Bayesian methods for evidence synthesis in cost-effectiveness analysis

Abstract

Recently, health systems internationally have begun to use cost-effectiveness research as formal inputs into decisions about which interventions and programmes should be funded from collective resources. This process has raised some important methodological questions for this area of research. This paper considers one set of issues related to the synthesis of effectiveness evidence for use in decision-analytic cost-effectiveness (CE) models, namely the need for the synthesis of all sources of available evidence, although these may not ‘fit neatly’ into a CE model.

Commonly encountered problems include the absence of head-to-head trial evidence comparing all options under comparison, the presence of multiple endpoints from trials and different follow-up periods. Full evidence synthesis for CE analysis also needs to consider treatment effects between patient subpopulations and the use of nonrandomised evidence.

Bayesian statistical methods represent a valuable set of analytical tools to utilise indirect evidence and can make a powerful contribution to the decision-analytic approach to CE analysis. This paper provides a worked example and a general overview of these methods with particular emphasis on their use in economic evaluation.

Appendix

1. Connectedness in Mixed Treatment Comparison Evidence Networks

The statistical approach to synthesising mixed treatment comparison evidence requires that the treatment comparisons are connected. For example, in a structure AB, BC, AC, EF, FG, the EFG group of treatments are not connected to the ABC group. The existence of AG trials would connect the two groups. The methods we propose cannot be applied to disconnected groups without further assumptions, and these assumptions would inevitably mean that inferences about all the relative treatment effects between the two groups would be based on nonrandomised evidence.

2. WinBUGS Programmes and Data Listing

We show here the WinBUGS code for the problem, partly because it clarifies some of the results in table II and also because it throws light on the nature of the synthesis. The use of this package requires some basic familiarity with Bayesian theory and MCMC methods, as well as careful attention to certain technical issues such as convergence of the MCMC chains,[17,76] and sensitivity to the way vague prior distributions are specified.[78] The WinBUGS code uses a neater notation, in which the d _AB , d _AC , d _AD notation is replaced by d[2], d[3] and d[4]. This allows us to write the entire core model efficiently, regardless of how many treatments there are. The full data listing and all the programmes are available from http://www.hsrc.ac.uk/Current_research/research_programmes/mpes.htm (figure A1).

Fig. A1

Sample WinBUGS data listing for the mixed treatment comparison dataset (see also section 2.2.2).

A sample data listing, shown below, shows 4 of the 50 randomised trials, indexed i, giving numerators r[i], denominators n[i], treatments t[i] and study numbers s[i]. There are 24 studies. The vector b[i] indicates which treatment is the effective trial-specific ‘baseline’ treatment in that study.

3. WinBUGS Programmes for Mixed Treatment Comparison

The core fixed and random effect programmes are shown here. The latter assumes that the degree of between-trials variation in random effect models is the same for all the pair-wise comparisons. This can be relaxed,[75] although the current data set is clearly much too sparse to obtain meaningful estimates of six different variances (figure A2).

Fig. A2

WinBUGS code for the (a) fixed and (b) random effect mixed treatment comparison models. LOR = log odds ratio.

Technically, a three-arm trial comparing, for example, A, B and C provides estimates of δj _AB and δj _AC which are correlated. Strictly speaking, this correlation should be reflected in the analysis,[39,75,79] although the effect will probably be small unless the proportion of multi-arm studies or their size is large. WinBUGS code that takes account of these correlations is available at the website given above.

4. Absolute Effects, Ranking of Treatments and Pair-wise Odds Ratios

Most CE analysis models require an explicit baseline effect on the absolute scale. Here we derive an estimate based on the average log odds of smoking cessation in the ‘no treatment’ condition A in the 19 studies in which A was involved. Investigators could alternatively take the baseline from a cohort study, or a single trial considered to represent a contemporary relevant estimate. Given an absolute estimate for treatment A we can derive absolute estimates of all other treatments by adding the LORs on the log odds scale and converting back to the natural scale. On each MCMC cycle the four interventions can be ranked, and an indicator variable, best[k], takes the value one when treatment k is best and zero otherwise. The average of the sequence of zeros and ones, given in the posterior summary, estimates the probability that treatment k is best. A further line of code generates posterior distributions for all the 6 pair-wise odds ratios (figure A3).

Fig. A3

Additional WinBUGS code for absolute effects, ranking of treatments and posterior distribution of odds ratios.