Survival Modeling for the Estimation of Transition Probabilities in Model-Based Economic Evaluations in the Absence of Individual Patient Data: A Tutorial Vakaramoko Diaby, Georges Adunlin & Alberto J. Montero
PharmacoEconomics ISSN 1170-7690 PharmacoEconomics DOI 10.1007/s40273-013-0123-9
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Author's personal copy PharmacoEconomics DOI 10.1007/s40273-013-0123-9
PRACTICAL APPLICATION
Survival Modeling for the Estimation of Transition Probabilities in Model-Based Economic Evaluations in the Absence of Individual Patient Data: A Tutorial Vakaramoko Diaby • Georges Adunlin Alberto J. Montero
•
Springer International Publishing Switzerland 2013
Abstract Background Survival modeling techniques are increasingly being used as part of decision modeling for health economic evaluations. As many models are available, it is imperative for interested readers to know about the steps in selecting and using the most suitable ones. The objective of this paper is to propose a tutorial for the
Electronic supplementary material The online version of this article (doi:10.1007/s40273-013-0123-9) contains supplementary material, which is available to authorized users. V. Diaby (&) Programs for Assessment of Technology in Health (PATH) Research Institute, St Joseph’s Healthcare Hamilton, 25 Main St. W., Suite 2000, Hamilton, ON L8P 1H1, Canada e-mail:
[email protected] V. Diaby Department of Clinical Epidemiology and Biostatistics, McMaster University, Hamilton, ON, Canada V. Diaby G. Adunlin Division of Economic, Social and Administrative Pharmacy, College of Pharmacy and Pharmaceutical Sciences, Florida A&M University, Tallahassee, FL, USA e-mail:
[email protected] A. J. Montero Cleveland Clinic, Taussig Cancer Institute, Cleveland, OH, USA e-mail:
[email protected]
application of appropriate survival modeling techniques to estimate transition probabilities, for use in modelbased economic evaluations, in the absence of individual patient data (IPD). An illustration of the use of the tutorial is provided based on the final progression-free survival (PFS) analysis of the BOLERO-2 trial in metastatic breast cancer (mBC). Methods An algorithm was adopted from Guyot and colleagues, and was then run in the statistical package R to reconstruct IPD, based on the final PFS analysis of the BOLERO-2 trial. It should be emphasized that the reconstructed IPD represent an approximation of the original data. Afterwards, we fitted parametric models to the reconstructed IPD in the statistical package Stata. Both statistical and graphical tests were conducted to verify the relative and absolute validity of the findings. Finally, the equations for transition probabilities were derived using the general equation for transition probabilities used in modelbased economic evaluations, and the parameters were estimated from fitted distributions. Results The results of the application of the tutorial suggest that the log-logistic model best fits the reconstructed data from the latest published Kaplan–Meier (KM) curves of the BOLERO-2 trial. Results from the regression analyses were confirmed graphically. An equation for transition probabilities was obtained for each arm of the BOLERO-2 trial. Conclusions In this paper, a tutorial was proposed and used to estimate the transition probabilities for modelbased economic evaluation, based on the results of the final PFS analysis of the BOLERO-2 trial in mBC. The results of our study can serve as a basis for any model (Markov) that needs the parameterization of transition probabilities, and only has summary KM plots available.
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Key Points for Decision Makers
• This is the first application of a step-by-step approach to estimate transition probabilities for model-based economic evaluations based on published Kaplan– Meier (KM) curves. • In the absence of individual patient data (IPD), researchers can reconstruct the IPD from published KM curves, using an algorithm implemented in the statistical package R. • In selecting the best parametric model to fit their data, researchers should use both statistical and graphical tests. • Parametric survival modeling techniques are suitable for developing equations for transition probabilities for use in model-based economic evaluations.
1 Introduction Nowadays, survival modeling is required for economic evaluations that use data from clinical trials as input parameters, especially for treatments or interventions that impact life expectancy and/or quality of life. This is owing to the fact that clinical trials are usually shorter term in duration and may not be adequate for determining the longterm costs and outcomes of competing options [1]. As a result, decision analytic modeling techniques are often used as an approach to implementing economic evaluations [2]. A commonly used decision analytic modeling technique is the Markov model. An important feature of this model is the ‘transition probabilities’. These probabilities represent the likelihood of the occurrence of an event in the future [1]. In economic evaluations, especially model-based, these probabilities can be estimated following the extrapolation of Kaplan–Meier (KM) curves. However, the process of extrapolating data from published clinical trials is not without pitfalls. In fact, the literature can be confusing as it presents several approaches whose applications are contingent upon structural assumptions. There are hardly any documented guidelines on the use of survival modeling techniques for model-based economic evaluations. This was recently confirmed by the review of Latimer [3], who analyzed 45 Health Technology Assessments (HTAs) in oncology. Based on his review, Latimer [3] proposed a framework to apply survival analysis required for economic evaluations. Drawing upon his framework and the literature of model-based economic evaluations in the cancer area, we propose a tutorial that illustrates the application of appropriate survival modeling techniques to estimate transition probabilities in model-based economic evaluations. The illustration is based on the final progression-free survival (PFS) analysis of the BOLERO-2 trial [4].
The paper is outlined as follows. The second section provides a step-by-step guide (tutorial) on the selection of appropriate survival models representing the final PFS analysis of the BOLERO-2 trial [4], as well as the estimation of transition probabilities to be used in a modelbased economic evaluation. The third section deals with the presentation of the results following the application of survival modeling techniques to individual patient data (IPD) obtained from the final PFS analysis of the BOLERO-2 trial [4]. Finally, the fourth section discusses the findings of the study and announces a research agenda.
2 Methods Different approaches can be utilized to estimate transition probabilities based on KM curves from published clinical trials. One approach is to set all transition probabilities to those obtained directly from the KM curves of published trials [5]. The main limitation with this approach is that KM curves tend to overfit the empirical data, which in turn is likely to impact the generalizability of the estimated transition probabilities [5]. An alternative to this approach, which is commonly used, is to fit parametric models to IPD used to create the KM curves. Parametric models, compared to semiparametric and non-parametric models, are more convenient for modeling since equations that translate the model parameters into transition probabilities are well-known [6]. The implementation of this alternative is contingent upon the availability of IPD. However, most trials do not publish IPD corresponding to KM curves [7, 8]. Guyot and colleagues [9] proposed a solution to this problem. These authors developed an algorithm for reconstructing IPD based on published KM curves from clinical trials. They implemented their algorithm in the statistical package R. In our study, this algorithm was used to reconstruct the IPD from the PFS KM curves of the BOLERO-2 trial [4]. After reconstruction of the IPD, parametric distributions were fitted to data and transition probabilities were estimated. It should be emphasized that the reconstructed IPD represent an approximation of the original data. The parametric distribution fitting was done in Stata since the authors were more conversant with the use of this statistical package. However, readers are free to choose other statistical packages to replicate the method based on their ‘hands-on’ experience with the selected packages, while keeping in mind that each package has its own unique style, strengths and weaknesses. 2.1 Reconstructing Individual Patient Data (IPD) Based on Published Kaplan–Meier Curves The reconstruction of IPD from the final PFS KM curves of the BOLERO-2 trial [4] was done in the statistical package
Author's personal copy Survival Modeling for the Estimation of Transition Probabilities
R version 3.0.1, based on the algorithm developed by Guyot and colleagues [9]. The BOLERO-2 trial [10] is an international, doubleblind, phase III trial that compared two treatment arms: exemestane plus placebo, referred to as treatment arm 0, and everolimus plus exemestane, referred to as treatment arm 1. The disease being treated was advanced hormone receptor positive, human epidermal growth factor receptor 2 (HER2) negative metastatic breast cancer (mBC). The primary endpoint was PFS, based on radiographic studies assessed by the local investigators. Central assessment was done by an independent radiology committee to support the analysis. The overall process conducive to the reconstructed data, for each treatment arm, can be summarized in four steps, as shown in Fig. 1. The first step is defined as the creation of the initial input datasets. This consists of, for each treatment arm, extracting the coordinates [survival data (y axis) and corresponding time (x axis)] of the final PFS KM curves of the BOLERO-2trial (see Fig. 2) [4]. The extraction of coordinates can be achieved through the use of computer digitization programs such as ‘Plot digitizer’, ‘Engauge digitizer’ or ‘Digitizeit’. The computer digitization program used for the illustration of the tutorial was ‘Digitizeit’. Readers should bear in mind that the extraction of coordinates does not significantly differ by method. Nonetheless, readers may conduct sensitivity analysis to compare the outputs of the digitization programs. Prior to using the computer digitization program, the figure representing the KM curves should be scanned. The scanned figure is imported in the computer digitization program. The KM curves are digitized either manually or automatically, and the extracted coordinates can then be exported. The second step consists of checking the accuracy of the
Fig. 1 Steps in reconstructing individual patient data based on Kaplan–Meier curves. *Algorithm developed by Guyot and colleagues [8]
extracted coordinates. The analyst should ensure that survival data decrease over time, otherwise the statistical package R will return error codes when implementing the algorithm. It is also important to ensure that the survival data, obtained following the first step, are expressed in proportions rather than in percentages. The third step consists of creating a second dataset containing a series of 6-week intervals composing the Bolero-2 trial follow-up time (a total of 120 weeks followup time), the upper and lower bounds in terms of the number of digitized points corresponding to the interval times, and the number of individuals at risk for each interval. The last step consists of implementing the algorithm in R. The latter finds numerical solutions to the inverted KM equations, based on available information on number of events and numbers at risk [9]. Following the implementation of the algorithm, R will produce the summary of KM estimates and an approximation of the original censoring times (time variable) and failure events (failure variable). 2.2 Fitting Parametric Distributions to Reconstructed Data Parametric distributions can be categorized into two groups: ‘standard’ and ‘flexible’. The standard parametric distributions consist of exponential, Weibull, Gompertz, log-normal, and log-logistic distributions, and the flexible parametric models include the generalized gamma and F distributions [3]. Latimer [3] recommended considering, first, the standard parametric models to fit IPD. In case these models are not suitable, flexible parametric distributions should be used. Therefore, the standard parametric distributions were compared for goodness-of-fit to the reconstructed IPD. An initial step in the selection of the appropriate models to be fitted to survival data consists of graphically assessing the proportional-hazards (PH) assumption [3]. The PH assumption stipulates that the hazard ratio (HR) obtained from the comparison of KM curves is constant over time [11]. Testing the PH assumption allows analysts to assess whether or not researchers can estimate the equation of one of the KM survival curves and then apply the HR obtained from the KM survival analysis as a factor to derive the equation of the second KM curve (comparator). If the PH holds, then researchers can apply the HR as a factor. If the PH does not hold, then researchers will have to estimate separate equations for the KM curves. The graphical assessment of the PH assumption can be done by comparing the log-cumulative hazard plots of the KM curves [3]. If plots are parallel, then the PH holds. Conversely, if plots are not parallel then the PH assumption should be rejected. In that case, consideration should be
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Fig. 2 Final Kaplan–Meier curves of progression-free survival (local assessment) of the Bolero-2 trial (adapted from Piccart et al. [4]). CI confidence interval, EVE everolimus, EXE exemestane, HR hazard ratio, PBO placebo
given to parametric accelerated failure time (AFT) models as these models are not subject to the PH assumption. A quick assessment of the PH assumption, in Stata 12, shows clearly that the PH assumption should be rejected (see Fig. 3). Therefore, we fitted individual parametric AFT models to the reconstructed IPD in Stata 12. These models are the exponential, Weibull, log-normal, and loglogistic models. The general steps of the parametric AFT model fitting are described below. The full Stata commands for parametric AFT model fitting and selection can be accessed in the electronic supplementary material (ESM) Appendix 1. For each treatment arm, the censoring times (time variable) and failure events (failure variable) were imported in
Stata 12. These data were declared as survival-time data using the command stset. Afterwards, we used the Stata command Streg to create different regression models based on the distributions to fit. The regression outputs are presented in AFT metric. Table 1 summarizes the parameters tested for significance for each distribution. The hypothesis test (a = 0.05) conducted on these parameters is presented as follows: H0 H1
The parameters tested are not significantly different from zero; The parameters tested are significantly different from zero.
Only distributions with significant parameters were considered for selection. Information criteria were used to select the distribution that best fits the observed data (goodness-offit). These criteria are known as the Akaike information criterion (AIC) [12] and the Bayesian information criterion
Table 1 Parameters to be estimated and tested for significance
Fig. 3 Graphical proportional hazards assumption test
Distribution
Parameters to be estimated and tested for significance (a = 0.05)
Exponential
k
Weibull
kc
Log-normal
r
Log-logistic
c
k scale of the distribution, c shape of the distribution, r standard deviation of the distribution
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(BIC) [13]. Selecting the distribution that represents the best fit to the data consists of identifying the distribution that exhibits the lowest AIC and BIC values. In Stata 12, the commands estat ic or estimates store can be used to invoke these criteria. The results suggested by the comparison of information criteria were confirmed by the graphical analysis of the Cox–Snell residuals [14] obtained after each regression (i.e. for each fitted model), using the Stata command predict. Indeed, for each fitted distribution, the empirical estimate of the cumulative hazard function was plotted against the Cox–Snell residuals and compared with a diagonal line (45 line). If the hazard function follows the 45 line (slope equal 1) then we would conclude that the tested distribution fits the IPD. As a consequence, the distribution that best fits the IPD would be the one whose cumulative hazard function follows best the diagonal line.
treatment arm 1, the number of events estimated is 310, with an estimated median PFS time of 34.4 weeks (30.2; 37.3). These figures are very close to those reported in the poster showing the final PFS analysis of BOLERO-2 [4] (see Table 2). This confirms the face validity of the obtained results. For each treatment arm, we also obtained the reconstructed censoring times and failure events. These data, used for parametric model fitting, can be accessed online in ESM resources C and D, respectively, for treatment arms 0 and 1.
2.3 Estimating Transition Probabilities for Economic Analysis
3.2.1 Treatment Arm 0: Exemestane Plus Placebo
The last phase of this work consisted of substituting the parameters of the general equation for transition probabilities [1] by the parameters estimated from the selected distribution, following the regression analysis. It is then possible to estimate the transition probabilities for each cycle considered in an economic model (Markov model).
3 Results 3.1 Reconstruction of IPD The input files (extracted coordinates and second dataset) created from step 1 to step 3 can be accessed online (ESM resources A and B, respectively, for treatment arms 0 and 1). After running the algorithm in R, we obtained the IPD outputs for each treatment arm (0 and 1). For treatment arm 0, the number of events estimated is 197.0, with an estimated median PFS time of 14.1 weeks (12.1; 18.1). For
Table 2 Comparison of the results of the final PFS Kaplan–Meier curves of the BOLERO-2 trial to those reconstructed following the use of the algorithm in R Treatment arm 0
Number of events
Original
Reconstructed
200
197
Median PFS timeb (CI)a
3.2
3.29 (2.82–4.22)
Treatment arm 1 a
Original
Reconstructeda
310
310
7.8
8.02 (7.05–8.7)
3.2 Fitting Parametric Models to Reconstructed Data The results regarding parametric models fitting are presented separately for each treatment arm of the Bolero-2 trial under Sects. 3.2.1 and 3.2.2.
Out of the four models fitted, three have significant parameters. These distributions are exponential, Weibull, and log-logistic. Having significant parameters implies that the time-dependent parameters tested are significantly different from zero. Based on the respective AIC and BIC of the competing distributions (see Table 3), the loglogistic distribution seems to be the best fit to the observed data. Looking at the graph of Cox–Snell residuals, we see that, among the tested distributions, the hazard function that follows the 45 line very closely is that of the loglogistic (see Fig. 4). As a result, the distribution that best fits the data is the log-logistic. 3.2.2 Treatment Arm 1: Everolimus Plus Exemestane The results obtained for treatment arm 1 were similar to those of treatment arm 0. Indeed, three of the four models fitted have significant parameters. These models are exponential, Weibull, and log-logistic. The comparison of the AIC and BIC of these distributions (see Table 4) suggests that the log-logistic distribution is the best fit to the observed data. The analysis of the Cox–Snell residuals (see Fig. 5) suggests that the distribution that best fits the observed data is the log-logistic distribution. Indeed, the graph of Cox–Snell residuals shows that, among the tested distributions, the hazard function that follows the diagonal line very closely is that of the log-logistic (see Fig. 5). Otherwise said, the results of the residual analysis confirm those suggested by the AIC and BIC analysis. 3.3 Deriving the Transition Probabilities Formula
PFS progression-free survival, CI confidence interval a
Estimated data
b
Time in months
The general equation for transition probabilities [1] is given by Eq. 1.
Author's personal copy V. Diaby et al. Table 3 Comparison of models in terms of AIC and BIC for treatment arm 0 Model
Obs
ll(null)
ll(model)
df
AIC
BIC
Exponential
239
-323.9798
-323.9798
1
649.9597
653.4362
Weibull
239
-321.0848
-321.0848
2
646.1696
653.1225
Log-logistic
239
-305.3614
2
614.7228
621.6757
AIC Akaike information criterion, BIC Bayesian information criterion, Obs observed, ll(null) log likelihood (null), ll(model) log likelihood (model), df degree of freedom Italic values represent the lowest values respectively for the AIC and the BIC. The model with the lowest AIC and BIC values is the one that represents the best fit to the data
Fig. 4 Analysis of Cox–Snell residuals for fitted distributions for treatment arm 0
tp ðtu Þ ¼ 1 expfHðt uÞ HðtÞg
ð1Þ
where tp indicates the transition probability, tu the cycle for which the transition probability is estimated, u the cycle length and H(t) the cumulative hazard function of the parametric distribution. The form of the cumulative hazard function for the log-logistic distribution is given by Eq. 2. HðtÞ ¼ 1 þ ðktÞðcÞ ; 1
ð2Þ
with k being the scale of the distribution and c being the shape of the distribution. Based on Eq. 2, Eq. 1 can be rearranged as Eq. 3. tp ðtu Þ ¼ 1 expf½kðt uÞð Þ ðktÞð Þ g 1 c
1 c
ð3Þ
It is important to emphasize that, in Stata, the scale (k) of
the log-logistic distribution is parametrized as k ¼ expðxj bÞ, with b being the vector of regression coefficients estimated from the regression analysis. As for the shape of the distribution (c), this is estimated from the regression analysis conducted in Stata when fitting the loglogistic distribution to the data. After replacing the parameters k and c by their values (based on Stata streg outputs), the transition probabilities can be estimated using Eqs. 4 and 5, respectively, for the treatment arms 0 and 1. tp ðtu Þ ¼ 1 expf½0:068025 ðt uÞð0:5583247Þ 1 ð0:068025tÞð0:5583247Þ g
ð4Þ
tp ðtu Þ ¼ 1 expf½0:030142 ðt uÞð0:6187177Þ 1 ð0:030142tÞð0:6187177Þ g
ð5Þ
1
1
Author's personal copy Survival Modeling for the Estimation of Transition Probabilities Table 4 Comparison of models in terms of AIC and BIC for treatment arm 1 Model
Obs
ll(null)
ll(model)
df
AIC
BIC
Exponential
485
-590.5286
-590.5286
1
1,183.057
1,187.241
Weibull
485
-582.4452
-582.4452
2
1,168.89
1,177.259
Log-logistic
485
-576.1198
2
1,156.24
1,164.608
AIC Akaike information criterion, BIC Bayesian information criterion, Obs observed, ll(null) log likelihood (null), ll(model) log likelihood (model), df degree of freedom Italic values represent the lowest values respectively for the AIC and the BIC. The model with the lowest AIC and BIC values is the one that represents the best fit to the data
Fig. 5 Analysis of Cox–Snell residuals for fitted distributions for treatment arm 1
4 Discussion In this paper we have conducted a step-by-step survival analysis for the estimation of transition probabilities in economic evaluation, based on the final PFS KM curves of the BOLERO-2 trial [4]. As IPD were not readily available from the BOLERO-2 trial, we used an algorithm to approximate the original data. Parametric distributions were then fitted to the reconstructed data. Based on the outputs of the regression analyses conducted on these IPD, two log-logistic models were selected as the best-fit models to the data for the treatment arms 0 and 1. Finally, for each treatment arm, the equations for estimating the transition probabilities for an economic model were presented in the Results section of the current paper. These equations made use of the parameters of the log-logistic distributions,
estimated from the observed data. In this study, the proposed tutorial with the findings can serve as a basis for any model (Markov) that needs the parameterization of transition probabilities, and only has summary KM plots available. As uncertainty is inherent in the estimation of parameters (following parametric extrapolation of survival estimates) that are used in any model-based economic evaluation, it is imperative to assess the impact of uncertainty on the base-case results of that evaluation. In this regard, we recommend researchers conduct sensitivity analyses to test the use of the remaining standard parametric models, considered as part of the model selection, to estimate transition probabilities. Doing so will allow researchers to determine the range of variation of the incremental cost-effectiveness ratio estimated following the change of the selected parametric model.
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Elaboration of the Methods section of the paper was mainly done in light of two papers, Guyot and colleagues [9] and Latimer [3]. Guyot and colleagues [9] developed an algorithm to reconstruct IPD. This novel research significantly eases the ability to perform survival analysis in the absence of IPD. In fact, most clinical trials do not publish patient-level data, and pharmaceutical companies (promoters) do not always grant researchers access to their data. Latimer [3] proposed a framework for survival modeling for economic evaluations. His study attempts to fill the gap in the literature because, to the best of our knowledge, there are no detailed method papers that provide guidance on selecting appropriate distributions to fit censored data from clinical trials. We concur with Latimer [3] that different ways of fitting and selecting appropriate distributions for censored data exist. As an example, instead of using parametric AFT models when the PH assumption does not hold, the analyst can explore the use of the Cox PH model with time-dependent covariates [11]. Additionally, there are a number of new techniques for survival analysis that necessitate refinement to be easily implemented as part of model-based economic evaluations. These include flexible parametric models proposed by Royston and Lambert [15] and Bayesian parametric models [16]. It would be worthwhile developing guidelines for survival modeling in order to guarantee consistency across model-based economic evaluations. In this paper, the authors provided insights into the practical application of survival modeling techniques required for model-based economic evaluation, especially when patient-level data are not available. We believe the tutorial proposed and illustrated would appeal to readers and researchers who have interest in pharmacoeconomics. Acknowledgments The author contributions are presented below. Study concept and design: Vakaramoko Diaby, Georges Adunlin, and Alberto J. Montero. Data acquisition: Vakaramoko Diaby, Georges Adunlin. Data analyses and interpretation: Vakaramoko Diaby. Drafting of the article: Vakaramoko Diaby, Georges Adunlin, and Alberto J. Montero drafted the manuscript. Revision for intellectual content: All Authors. Guarantor: Vakaramoko Diaby. The authors are grateful to Dr. Patricia Guyot for her help in the implementation of the algorithm in the statistical package R. The authors would also like to thank Moussa K. Richard, Gordon Blackhouse, Dr. Robert Hopkins, and Askal Ali for their insightful comments on earlier versions of the paper. Conflict of interests Dr. Vakaramoko Diaby, Georges Adunlin, and Dr. Alberto J. Montero certifies that they have no conflicts of interest with any financial organization regarding the material discussed in the manuscript.
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