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Simulating recurrent event data with hazard functions defined on a total time scale
 Antje JahnEimermacher^{1}Email author,
 Katharina Ingel^{1},
 AnnKathrin Ozga^{1},
 Stella Preussler^{1} and
 Harald Binder^{1}
https://doi.org/10.1186/s1287401500052
© JahnEimermacher et al.; licensee BioMed Central. 2015
Received: 30 July 2014
Accepted: 5 February 2015
Published: 8 March 2015
Abstract
Background
In medical studies with recurrent event data a total time scale perspective is often needed to adequately reflect disease mechanisms. This means that the hazard process is defined on the time since some starting point, e.g. the beginning of some disease, in contrast to a gap time scale where the hazard process restarts after each event. While techniques such as the AndersenGill model have been developed for analyzing data from a total time perspective, techniques for the simulation of such data, e.g. for sample size planning, have not been investigated so far.
Methods
We have derived a simulation algorithm covering the AndersenGill model that can be used for sample size planning in clinical trials as well as the investigation of modeling techniques. Specifically, we allow for fixed and/or random covariates and an arbitrary hazard function defined on a total time scale. Furthermore we take into account that individuals may be temporarily insusceptible to a recurrent incidence of the event. The methods are based on conditional distributions of the interevent times conditional on the total time of the preceeding event or study start. Closed form solutions are provided for common distributions. The derived methods have been implemented in a readily accessible R script.
Results
The proposed techniques are illustrated by planning the sample size for a clinical trial with complex recurrent event data. The required sample size is shown to be affected not only by censoring and intrapatient correlation, but also by the presence of riskfree intervals. This demonstrates the need for a simulation algorithm that particularly allows for complex study designs where no analytical sample size formulas might exist.
Conclusions
The derived simulation algorithm is seen to be useful for the simulation of recurrent event data that follow an AndersenGill model. Next to the use of a total time scale, it allows for intrapatient correlation and riskfree intervals as are often observed in clinical trial data. Its application therefore allows the simulation of data that closely resemble real settings and thus can improve the use of simulation studies for designing and analysing studies.
Keywords
Background
Recurrent event data are multivariate failure time data where the individuals experience repeated occurrences of the same type of event. In clinical applications, recurrent events are often particular medical conditions, such as hospitalizations due to a particular disease, cardiovascular events, epileptic seizures, episodes of multiple sclerosis or falls in elderly people. Many survival models have been proposed to handle recurrent event data [1], and simulations are commonly used to investigate statistical methods or to plan the sample size of a clinical trial [2]. A survey of PubMed indicates 36 articles within the last five years that feature the term recurrent event and simulation in the title or abstract.
Whereas some authors have demonstrated the need to investigate statistical methods under different event generation processes and correlation structures [3,4], little attention has been given to the time scale that is applied for subsequent events. In particular, simulation studies often generate data from a gap time perspective, where the time and thus the risk process is reset after each event, which simplifies the simulation process. Conversely, in many clinical applications a total time perspective is appropriate, i.e. where the hazard for experiencing a particular recurrent medical condition depends on the time since some starting point. To improve the accordance between the simulation models applied to investigate statistical techniques and the statistical models used to analyze clinical data, we have derived a flexible simulation algorithm that implements a total time perspective.
The definition of time that is used as the argument of the hazard function is a fundamental step in modeling data and affects the statistical results and their interpretation [5,6]. The starting point 0 in a total time perspective may be the onset of a disease, the beginning of some treatment, entering a clinical trial or birth (agedependent risks) [7,8]. A total time scale is, for example, commonly applied when analyzing the efficacy of pneumococcal vaccines to prevent recurrent episodes of acute otitis media [912] because the risk of acute otitis media is known to be agedependent [13,14]. Schneider et al. encourage the use of a total time scale when reestimating the sample size for trials in relapsing multiple sclerosis [15], as total time trends of the event rates seem to be present [8]. Some authors use the term counting process time [6] or calendar time [5] to further specify the risk intervals that are to be used in the regression analysis. The time scale is the same as total time. A total time scale (counting process) also underlies the AndersenGill model, probably the most frequently applied model used to analyze recurrent failure time data in medical science. Conversely, in simulation studies often a gap time scale is used [3,4,6,16], sometimes by defining constant hazards. This may be due to a lack of information on how to simulate recurrent event data using a total time scale as opposed to published simulation algorithms for gap time [3,4] and other multivariate [17,18] or univariate [19] failure time data.
Furthermore, in simulation designs, the individuals are usually assumed to be continuously at risk for experiencing recurrent events. In reallife situations, individuals might be temporarily insusceptible to a recurrent incidence of the event (riskfree intervals). An example are relapsing diseases where effective treatments are available that prevent disease progression, but the treatments can only be discontinuously administered because of adverse side effects. Individuals experience disease relapses outside the treatment courses but are considered to be insusceptible to events under treatment.
The aim of this article is to provide a general approach for simulating recurrent event data when considering an AndersenGill model with a total time scale. We allow for arbitrary timevarying hazard functions, riskfree intervals, and incorporate interpatient heterogeneity by including covariates and frailty terms.
The article is organized as follows: We start with clinical examples that characterize settings where total time models should be considered. Then, in the Methods section, we introduce the recurrent event model, derive the simulation algorithm and provide closed form solutions. In the Results section we demonstrate the implementation and illustrate the use of our methods on power and sample size planning for trials with complex recurrent event data. We discuss our methods and results and finally we finish the article with concluding remarks.
Examples for total time scale settings
We illustrate the rationale behind the use of a total time scale on two clinical examples. In addition to a total time model, we will further assume that increments in the frequency of events over time do not depend on previous events (AndersenGill model). Only in the case of a constant hazard rate does this model reduce to a gap time model due to the memorylessness of the exponential distribution. When analyzing recurrent failure time data, individuals usually are assumed to be continuously at risk for experiencing events during followup. Different circumstances can prevent individuals from experiencing events within certain intervals within the observational period. These riskfree intervals have to be considered in the statistical model. In both examples riskfree intervals are present.
Episodes of otitis media
Falls in elderly people
According to the World Health Organization (WHO) falls are the second leading cause of accidental or unintentional injury deaths worldwide. Adults older than 65 suffer the greatest number of fatal falls. Recently, the fall rate in elderly Australian adults was evaluated over a three year followup period [20]. The fall rate increased within three years from 0.33 falls per personyear to 0.55 falls per personyear. The difference in fall rates demonstrates the agedependent risk to fall and thus the need of a total time modeling approach. In contrast, in a gap time model, the yearly fall rate would be expected to not differ between both observational periods. If a fall is followed by a period of hospitalization or bed rest, further falls will most probably not occur within these periods. As a consequence, patients are to be considered not at risk for further falls until they have restarted usual living.
Methods
with \(\phantom {\dot {i}\!}F_{t^{}}\) being the history up to time t. The cumulative hazard function is defined by \(\Lambda (t)={\int _{0}^{t}} \lambda (s) \mathrm {d}s\).
Distributional derivations
and \(\tilde {\lambda }^{1}(u)=\lambda (u)\).
For a specific time to recurrent event model, closed form solutions can be found for \(\tilde {\Lambda }_{t}\) derived from λ. Examples will be presented in the section ‘Closed form solutions for common distributions’.
with A∼U[0,1]. For i=1, U_{1}=T_{1}∼Λ^{−1}(− log(A)).
The recursive simulation algorithm

Specify the hazard λ(t) as a function of total time.

Derive \(\tilde {\Lambda }_{t}\) and \(\tilde {\Lambda }_{t}^{1}\) according to (2).

Simulate independent random numbers a_{ i } following a uniform distribution on the interval from 0 to 1.

Apply the recursive algorithm

i=1: t_{1}=Λ^{−1}(− log(a_{1}))

\(i\rightarrow i+1: \;\;t_{i+1} = t_{i} + \tilde {\Lambda }_{t_{i}}^{1}(\log (a_{i+1}))\).


\(i\rightarrow i+1: \;\; t_{i+1} = t_{i}+d + \tilde {\Lambda }_{t_{i}+d}^{1}(\log (a_{i+1}))\)
in step 4 of the algorithm. Here, in analogy to (2), \(\tilde {\Lambda }_{t_{i}+d}^{1}(u)\) is defined as Λ(u+t_{ i }+d)−Λ(t_{ i }+d).
Closed form solutions for common distributions
Simulation in the AndersenGill model
respectively. In combination with the AndersenGill model, we do not use the Log normal parametrization due to its nonproportional behavior of the hazards.
When fitting a misspecified AndersenGill model (3) to data that follow the frailty model model (6) (and thus are subject to unobserved interpatient heterogeneity) regression coefficients are estimated unbiasedly [16]. However, to account for the intrapatient correlation that is caused by the frailty term, standard errors used for calculating statistical tests and confidence intervals have to be replaced by robust versions [1,24] to control level α.
Simulation of T_{ i } for different realizations of (x,z) can therefore be performed by first deriving \(\tilde {\Lambda }_{0,t}(u)\) and \(\tilde {\Lambda }_{0,t}^{1}(u)\), then for each (x,z) calculating \(\tilde {\Lambda }_{t}^{1}(u)\) by (7) and afterwards applying step 3 and step 4 of the recursive simulation algorithm.
This approach will in particular simplify the simulation of data representing a sample of individuals each experiencing recurrent events and each differing in their covariate values. Examples will be presented in the section ‘Results’.
Results
First, we will illustrate the implementation of the derived algorithm on two exemplary simulation studies. Thereafter, we will provide an example, that demonstrates the use of our methods for sample size planning.
Implementation
As already noted by Bender et al. [19] random numbers following a U[0,1] distribution are frequently available in statistical program packages. Therefore, the recursive algorithm can easily be implemented within common software. We implemented the simulation algorithm in the opensource statistical environment R, version 3.1.0 (20140410) [25] and provide the standalone R function as Additional file 1 with a detailed description as Additional file 2. The output dataset is provided in counting process style as required by most software packages when analysing data applying the AndersenGill model. We illustrate this implementation on two simulation studies. In the first study, we evaluate the effect of discontinuous risk intervals on the precision of cumulative hazard estimation. We randomly generate 1000 datasets, each with N=100 observational units experiencing events over a period of two years, that follow a Weibull distribution with scale \(\lambda =4/\sqrt {2}\) and shape ν=0.5 by multiple calls of
A further 1000 datasets are generated with the same underlying distributional assumptions but incorporating the presence of riskfree intervals that follow events and last three months by multiple calls of
In a second simulation study, we evaluate the impact of unadjusted interpatient heterogeneity on the bias and precision of regression parameter estimation when applying the AndersenGill model (3) for data analysis. Interpatient heterogeneity is realized by including a random frailty term when simulating data. Independent datasets are randomly generated, each with N=100 observational units that experience events over time [0,2] following an event generation process according to model (6). We underlie a Weibull baseline hazard with scale \(\lambda =4/\sqrt {2}\) and shape ν=0.5, incorporate a random Bernoulli distributed covariate X ∼B(1,0.5) with regression coefficient β=1 and a Gammadistributed frailty term with mean 1 and variance θ [21]. For each θ∈{0,0.1,0.2,0.3,0.4,0.5} we randomly generate 1000 datasets by multiple calls of
Application: Sample size determination for complex recurrent event data
In the planning phase of a clinical trial the sample size N_{0} has to be determined, that is required to obtain a specific power under a prespecified clinically relevant effect size. Sample size formulas exist and can safely be applied for many trial designs and statistical tests. However, with increasing complexity of trial data, appropriate sample size formulas are often missing. For these situations simulation algorithms provide a useful statistical tool to derive the required sample size N_{0}. Random samples each of size N are simulated and statistically analyzed while N varies iteratively until the sample power approximately coincides with the targeted power.
We will illustrate the use of the recursive simulation algorithm for sample size determination in a clinical trial with complex time to recurrent event data. As a hypothetical illustrating example, we consider a balanced randomized controlled trial aimed at investigating whether a particular intervention will reduce the incidence of falls in elderly people (section ‘Falls in elderly people’). Individuals are to be followed for a period of two years after starting the intervention. As the fall rate is expected to change with total time, the AndersenGill model with a single binary covariate indicating the randomized intervention will be applied, and the null hypothesis of no intervention effect, H_{0}:{β=0}, is to be tested at a twosided significance level of 5%. It is assumed that the twoyearincidence can be decreased from 3.72 in the control group to 2.74 in the intervention group. These assumptions are based on results observed in a controlled trial on vitamin D supplementation [26], where personyears rates of 1.86 and 1.37 have been observed in the control and vitamin D group, respectively. Furthermore, we assume that the risk to fall is expected to increase with time (see section ‘Falls in elderly people’), and therefore we assume the failure time data to follow a Weibulldistribution with shape parameter ν>1. We illustrate sample size derivations for an assumption of ν=2. The scale parameter is derived as 3.72/2^{ ν }=0.93 in the control group and 2.74/2^{ ν }=0.69 in the intervention group to be consistent with the twoyearincidence rates of 3.72 and 2.74. The hazard ratio between intervention and control group is 0.69/0.93=0.74. Furthermore, it has to be considered that some of the individuals will dropout early for example due to death or the onset or deterioration of a disease that prevents further participation in the trial. For illustrative reasons, we assume a high lost to followup rate of 50% and assume this early dropout to follow a uniform distribution on the interval [ 0,2]. If data follow this distributional pattern, known sample size formulas, for example as proposed by Bernardo and Harrington [27], can be applied indicating that a sample size of 160 individuals is required for the detection of a statistically significant intervention effect with a power of 80%.
However, closed and simple to apply sample size formulas often fail if the complexity of the data distribution increases. In the present clinical trial example, it must be further considered that falls can be followed by a period of hospitalization or bed rest. As a consequence, these patients are considered insusceptible to subsequent falls until they have restarted usual living (riskfree intervals). For illustrative reasons we consider two scenarios. In the first one we assume that falls are followed by relatively short riskfree intervals of length d=2 weeks with probability p=0.2. In the second scenario prolonged riskfree intervals of d=8 weeks are expected, which arise with a higher probability of p=0.5 after each event. According to the results of the section ‘Implementation’ (Figure 2), the presence of riskfree intervals affects the precision of parameter estimation and, as a consequence, the power of a trial. We do not know of a sample size formula that takes riskfree intervals into account and therefore apply the simulation algorithm to determine the required sample size. For each setting, 10000 datasets are randomly generated and statistically analyzed using the Wald test statistic within the AndersenGill approach. Simulation results indicate that a few small riskfree intervals only marginally affect the required sample size, whereas the presence of longer and more frequent riskfree intervals increases the sample size requirements to N=184 individuals.
Sample size calculation results
Number of patientsN_{0}  

θ  d=2,p=0.2  d=8,p=0.5 
0  160  184 
0.1  204  226 
0.2  252  274 
0.3  296  320 
0.4  340  366 
0.5  380  422 
Example code for derivation of these results is given in the Appendix along with the standalone R function simrec.
Discussion
We have presented a general method for simulating recurrent event data when the hazard is defined on a total time scale. In particular, this method allows the simulation of recurrent event data following an AndersenGill model by incorporating fixed and random covariates. Our application demonstrates the use of the simulation design for planning a clinical trial, in particular if trial data are expected to be complex. We have also applied the simulation algorithm to evaluate the impact of riskfree intervals and unobserved interpatient heterogeneity on time to event estimates. The proposed simulation algorithm can also be applied to simulate the recurrence process in joint models for recurrent events and a terminal event [28,29]. We have considered models with a total time scale as this will be reasonable in many clinical settings. Gap time models are defined by the distribution of the interevent times, thus simplifying their simulation. Pietzner and Wienke [30] proposed a trendrenewal process to include both a total time scale and a gap time scale and Duchateau et al. [5] applied both time scales to analyze recurrent asthma events in a parametric frailty model. It will be interesting for further research to extend our simulation algorithm to incorporate more than one time scale. It should be noted that there is a long history of methods for simulating inhomogeneous Poisson processes, with timescaletransformations of homogeneous Poisson processes [31] and thinning [32] being two prominent methods. Conversely, we have derived an algorithm to simulate the interevent times. We think that this approach has two advantages that are relevant for medical science: first, riskfree intervals can be incorporated in the simulation design, which will be useful for many applications where individuals are temporarily insusceptible to a recurrent incidence of the event; and second, simulating interevent times allows for simulating hazards that change with the number of previous events. Metcalfe [3] demonstrated the need to randomly generate these kind of data for the evaluation of statistical methods. We provided an R function simrec for simulating recurrent event data under a total time scale in the counting process style. This function fills the gap between existing R packages for simulating gap time models (covered by complex.surv.dat.sim [33]) and for simulating inhomogeneous Poisson processes using time scale transformations (covered by NHPoisson [34]). We aim to extend this function to an R package that will be available from the standard CRAN repository for the R environment.
Conclusions
We have derived an algorithm for simulating data following an AndersenGill model defined on a total time scale. The use of a total time scale provides a better fit to many medical trial data than the commonly applied gap time simulation models. Furthermore, our method allows for a complex data structure by incorporating intrapatient correlation and/or riskfree intervals. Its application therefore allows the simulation of data that closely resemble real settings and thus can improve the use of simulation studies for designing and analysing studies.
Declarations
Acknowledgements
We thank the reviewers for their comments and suggestions that greatly helped to improve the paper. Furthermore, we thank GlaxoSmithKline Biologicals for providing data of the POET trial. We would also like to thank Leesa Wockner for constructive comments improving the publication and Kathy Taylor for proofreading. This research was supported by a grant of the Deutsche Forschungsgemeinschaft (DFG) grant number JA 1821/1.
Authors’ Affiliations
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