Bayesian Analysis CEA

Bayesian Analysis CEA with Posterior Net Benefit, Priors and the Probability of Cost-Effectiveness

A Bayesian cost-effectiveness analysis treats every uncertain input, from relative treatment effects to costs and utilities, as a quantity with a probability distribution, so that expected net benefit, the probability of cost-effectiveness and the value of further research are read from one updated distribution. It matters because most outputs of a probabilistic decision model already have a Bayesian reading. Posterior distributions of incremental cost and effect are formed from the model parameters, probabilistic sensitivity analysis works as an approximation to a full Bayesian analysis, evidence synthesis feeds model inputs and the cost-effectiveness acceptability curve is a posterior probability. An illustrative example then shows how the choice of prior can change the decision.

From posterior parameters to posterior net benefit

The starting point is general Bayesian analysis applied to the parameters of an economic evaluation. A prior distribution states what is known before the current evidence, and a likelihood describes how that evidence arises. Bayes' theorem combines the two as follows.

$$p(\theta \mid D) \propto p(D \mid \theta)\, p(\theta)$$

where $\theta$ is the vector of model parameters (event rates, relative effects, costs, utility weights), $D$ is the evidence, $p(\theta)$ is the prior, $p(D \mid \theta)$ is the likelihood and $p(\theta \mid D)$ is the posterior distribution.

Costs and health effects are functions of the parameters, so the posterior for $\theta$ induces a posterior for the incremental cost and effect of a new option against its comparator. On one monetary scale this is the incremental net monetary benefit, defined below.

$$INB(\theta) = \lambda \, \Delta e(\theta) - \Delta c(\theta)$$

where $\lambda$ is the cost-effectiveness threshold per unit of health effect, $\Delta e(\theta)$ is the incremental health effect, for example in QALYs, and $\Delta c(\theta)$ is the incremental cost.

Claxton argued that adoption should rest only on the mean of this distribution, whether or not the difference is statistically significant, and that its spread matters only for deciding whether more evidence is needed. The rule is to adopt the new option when $\mathrm{E}[INB(\theta) \mid D] > 0$, with the expectation taken over the posterior.

Integrated and two-stage routes through a decision model

There are two broad ways of carrying Bayesian uncertainty through a decision model. Spiegelhalter describes a two-stage approach, in which input distributions are produced first and then propagated by Monte Carlo simulation, and an integrated approach, in which priors are updated and the posterior propagated in one analysis, generally by Markov chain Monte Carlo (MCMC).

Probabilistic sensitivity analysis is the familiar two-stage form. Baio and Dawid describe it as clearly in line with Bayesian analysis, except that uncertainty is handled by simulation rather than integrated out. For frequentist evidence syntheses that involve multi-arm trials, loops in the evidence network, baseline models or covariates, the NICE Decision Support Unit notes that a multivariate normal distribution built from the maximum likelihood estimates and their covariance matrix should be approximately the same as a Bayesian posterior. The integrated route avoids forcing posteriors into convenient parametric shapes and carries the dependence between parameters through to the outputs, at the cost of extra complexity and MCMC software.

Whichever route is used, the NICE manual for technology appraisal (PMG36) states that the committee's preferred cost-effectiveness estimate should come from a probabilistic analysis when possible, unless the model is linear.

For trial-based analyses, O'Hagan and Stevens set out a general Bayesian framework linking mean cost and mean efficacy to the parameters of whatever model suits the data, including log-normal and mixture models for costs. This matters because cost data are typically highly skewed, and the same authors later argued that the bootstrap and methods based on asymptotic normality can then give inefficient or even misleading inferences.

Evidence synthesis feeding model inputs

Relative treatment effects in most appraisals come from a synthesis of several trials, and Bayesian methods are a standard way of carrying out that evidence synthesis. The Decision Support Unit's technical support document on pairwise and network meta-analysis calls Bayesian MCMC the long-standing mainstay of comprehensive decision analysis, because posterior simulation supplies estimation, inference and a platform for probabilistic decision making at once.

The same document notes that posterior sampling retains the correlation between parameters estimated jointly from the same data. When the evidence network has closed loops, the posterior samples should be used in the decision model, or the correlations propagated by other means, because distributions built from marginal posterior summaries are not adequate. PMG36 states that NICE prefers the network meta-analysis methods in this evidence synthesis series.

Acceptability curves as posterior probabilities

With the posterior of incremental net benefit in hand, the probability that the new option is cost-effective follows directly. Plotted across thresholds, this probability forms the acceptability curve.

$$Q(\lambda) = \Pr\left( INB(\theta) > 0 \mid D \right) \approx \frac{1}{S} \sum_{s=1}^{S} \mathbf{1}\left[ INB(\theta^{(s)}) > 0 \right]$$

where $Q(\lambda)$ is the probability of cost-effectiveness at threshold $\lambda$, $S$ is the number of draws, $\theta^{(s)}$ is the parameter set in draw $s$, and $\mathbf{1}[\cdot]$ equals 1 when the condition holds and 0 otherwise.

Gabrio, Baio and Manca note that this has a natural interpretation within a Bayesian approach as a posterior probability given the observed data. O'Hagan, Stevens and Montmartin gave a simple Bayesian computation of the curve from trial data. PMG36 asks models to quantify decision uncertainty, the probability that a different decision would be reached if true cost effectiveness were known. It lists acceptability curves among appropriate presentations, asks for the cost-effectiveness acceptability frontier alongside them, and asks for the probability of cost-effectiveness at maximum acceptable ICERs of £25,000 to £35,000 per QALY gained.

Choosing priors and showing their influence

Priors are the most visible difference from conventional analysis, and the worry in public decision making is that they could be chosen to favour a result. The review of Bayesian methods in health technology assessment by Spiegelhalter and colleagues concluded that the prior is important and not unique, so a range of options should be examined in a sensitivity analysis.

Spiegelhalter describes a community of archetypal priors for this. A sceptical prior has a mean of zero effect and a spread that sets how much doubt is expressed about large effects; an enthusiastic prior is centred on the effect the trial was designed to detect, with a low chance, for example 5%, that the true benefit is negative. In trial monitoring, a positive conclusion that survives the sceptical prior should persuade a reasonable doubter.

PMG36 draws a line between parameters. In network meta-analyses with few studies, informative priors for between-study heterogeneity may be preferable, with their source justified and alternative priors tested. Informative priors for relative effectiveness are not recommended except in very specific circumstances, such as very sparse adverse event data. More generally, input distributions should represent the available evidence, and formal elicitation can be used when data are lacking.

Worked example with vague and sceptical priors on the QALY gain

The figures are illustrative. A trial estimates an incremental effect of 0.20 QALYs (standard error 0.06) and an incremental cost of £3,000 (standard error £800) for a new treatment against usual care. For simplicity, cost and effect are treated as independent and approximately normal, and the cost estimate is used without prior information. The sceptical prior is a sensitivity analysis drawn from a community of priors, not a base case, since PMG36 does not recommend informative priors for relative effectiveness as standard.

1. Vague prior. With a prior that is flat relative to the data, the posterior for the QALY gain has mean 0.20 and variance 0.0036. At £25,000 per QALY the expected incremental net benefit is as follows.

$$\mathrm{E}[INB] = 25000 \times 0.20 - 3000 = 2000$$

where $\mathrm{E}[INB]$ is the posterior mean incremental net monetary benefit in pounds.

$$\mathrm{Var}[INB] = 25000^2 \times 0.0036 + 800^2 = 2250000 + 640000 = 2890000$$

where the first term is the variance of the monetary value of the QALY gain and the second is the variance of the incremental cost. The standard deviation is £1,700, so the probability of cost-effectiveness is the standard normal probability below 2,000 / 1,700 = 1.18, which is 0.88.

2. Sceptical prior. A prior centred on no QALY gain, with standard deviation 0.10 (variance 0.01), is combined with the data by the normal conjugate formulae, as follows.

$$\mu_1 = \frac{0.01 \times 0.20 + 0.0036 \times 0}{0.01 + 0.0036} = \frac{0.002}{0.0136} = 0.147$$

where $\mu_1$ is the posterior mean QALY gain, 0.01 is the prior variance, 0.0036 is the variance of the trial estimate and 0 is the prior mean.

$$\tau_1^2 = \frac{0.01 \times 0.0036}{0.01 + 0.0036} = \frac{0.000036}{0.0136} = 0.002647$$

where $\tau_1^2$ is the posterior variance. The trial estimate is pulled about a quarter of the way towards zero.

3. Net benefit under the sceptical prior. Using the unrounded mean of 0.14706 and rounding to the nearest pound, the result is as follows.

$$\mathrm{E}[INB] = 25000 \times 0.14706 - 3000 = 676$$

where the terms are as in step 1.

$$\mathrm{Var}[INB] = 25000^2 \times 0.002647 + 800^2 = 1654375 + 640000 = 2294375$$

where the terms are as in step 1. The standard deviation is about £1,515, so the probability of cost-effectiveness is the standard normal probability below 676 / 1,515 = 0.45, which is 0.67.

4. Across thresholds. Repeating the steps gives the two acceptability curves in the table below.

5. Interpretation. At £25,000 and £30,000 per QALY the new treatment has the higher expected net benefit under either prior, although the sceptical prior cuts the probability of cost-effectiveness at £25,000 from 0.88 to 0.67. At £20,000 the priors lead to different decisions, as expected net benefit is positive under the vague prior and slightly negative under the sceptical one, so usual care would be retained. A threshold of £20,000 per QALY is below the £25,000 to £35,000 range in PMG36, so this flip illustrates how a prior can matter rather than a NICE decision. Reporting only one prior would hide the fact that the decision at that threshold depends on how much doubt about the trial effect is accepted.

Posterior uncertainty and the value of further research

The spread of the posterior drives decisions about further research. The expected value of perfect information compares the expected net benefit of choosing with perfect knowledge of the parameters against that of the best choice under current evidence. Gabrio, Baio and Manca explain that, in a Bayesian setting, the possible futures in this calculation are samples from the posterior, conditional on the model being correct, which is the formal link to value of information analysis. They also point out that an acceptability curve reports how likely a wrong decision is but not the size of its consequences. Baio and Dawid accordingly argue that, where more data can be gathered, probability summaries should give way to value of information analysis.

Where Bayesian CEA ends and neighbouring methods begin

Bayesian cost-effectiveness analysis sits between general Bayesian statistics, which covers any unknown quantity, and routine probabilistic sensitivity analysis, which is approximately Bayesian when its input distributions reflect the evidence. Bootstrapping of patient-level trial data is a common frequentist alternative for trial-based analyses, and it describes sampling variability rather than belief about parameters. A Bayesian credible interval states where a parameter lies given the evidence and prior. Only a posterior distribution licenses a direct probability statement such as "the probability that the treatment is cost-effective is 0.88".

Misreadings and practical limits

The commonest misreading concerns the acceptability curve. Fenwick, O'Brien and Briggs showed that it is not a cumulative distribution function, because it transforms a joint distribution on the cost-effectiveness plane that can straddle quadrants. With more than two options, the option most likely to be cost-effective need not have the highest expected net benefit, which is why the frontier is used for decisions.

A posterior is conditional on the model as well as the data, so structural uncertainty needs separate treatment, for example by scenario analysis, or by model averaging within the probabilistic model as PMG36 allows. Treating correlated inputs as independent, as the worked example does for simplicity, can misstate the spread of net benefit. MCMC output needs convergence checks, and PMG36 asks for enough simulations to minimise Monte Carlo error.

Gabrio, Baio and Manca suggest reluctance to use prior opinions as one reason health technology assessment has been slow to adopt Bayesian methods. Stating each prior, its evidential basis and how results change under sceptical and enthusiastic alternatives lets a committee see which conclusions rest on judgement and which on data.

Sources

• Baio G, Dawid AP. Probabilistic sensitivity analysis in health economics. Research Report No. 292. London: Department of Statistical Science, University College London; 2008. https://discovery.ucl.ac.uk/1472742/1/Baio_rr292.pdf
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• Dias S, Welton NJ, Sutton AJ, Ades AE. NICE DSU Technical Support Document 2: A generalised linear modelling framework for pairwise and network meta-analysis of randomised controlled trials. Sheffield: Decision Support Unit, ScHARR; 2011, last updated September 2016. Sections 1 and 7.1.
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