Estimation of Survival Probabilities for Use in Cost-effectiveness Analyses: A Comparison of a Multi-state Modeling Survival Analysis Approach with Partitioned Survival and Markov Decision-Analytic Modeling

Modeling of clinical-effectiveness in a cost-effectiveness analysis typically involves some form of partitioned survival or Markov decision-analytic modeling. The health states progression-free, progression and death and the transitions between them are frequently of interest. With partitioned survival, progression is not modeled directly as a state; instead, time in that state is derived from the difference in area between the overall survival and the progression-free survival curves. With Markov decision-analytic modeling, a priori assumptions are often made with regard to the transitions rather than using the individual patient data directly to model them. This article compares a multi-state modeling survival regression approach to these two common methods. As a case study, we use a trial comparing rituximab in combination with fludarabine and cyclophosphamide v. fludarabine and cyclophosphamide alone for the first-line treatment of chronic lymphocytic leukemia. We calculated mean Life Years and QALYs that involved extrapolation of survival outcomes in the trial. We adapted an existing multi-state modeling approach to incorporate parametric distributions for transition hazards, to allow extrapolation. The comparison showed that, due to the different assumptions used in the different approaches, a discrepancy in results was evident. The partitioned survival and Markov decision-analytic modeling deemed the treatment cost-effective with ICERs of just over £16,000 and £13,000, respectively. However, the results with the multi-state modeling were less conclusive, with an ICER of just over £29,000. This work has illustrated that it is imperative to check whether assumptions are realistic, as different model choices can influence clinical and cost-effectiveness results.

also used.

The particular assumptions made for each transition by the manufacturer in their
Markov decision-analytic model were described in the "Markov decision-analytic modelling approach adopted by the manufacturer" section on page 7 of the main article. Therefore the assumptions used for each of the transitions in the multi-state modelling were similar and are described below. The assumptions for each transition are expressed in terms of cumulative hazards as this was the basis for the calculation of state occupancy probabilities in the multi-state modelling. When transition hazards were used in the multi-state modelling these were derived from the difference in cumulative hazards between two consecutive time points. When transition probabilities were used these were derived using S(t) = exp(-H(t)), where S is survival, i.e. 1-the transition probability, H is the cumulative hazard and t is time.
The only exception to this was for progression-free  progression which is detailed below.
 progression  death for each treatment arm, an exponential distribution with a hazard rate of 1/24.1791 and corresponding cumulative hazard H(t)=1/24.1791 × t  progression-free  death the cumulative hazard was based on the maximum of the observed rate of death whilst progression-free and an age-specific background mortality rate.
The observed rates of death whilst progression-free were based on monthly probabilities of 0.0012 and 0.0039 for the RFC and FC arms respectively  progression-free  progression the cumulative hazard was based on the Weibull cumulative hazard of progression or death (the compliment of staying in the progression-free state) minus the exponential cumulative hazard for progression-free -> death. Since this cumulative hazard was not based on a known standard distribution it was not possible to convert it to a transition probability in a standard way.
Therefore, for the purpose of using the transition probability as input in the multi-state modelling, a similar approach to that used by the manufacturer in their Markov decision-analytic model was used for the calculation of the transition probability. This involved using the probabilities of staying in the progression-free state and that for the progression-free  death transition.
The probability of staying in the progression-free state was based on the same Weibull fit to the progression-free survival data as used by the manufacturer. Table A1 shows the incremental mean Life Years in each of the relevant health states using each of the approaches. Two methods of calculating the mean Life Years are shown. Firstly, the trapezoidal rule is used to calculate the area under the curve, the approach used in the main paper for the multi-state modelling. Secondly, the probabilities at each time point were summed together, the approach used by the manufacturer in their Markov decision-analytic modelling. The time points (measured in years) were at 1/12 increments equivalent to the monthly cycles used in the  It can be seen in Table A1 that the results for mean Life Years in Progression were very similar. It can be seen that the results for the individual treatments from the multi-state modelling with transition probabilities were closer to that for the Markov decision-analytic modelling than the corresponding results using transition hazards.
However, in terms of incremental results, the transition hazards approach produced means that were nearer the Markov decision-analytic modelling results. This was because, whilst there was more discrepancy between the individual treatment means, the differences were comparable for each treatment resulting in a similar incremental effect. Table A1 shows there was more of a discrepancy in results for mean Life Years Progression-free, although the results were still very similar. Regardless of the method used to calculate the means, the multi-state modelling approach using transition probabilities produced incremental results which were higher than any of the (similar) corresponding results from the other approaches. It can be seen that the transition hazards approach was most comparable to the actual Markov decisionanalytic modelling when the summing of probabilities method was used. When the trapezoidal rule was used, the results using the multi-state modelling with transition hazards and the Markov decision-analytic modelling were similar. However it was the Markov decision-analytic modelling without the half-cycle correction that most represented the actual Markov decision-analytic modelling when the trapezoidal rule was used. This suggests that using the trapezoidal rule for calculating the means (based on probabilities without a half-cycle correction) is equivalent to the summing of probabilities for which there was a half-cycle correction.
For the Markov decision-analytic modelling in the main paper, which uses a halfcycle correction, the means are calculated using the sum of the probabilities. For all other approaches no half-cycle correction is involved and the means are calculated using the trapezoidal rule. Therefore the calculation of means used in the comparison presented in the main paper would appear to be reasonable. 1

Appendix 2 Mean Life Years and QALYs: trial observation period of 0-4 years and extrapolation period of 4-15 years
In Tables A2.1   It can be seen in Table A2.1 that the approaches were reasonably comparable over the observed period of the trial. However in the unobserved extrapolation period