
Table 4. Simulation results: estimations of bias and mean squared error for the loglogistic model
Proportion of replications where the estimator is larger than the true value
For both approaches, for all distributions and for both parameters, Tables 5, 6 and 7 show that the naive estimator of λ appears to be almost always larger than the theoretical value λ, and that this is not far from being true for the naive estimator of β. This suggests that the naive estimator of λ might be almost surely larger than the true value of the parameter, which would be a  non desirable  statistical feature of the naive estimator.
Table 5. Simulation results: proportion of replications where the maximum likelihood estimator is larger than the true value of the parameter for the exponential model
Table 6. Simulation results: proportion of replications where the maximum likelihood estimator is larger than the true value of the parameter for the Weibull model
Table 7. Simulation results: proportion of replications where the maximum likelihood estimator is larger than the true value of the parameter for the loglogistic model
Application study
Table 8 presents the estimates of the parameters for the three models and both approaches. There was no problem of maximization. The naive estimates are always larger than the truncationbased estimates. From the simulation results, it might be thought that the naive estimator overestimates the true values of parameters λ and β, and that the size of the bias is related to the unknown probability p. Estimations of the parameters for the truncationbased approach make it possible to estimate p by calculating . However, estimates of p are different according to the model (Table 8). In particular, for the Weibull model, the estimate is large ( = 0.98). The larger is , the closer are the naive and the truncationbased estimates.
Table 8. Parameter estimation and estimated mean timetoonset for 64 cases of lymphoma that occurred after anti TNF α treatment
Figure 2 shows the nonparametric maximum likelihood estimation of the conditional survival function, , and the parametric maximum likelihood estimation of the conditional, , and unconditional, , survival functions for the truncationbased approach for these data. The estimations of the conditional survival functions are always closer to the nonparametric estimation than the estimations of the unconditional survival functions. The conditional and unconditional estimations of the Weibull survival functions are almost similar because the estimate of p is about 1. This figure shows that the estimation of the conditional Weibull survival function is closer to the nonparametric maximum likelihood estimation of the conditional survival function than the estimations of the conditional exponential and conditional loglogistic survival functions. Thus, Weibull could be a reasonable candidate model to describe the data.
Figure 2. Right truncationbased estimations of timetoonset of lymphoma that occurred after anti TNFα treatment. Data include 64 cases. Three models are fitted: exponential, Weibull and loglogistic. Estimations of the conditional survival function (C), estimations of the unconditional survival function (U) and the nonparametric maximum likelihood estimation of the survival function (NPMLE) are displayed.
Figure 3 shows the parametric maximum likelihood estimation of the unconditional survival function for both approaches. The distance between both survivals, naive and truncationbased, decreases with the estimated probability (in the order: exponential, loglogistic and Weibull). Furthermore, the survival functions from the truncationbased estimates are always above the survival functions from the naive estimates, which is consistent with the naive estimator overestimating the true values of the parameters λ and β. Even for the Weibull model, i.e. the model with the largest , the estimated expected timetoonset would be 135 weeks with the naive approach and 193 weeks with the truncationbased estimates, which corresponds to a markedly large gap (Table 8). For completeness, we also calculated the 95% simple bootstrap confidence intervals of the expected time (BCa method) [26,27] based on 5000 bootstrap samples, for the truncationbased approach. They do not include the naive estimated mean time, whatever the fitted model, and even though these confidence intervals are extremely wide.
Figure 3. Naive and right truncationbased estimations of timetoonset of lymphoma that occurred after anti TNFα treatment. Data include 64 cases. Three models are fitted: exponential, Weibull and loglogistic. Estimations of the unconditional survival function for the naive approach (Naive) and for the truncationbased approach (TBE) are displayed.
Discussion and conclusions
In drug safety assessment, the temporal relationship between drug administration and timetoonset is of utmost relevance. A better understanding of the underlying mechanism of the occurrence of an adverse effect is crucial, as it could allow the identification of particular groups of patients at risk and of particular risk timewindows in the course of a treatment and lead to preventing or diagnosing earlier the occurrence of adverse reactions. In this framework, the timetoonset of an adverse drug reaction constitutes an essential feature to be analyzed. Its accurate estimation and modeling could help in understanding the mechanism of a drug’s action.
As rare adverse effects are not generally identified by cohort studies of exposed patients but from spontaneous reporting systems, we investigated with a simulation study the accuracy of estimates that can be obtained from these data in a parametric framework. As one can only estimate a conditional distribution function in a nonparametric setting, the nonparametric maximum likelihood estimator is of rather little interest for pharmacovigilance people. For a finite sample size, the simulations show that, whatever the approach, naive or truncationbased, the parametric maximum likelihood estimator may be positively biased and that this bias and the corresponding mean squared error increase when the theoretical probability p for the timetoonset to fall within the observable values interval decreases. However, for a fixed value of p, the bias and the mean squared error are always larger when the right truncation is not considered than when it is, and the gap may be large. In addition, bias and mean squared error might in some instances (Weibull, loglogistic) be unacceptably large for the naive approach, even for a large value of p, while with a probability p of 0.8, or sometime even less, the TBE shows good performances. Asymptotically, the naive estimator may not be unbiased because the bias and the mean squared error seem to be constant with the sample size and the maximization is based on a misleading likelihood, while the bias and the mean squared error for the TBE decrease as the sample size increases. Therefore, even if the sample size is large, the gap between both estimators does not disappear and the truncationbased approach should be used.
The probability p plays an important role in the estimation of the distribution of the timetoonset of adverse reaction for righttruncated data. Knowledge exists on a range of possible pharmacological mechanisms. It is thus possible to get a rough idea of the fraction of potentially missed cases (the adverse reactions of treated patients that have yet to occur) and then to decide on the relevance of the time of analysis. Spontaneous reports result from three processes: the occurrence case process, its diagnosis and the reporting process. It is well known that underreporting is widespread, even for serious events. In addition, factors of underreporting include the seriousness of the effect, the age of the patient and the novelty of the effect, but also timerelated variables such as the length of marketing or the time since exposure [2833]. In the approach proposed here, it is assumed that the underreporting is uniform. Such a hypothesis might not always be acceptable. However, with longterm effects such as lymphoma and a homogeneous observation period within the marketing life of the product, nonstationarity of reporting is unlikely.
Problems of maximization may arise when right truncation is taken into account. The smaller is p, the more the iterative algorithm is likely to fail. Some papers mentioned the existence of a problem in the parametric likelihood maximization and explained that, because of right truncation, the likelihood may be flat and the maximum may be difficult to find [21,3436].
For the 64 cases of lymphoma after anti TNF α treatment, there was no problem of convergence of the iterative algorithm. Both estimates, naive and truncationbased, were available for each fitted model. From the truncationbased estimates, it is possible to estimate p. Here it ranges from 0.98 (Weibull) to 0.60 (exponential). Since this probability is unknown, the nonparametric maximum likelihood estimation estimates only the distribution function conditional on the timetoevent being less than the maximum observed truncation time. However, although conditional, the nonparametric estimate is a reference that provides an idea of how the data fit a given model. We followed the graphical procedure for checking goodnessoffit for righttruncated data suggested by Lawless (2003) that is based on the nonparametric maximum likelihood estimator and consists in plotting the conditional fitted parametric survivals together with the nonparametric estimation [36]. Here, the conditional Weibull survival function seems the closest to the nonparametric estimation. This finding underlines the interest for developing goodnessoffit tests adapted to righttruncated data. While only three families of distributions were considered for the present simulation study, other families could be explored such as the gamma or the lognormal families or mixture models. For instance, in more complex situations, the treatment might be a combination of drugs, each of them inducing the effect but in a different time window. In that case, the hazard function may vary several times and a family of more complex distributions could be of greater interest. Additionally, we chose to consider the truncation times as deterministic, which is equivalent to working on conditional distributions for the likelihood. However, another possible approach is to consider the truncation time as a random variable and to study the random pair (X,T) where X is the survival time and T is the truncation time [3739].
Finally, improvement of timetoonset distribution assessment could make it possible to compare two drug profiles or more generally to assess risk factors with regression models.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
FL, JYD and PTB conceived and designed the work. FL implemented the simulations, performed data analysis and wrote the initial draft of the manuscript. HT and FH made the extraction of the data from the national pharmacovigilance database. All authors contributed to the interpretation of the results of the data analysis. All authors reviewed and revised the draft version of the manuscript. All authors read and approved the final version of the manuscript.
Acknowledgements
This work was supported by the Fondation ARC (fellowship DOC20121206119 to Fanny Leroy).
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