Bayesian Multiparameter Evidence Synthesis to Inform Decision Making: A Case Study in Metastatic Hormone-Refractory Prostate Cancer

In health technology assessment, decisions are based on complex cost-effectiveness models that require numerous input parameters. When not all relevant estimates are available, the model may have to be simplified. Multiparameter evidence synthesis combines data from diverse sources of evidence, which results in obtaining estimates required in clinical decision making that otherwise may not be available. We demonstrate how bivariate meta-analysis can be used to predict an unreported estimate of a treatment effect enabling implementation of a multistate Markov model, which otherwise needs to be simplified. To illustrate this, we used an example of cost-effectiveness analysis for docetaxel in combination with prednisolone in metastatic hormone-refractory prostate cancer. Bivariate meta-analysis was used to model jointly available data on treatment effects on overall survival and progression-free survival (PFS) to predict the unreported effect on PFS in a study evaluating docetaxel with prednisolone. The predicted treatment effect on PFS enabled implementation of a 3-state Markov model comprising stable disease, progressive disease, and dead states, while lack of the estimate restricted the model to a 2-state model (with alive and dead states). The 2-state and 3-state models were compared by calculating the incremental cost-effectiveness ratio (which was much lower in the 3-state model: £22,148 per quality-adjusted life year gained compared to £30,026 obtained from the 2-state model) and the expected value of perfect information (which increased with the 3-state model). The 3-state model has the advantage of distinguishing surviving patients who progressed from those who did not progress. Hence, the use of advanced meta-analytic techniques allowed obtaining relevant parameter estimates to populate a model describing disease pathway in more detail while helping to prevent valuable clinical data from being discarded.


Appendix A: Summary of synthesis of evidence conducted by Collins et al
As part of the Health Technology Assessment (HTA) (Collins et al., 2007) evaluating the clinical-and cost-effectiveness of docetaxel in combination with either prednisone or prednisolone (D+P) for the treatment of metastatic hormone-refractory prostate cancer (mHRPC), a scoping search for studies evaluating the clinical-and cost-effectiveness of D+P was conducted. As only one RCT was identified to have compared D+P with mitoxantrone plus prednisone (M+P) and no other RCT compared D+P with any other possible interventions, RCTs that assessed mitoxantrone in combination with a corticosteroid compared with any chemotherapy regimen or best supportive care or placebo were also included in the scoping search. Extension of the studies selection to include studies that evaluated mitoxantrone in combination with a corticosteroid was to allow for the comparison between D+P and other relevant interventions using mitoxantrone in combination with a corticosteroid as a common comparator in indirect comparison analysis. In total, seven RCTs were identified based on the inclusion criteria, of which three RCTs used docetaxel compared with M+P, three RCTs used mitoxantrone plus a corticosteroid (M+P/H) compared with a corticosteroid (P/H) and one RCT used M+P compared with mitoxantrone plus prednisone plus clodronate (M+P+Clo). The three RCTs that included docetaxel had docetaxel in the following combination: D+P, docetaxel with estramustine (D+E) and docetaxel with estramustine and prednisone (D+E+P). All studies with details of interventions and reported outcomes are presented in Table A1. †unlicensed interventions used by Collins et al. in sensitivity analysis (we use SWOG data to obtain some of the transition probabilities) .

A.1 Definitions of outcome measures
Definitions of OS were consistent for the four RCTs in HTA Set except for trial Berry, where OS was not explicitly defined (Berry et al., 2002). Overall survival was defined as the time from the date of randomisation to the date of death or censored at the date when the patient was last known to be alive. There were inconsistencies in the definition of progression across the three RCTs in HTA Set (excluding trial TAX327 which did not report PFS). Specifically, PFS reported for CALGB 918210 was defined as the time from the date of randomisation to the date of progression or death, whichever occurred first. Neither PFS nor time to progression (TTP) was reported for trial CCI-NOV2212; however, the HTA report presented TTP estimates for this trial. TTP was reported by Berry and colleagues (Berry et al., 2002) but no explicit definition for TTP was provided.

Appendix B: Cost-effectiveness analysis methods
For the cost-effectiveness assessment of D+P in the HTA report, two separate analyses were performed. This is due to the unlicensed status of some of the treatment regimens. For purpose of conducting the meta-analysis, Collins et al grouped both corticosteroids P and H together and we denoted them as P for presentation in the main manuscript and this supplement. The first analysis looked at three interventions (that are licensed at the time of the HTA report submission), namely D+P, M+P and P. The second analysis looked at eight interventions, including the three in the first analysis and the following five: D1+P, D+E, D70+E+P, D35+E+P, M+P+Clo. Due to the unlicensed status of the interventions in the second analysis, the economic decision making in the HTA report was based on the results of the first analysis. In the research that follows in this paper, the focus is on the costeffectiveness assessment of the interventions compared in the first analysis. Data from SWOG trial were used to inform the transition probabilities in the cost-effectiveness analysis.
To develop the economic models, data for the construction of transition probabilities, definition of costs and utilities for each of the interventions need to be extracted. These data can be extracted from reviews, single RCT or evidence synthesis from a number of trials/studies.
Specifications of the transition probabilities, cost and utilities are described in sections B.1, B.2 and B.3 respectively.

B.1 Transition probabilities
For the WinBUGS two-state model, the transition probabilities were estimated using the Weibull parameters reported in the HTA report, which used IPD from trial TAX 327. This was for consistency with the parameters used in the HTA model.
where gives the scale of the distribution and defines the shape. The hazard function for this distribution is therefore: with a cumulative hazard function of: where the survival function is related to the cumulative hazard function in the following form: Since hazards are instantaneous, these need to be converted to a transition probability for a given period, such as a Markov cycle. Using the survival function, transition probability between time-points ( − ) and , denoted as ( ) where is the cycle length, was defined as one minus the ratio of the survival function at the end of the interval to the survival function at the beginning of the interval. This function defined as: was re-written using the cumulative hazed function as: Therefore, transition probability was defined using the Weibull parameters as follows: In the HTA report, results of the Weibull survival analysis model were presented in the form of the regression coefficients of the intercept and scale parameters. These two parameters are expressed in terms of the Weibull parameters,  and , as follows: where is the intercept and is the scale regression coefficient parameters from the Weibull survival analysis.
When performing the probabilistic analysis, the covariance between the intercept and scale regression parameter from the Weibull survival analysis were also incorporated in the WinBUGS two-state model. This was achieved by using the Cholesky decomposition matrix where + and + are the intercept and scale regression coefficients for M+P presented in the HTA report; and ,D, + ~ (0, 1) and ,D, + ~ (0, 1).
Transition probabilities of interventions D+P and M+P for the WinBUGS two-state model were calculated using the regression coefficients from the HTA report (Table 28 ( Collins et al., 2007)). Transition probabilities for P were calculated by applying the HR of P versus M+P or HR of P versus D+P to the hazard rates of M+P and D+P in the transition probabilities respectively. Therefore, assuming that the transition probability for M+P is given by: and with a HR for P versus M+P, denoted as + ⁄ , the transition probability for P is given by: Uncertainty associated with the HR was incorporated in the model by assigning a normal distribution to the logarithm of the HR as follows: where ̅ and 2 are the mean and variance estimate of the log HR (LHR) from random-effects meta-analysis.
For the three-state model, the set of transition probabilities for intervention M+P was calculated using regression coefficients of the parameters of a Weibull survival model for PFS using re-constructed IPD from one of the RCTs in HTA Full Set selected based on the criteria outlined above. As no PFS patient-level data were available for the interventions D+P and P, transition probabilities for each of the interventions were calculated by applying their HR with respect to M+P to the transition probabilities of M+P. Similarly, uncertainty associated with each of the HRs was included in the respective models by assigning normal distribution to the LHRs.

B.1.2. Transition probabilities from PD state to dead state (three-state model)
Although IPD were reconstructed for PFS (together with OS) for the trial selected for estimating the transition probability from StD state to PD state, the reconstructed IPD for PFS and OS were not paired by patient. Hence, it would not be possible to estimate the transition probabilities from PD state to dead state using parametric survival analysis performed using reconstructed IPD as described in the previous section. To overcome this issue, transition probabilities were estimated by assuming the mean total survival time was equal to the weighted sum of combined survival time from stable disease to progression and then to death and the survival time when death occurred from other causes: where ( ) defines the mean time that patients stayed in the StD state before transition to the PD state; ( ) and ( ) define the mean time for PD state to dead state and StD state to dead state respectively; W defines the weight assigned to the mean time and is related to the number of patients who transition through the two potential pathways in the economic model as shown in Figure 2 (bottom), from stable disease state to dead state either with or without disease progression.
As the proportion of patients who died of causes unrelated to prostate cancer was expected to be small (<1%), we assumed that → 0, therefore → 1. Hence, and therefore, Assuming that the survival rates for patients from PD to death follow an exponential survival distribution, the transition probability between time-points ( − ) and , denoted as ( ) where is the cycle length, is defined as follows: where = 1 for the exponential survival model.
As the hazard rate, For M+P and D+P, the ( ) for each of the interventions were estimated using the mean survival time calculated from the reconstructed OS IPD of trial TAX 327. For P, the mean survival time was estimated by a random-effect meta-analysis of the log hazard rate of the three RCTs that had a P treatment regimen arm.
As PFS endpoint was not recorded for trial TAX 327, the mean number of cycles of drug reported in the HTA report was used to represent the mean time from stable disease to progression, ( ), based on the assumption that patients stopped drug treatment on the onset of disease progression. Mean number of cycles of drug P was not reported in the HTA report. Therefore, the mean time to progression was also estimated using meta-analysis of the log hazard rate of the two RCTs that reported PFS data for P.
Transition probabilities for transition from PD state to dead state for each intervention were therefore calculated using the equation: Uncertainty associated with the mean survival time or log hazard rate were also incorporated using normal distributions and propagated in the economic model. As the exponential survival model is a single parameter model, Cholesky decomposition was not required for defining the uncertainty.

B.2 Cost
Cost data comprises drug acquisition and administration cost for each interventions, cost of the management of adverse side effects and subsequent follow up cost that included cost of further chemotherapy after disease progression, management of side-effects and palliative cost. Cost for each of the interventions to be used in the WinBUGS two-state and three-state model were extracted from cost data presented in the HTA report. In the report, costs were categorised into three components: namely, (i) the drug cost, (ii) the follow up cost and (iii)  Costs of subsequent chemotherapy and hospitalisations accounted for between 70% and 80% of follow-up costs which most likely occurred post-progression and the remaining follow-up cost (20% to 30%) were related to side effects likely to occur prior to progression (but may also be associated with the subsequent chemotherapy post-progression). Therefore the follow up costs were divided into portions corresponding to StD state and PD state.
As the base case analysis for the three-state model, 75% ( = 0.75) of the follow-up costs were assigned to the PD state to account for the cost of subsequent chemotherapy, managing side-effects and hospitalisations post-progression. Computation of these costs were based on the number of patients who died per cycle while the remaining 25% of the costs that were assigned to the StD state were computed based on the number of patients who progressed per cycle. Follow-up costs were assigned as one-off cost in a similar way as the WinBUGS two-state model.
In the WinBUGS two-state model, follow-up costs were calculated based on patients died per cycle as: where , represents follow-up cost data for cycle i, and , represents the number of patients who died in cycle i.
For the three-state model, the follow-up costs were calculated as:

= +
where: represents the proportion of follow-up costs associated with the PD state (termed "division factor") and , represents the number of patients who progressed in cycle i.
An annual discount rate of 3.5% was used for discounting the cost after the first year.

B.3 Utility
EQ-5D values used in the cost-effectiveness analysis were obtained from Sandblom et al., who reported the EQ-5D values stratified according to whether the patients subsequently died due to cancer, due to other causes or were still alive, are listed in Table B1. These EQ-5D values, weighted by the transition probabilities amount to the utility for the StD state: where is the probability of remaining in StD state, the probability of transition from StD state to PD state and the probability of transition from StD state to dead state without progression. A utility value of zero was assigned to the dead state ( = 0).
The value of the utility for the StD state was calculated using the utility for progressed patients (as stated above for the two-state model) and two additional EQ-5D values extracted from the Sandblom study (Sandblom et al., 2004), they were the EQ-5D for patients who were still surviving at the time of analysis of the study, EQ-5D = 0.770 (95% CI: 0.755 to 0.785), and EQ-5D of patients who died of other non-prostate cancer-related death, EQ-5D = 0.564 (95% CI: 0.497 to 0.631). These two utility data were defined in the economic model using the following beta distributions, Beta(581.3, 173.6) and Beta(29.1, 22.5) respectively to derive the utility for the StD state. Eq5D score (± 95% CI) 0.538 ± 0.077 0.564 ± 0.067 0.770 ± 0.015 As the utilities defined were selected to reflect the general HRQoL of patients with advanced prostate cancer and were independent of the interventions administrated by the patients, the utilities were used in the model for all three interventions.

B.4 Cost effectiveness
Cost respectively.

C.1 Re-constructed IPD summary statistics
Tables C1 and C2 show HRs for OS and PFS respectively, for all seven studies listed in Table A1 for completeness.

C.2 Justification for choice of Trial SWOG
Overall survival curves for four RCTs in HTA Set 1 and Set 2 (excluding CALGB 9182 which used hydrocortisone instead of prednisone and CCI-NOV22 which did not report PFS) were compared to the OS curve of trial TAX 327 and are presented in Figure C1. The OS Kaplan-Meier curves suggested that trial SWOG has an OS profile closest to trial TAX 327.

C.3 Cost and utility
Costs of drug for interventions M+P and D+P were estimated using the mean cycle of chemotherapy administrated. The reported mean number of cycles for the interventions M+P and D+P were 5.9 (SE=0.17) and 7.3 (SE=0.18) respectively, Normal distributions were assigned to the number of treatment cycles to incorporate uncertainty around these values in the two economic models. Cost of drug for M+P was £347.73/cycle and £1253.92/cycle for D+P, including £177.46/cycle for outpatient attendance fees for both drug regimens. Cost of drug for P was calculated at £1.48 per patient per cycle. In the WinBUGS two-state model, the drug costs for a patient taking P was calculated for the number of cycles that the patient remains in the StD state before transition to the dead state. In the three-state model, the cost drug for a patient taking P was calculated for the number of cycles that the patient remains in the StD state before transition to the PD state. It was assumed that the patient would stop taking P after progression and hence, no drug cost would be calculated for the cycles postprogression before transition to the dead state.
Follow-up and terminal care costs for interventions M+P and D+P were estimated from trial TAX 327 as reported in the HTA report (Table 36 and Table 37 in (Collins et al., 2007)).
Uncertainties for the costs were applied in the two economic models using Gamma distributions. Follow-up and terminal care costs for drug P were not available and were estimated from the costs of intervention M+P from trial TAX 327. In order to estimate the costs for drug P, a cost ratio of drug P with reference to drug M+P was estimated using costing data of P and M+P from a review article (Bloomfield et al., 1998). The mean cost ratio estimated in the WinBUGS two-state model was 1.278 (95% CrI: 0.946 to 1.691) which suggested that the mean cost of P was higher than the mean cost of M+P. This cost ratio was calculated by assigning Gamma distributions [Gamma(α,β)] of Gamma(105,276) and Gamma(81,285) to the cost data of P and M+P respectively. The mean cost (drug, follow-up and terminal care) per patient at each state in the economic model for each of the interventions are presented in Table C4.
Mean EQ-5D HRQoL for all patients in the 12 months prior to death was 0.538 (95% CI: 0.461 to 0.615) as reported in the study by Sandblom and colleagues (Sandblom et al., 2004). Using this EQ-5D data, the utility for StD state in the WinBUGS two-state model was defined using a beta distribution with parameter values: Beta(21.1, 18.1). For the three-state model, the utility distribution, Beta(21.1, 18.1), for the StD in the WinBUGS two-state model was assigned as the utility distribution for the PD state. Two additional EQ-5D values as discussed in Section 2.3.3 of the main manuscript were extracted from the Sandblom study (Sandblom et al., 2004), they were the EQ-5D for patients who were still surviving at the time of analysis of the study, EQ-5D = 0.770 (95% CI: 0.755 to 0.785), and EQ-5D of patients who died of other nonprostate cancer-related death, EQ-5D = 0.564 (95% CI: 0.497 to 0.631). These two utility data were defined in the economic model using the following beta distributions, Beta(581.3,173.6) and Beta(29.1, 22.5) respectively. Utility for the StD state was calculated based on the method described in Section 2.3.3 of the main manuscript.
As the utilities defined were selected to reflect the general HRQoL of patients with advanced prostate cancer and were independent of the interventions administrated by the patients, the utilities were used in the model for all three interventions. Mean QALY per patient for each of the interventions in the economic models are presented in Table C4.  Table C5 shows the results of the case study where the treatment effect on PFS (hazard ratio and the corresponding 95% CrI) was predicted from the treatment effect on OS, conditional on the data for the treatment effects on both outcomes in the three studies in the HTA set (Table A1) reporting both outcomes. The predicted effects on PFS are obtained for TAX 327, used in the main manuscript and for SWOG trial. The SWOG trial was not part of the main analysis as it included an unlicensed intervention. However, as discussed in section C2, the trial was similar to TAX 327 and it reported the treatment effect on PFS. Therefore it was possible to compare the predicted effect on PFS for the SWOG trial (obtained from the BRMA without using the available PFS information) to the actual trial result (generated according to available PFS information). We conducted further investigation by including all trials from both sets: HTA Set1 and HTA Set2 (see Table A1) which reported treatment effects on both PFS and OS. This resulted in a new data set of five trials: CALGB 9182, CCI-NOV 22, Berry, SWOG and Ernst. We carried out a cross-validation procedure, taking in one study at a time the treatment effect on PFS (assuming not reported) and predicting this effect using the bivariate meta-analysis of the remaining data on the two outcomes. Results are included in Table C6. The predicted values in all studies were similar apart from study CCI-NOV 22. Investigating the results in Table C6 made it apparent that study CCI-NOV 22 was an outlier (the smallest HR for PFS) generating bias when making the predictions.  Table   C7.

C.5.1 Sensitivity analysis when using alternative predicted values for HR for PFS
We carried out three sets of sensitivity analysis using three sets of data when predicting HR for PFS in TAX 327. First we increased the evidence base containing the three HTA studies (denoted as "3 HTA" in Table C8) reporting both outcomes and the TAX 327 trial by adding SWOG trial to the data set. This was done to reduce the impact of the outlier (see section C.4). We then increased evidence base further by including Ernst study (SWOG and Ernst studies were the two studies in HTA Set2 reporting both outcomes). Both analyses resulted in reduced treatment effect difference on PFS (predicted HR closer to 1.00) and higher ICERs of £26,483 and £28,349 respectively. We then carried out the sensitivity analysis using reduced evidence base by removing the outlier from the data set in the main analysis (as in Table C7, denoted in Table C8 as "2 HTA"). This led to a higher ICER of £29,601. All results are shown in Table C8 with the original results (on the left-hand-side) listed for comparison. Differences were calculated for (D+P) versus (M+P)

C.5.2 Sensitivity analysis for 2-state model with alternative approach to utility and cost
We have carried out a sensitivity analysis to illustrate how the results of the economic model will change when the 2-state model is implemented, but the utility and cost is calculated to account for the differences between those patients who progressed and those who did not progress. To obtain the utility for the Alive state in the two state model, we use the values from Sandblom in Table B2 corresponding to patient who died due to prostate cancer and those who remained alive at the end of the Sandblom study averaged with weights corresponding to the transition probabilities for the transition out of the Alive state: similarly as in the three state model. The utility of patient who died of other causes is not included here separately, because in the two-state model transition probability to dead state corresponds to the all-cause mortality and ℎ had almost the same value as .
Therefore includes both set of patients.
In a similar manner we partition the cost of each treatment assigned to the StD state. Recalling from section B.2, between 70% and 80% of the costs of follow-up was assumed to be related to subsequent chemotherapy and hospitalisations (likely occurring post-progression) and the remaining follow-up cost (20% to 30%) were related to side effects (likely occurring prior to progression