PRACTICAL APPLICATION

Survival Modeling for the Estimation of Transition Probabilitiesin Model-Based Economic Evaluations in the Absenceof Individual Patient Data: A Tutorial

Vakaramoko Diaby • Georges Adunlin •

Alberto J. Montero

Published online: 13 December 2013

� Springer International Publishing Switzerland 2013

Abstract

Background Survival modeling techniques are increas-

ingly 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

application of appropriate survival modeling techniques

to estimate transition probabilities, for use in model-

based 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 met-

astatic 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 recon-

structed 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 model-

based 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 recon-

structed 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 model-

based 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.

Electronic supplementary material The online version of thisarticle (doi:10.1007/s40273-013-0123-9) contains supplementarymaterial, 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: diaby_v@yahoo.fr

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: george.adunlin@gmail.com

A. J. Montero

Cleveland Clinic, Taussig Cancer Institute, Cleveland, OH, USA

e-mail: MONTERO2@ccf.org

PharmacoEconomics (2014) 32:101–108

DOI 10.1007/s40273-013-0123-9

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 long-

term 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 proba-

bilities can be estimated following the extrapolation of

Kaplan–Meier (KM) curves. However, the process of

extrapolating data from published clinical trials is not with-

out pitfalls. In fact, the literature can be confusing as it pre-

sents several approaches whose applications are contingent

upon structural assumptions. There are hardly any docu-

mented guidelines on the use of survival modeling tech-

niques for model-based economic evaluations. This was

recently confirmed by the review of Latimer [3], who ana-

lyzed 45 Health Technology Assessments (HTAs) in oncol-

ogy. 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 appro-

priate 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 esti-

mation of transition probabilities to be used in a model-

based 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 transi-

tion 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 semi-

parametric 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, para-

metric distributions were fitted to data and transition proba-

bilities 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

102 V. Diaby et al.

R version 3.0.1, based on the algorithm developed by

Guyot and colleagues [9].

The BOLERO-2 trial [10] is an international, double-

blind, 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-2-

trial (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

extracted coordinates. The analyst should ensure that sur-

vival 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 follow-

up 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 implemen-

tation 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 distri-

butions 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

Fig. 1 Steps in reconstructing individual patient data based on

Kaplan–Meier curves. *Algorithm developed by Guyot and col-

leagues [8]

Survival Modeling for the Estimation of Transition Probabilities 103

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 log-

logistic 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 var-

iable) 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 pre-

sented 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 The parameters tested are not significantly different

from zero;

H1 The parameters tested are significantly different from

zero.

Only distributions with significant parameters were con-

sidered for selection. Information criteria were used to select

the distribution that best fits the observed data (goodness-of-

fit). These criteria are known as the Akaike information

criterion (AIC) [12] and the Bayesian information criterion

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

Fig. 3 Graphical proportional hazards assumption test

Table 1 Parameters to be estimated and tested for significance

Distribution Parameters to be

estimated and tested

for significance (a = 0.05)

Exponential k

Weibull k c

Log-normal r

Log-logistic c

k scale of the distribution, c shape of the distribution, r standard

deviation of the distribution

104 V. Diaby et al.

(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 regres-

sion (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 diag-

onal 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.

2.3 Estimating Transition Probabilities for Economic

Analysis

The last phase of this work consisted of substituting the

parameters of the general equation for transition probabil-

ities [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 esti-

mated median PFS time of 14.1 weeks (12.1; 18.1). For

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 treat-

ment arms 0 and 1.

3.2 Fitting Parametric Models to Reconstructed Data

The results regarding parametric models fitting are pre-

sented separately for each treatment arm of the Bolero-2

trial under Sects. 3.2.1 and 3.2.2.

3.2.1 Treatment Arm 0: Exemestane Plus Placebo

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 log-

logistic 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 log-

logistic (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) sug-

gests 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

The general equation for transition probabilities [1] is

given by Eq. 1.

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 Treatment arm 1

Original Reconstructeda Original Reconstructeda

Number of

events

200 197 310 310

Median PFS

timeb (CI)a3.2 3.29

(2.82–4.22)

7.8 8.02

(7.05–8.7)

PFS progression-free survival, CI confidence intervala Estimated datab Time in months

Survival Modeling for the Estimation of Transition Probabilities 105

tpðtuÞ 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 1þ ðktÞ1cð Þ; 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.

tpðtuÞ ¼ 1� expf½kðt � uÞ�1cð Þ � ðktÞ

1cð Þg ð3Þ

It is important to emphasize that, in Stata, the scale (k) of

the log-logistic distribution is parametrized as

k ¼ expð�xjbÞ, 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 log-

logistic 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.

tpðtuÞ ¼ 1� expf½0:068025� ðt � uÞ�1

0:5583247ð Þ

� ð0:068025tÞ1

0:5583247ð Þg ð4Þ

tpðtuÞ ¼ 1� expf½0:030142� ðt � uÞ�1

0:6187177ð Þ

� ð0:030142tÞ1

0:6187177ð Þg ð5Þ

Fig. 4 Analysis of Cox–Snell residuals for fitted distributions for treatment arm 0

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

106 V. Diaby et al.

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 pro-

posed tutorial with the findings can serve as a basis for any

model (Markov) that needs the parameterization of transi-

tion probabilities, and only has summary KM plots avail-

able. 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 uncer-

tainty 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 para-

metric 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.

Fig. 5 Analysis of Cox–Snell residuals for fitted distributions for treatment arm 1

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

Survival Modeling for the Estimation of Transition Probabilities 107

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 signifi-

cantly 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 (pro-

moters) 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 pro-

vide 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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