Bayesian Statistics in Health Economics

Transcription

1 Bayesian Statistics in Health Economics Tony O Hagan University of Sheffield, UK

2 What is Health Economics? Concerned with the question of how to use healthcare resources most efficiently Resources are limited We want to maximise health improvement We therefore consider the cost-effectiveness of competing healthcare technologies SISMEC

3 Cost and effectiveness Cost includes all resource use Cost of drug/treatment under consideration Costs of other medications and procedures General practitioner visits, etc. Effectiveness in terms of increased health Ideally measured by QALYs (Quality Adjusted Life Years) SISMEC

4 Cost-effectiveness Let C be the increment in mean per-patient cost Relative to a comparator or standard care Let E be the corresponding increment in mean perpatient effectiveness Let λ be the amount that the healthcare provider is willing to pay for one unit gain in effectiveness Then the incremental net benefit is INB = λ E C This treatment is cost-effective (relative to comparator) if INB > 0 SISMEC

5 Uncertainty Of course, there is uncertainty over the values of C and E Hence there is uncertainty over whether this treatment is cost-effective Uncertainty is the statistician s domain! There are many interesting problems and challenges for the statistician working in health economics SISMEC

6 Inferring cost-effectiveness E(INB) Choose this treatment if expected INB is positive Otherwise choose comparator P(INB > 0) Gives some measure of uncertainty in the choice May indicate need to get more evidence If choice of this treatment is not reversible, decisionmaker may demand a sufficiently high probability of cost-effectiveness SISMEC

7 Relevance of Bayesian methods C and E are population parameters Therefore INB is also an unknown parameter We can only contemplate E(INB) or P(INB > 0) in a Bayesian framework There are frequentist analogues, but they do not have the required interpretations SISMEC

8 Varying λ The healthcare provider s λ may not be given Consider relevant inferences over a range of λ values Expected INB is positive if λ > E( C)/E( E) Provided E( E) > 0, otherwise reverse the inequality Plot P(INB > 0) as a function of λ This is called a CEAC (cost-effectiveness acceptability curve) SISMEC

9 Example CEAC 0.96 Q(K) K ( / year) SISMEC

10 Cost-effectiveness analysis alongside a clinical trial The simplest statistical analysis arises when we measure both cost and effectiveness at patient level in a clinical trial Then we get sample values of incremental cost and incremental effectiveness We can estimate population mean values C and E But: 1. How to convert that into inference about cost-effectiveness? 2. In practice, a trial never gives measures of cost and effectiveness that we need SISMEC

11 The challenge of costs Analysis of cost-effectiveness trial data has all the technical challenges that we meet in regular clinical trial analysis plus costs 1. Costs are always skewed A few patients have very large costs We might transform (e.g. take logs) but we are interested in mean cost (not mean log-cost) 2. Cost and effectiveness are always correlated SISMEC

12 Example CEAC depends strongly on assumptions about cost distributions O Hagan & Stevens, Health Economics 12, Q CEACs for three different prior structures Exch Nonpar Weak K SISMEC

13 Trial design Design requires prior information Bayesian formulation is more natural Frequentist methods are based on power Probability of significant outcome conditional on assumed effect Bayesian approach is based on assurance Unconditional probability of desired outcome O Hagan & Stevens, Stat Meth Med Res 11, , 2002 O Hagan, Stevens & Campbell, Pharm Stat 4, , 2005 SISMEC

14 A trial is never enough The healthcare provider wants to know about the costs and effectiveness in normal use In practice, the conditions of a trial never represent normal use Restricted entry Short follow-up Short-term or surrogate outcomes Controlled conditions Wrong comparators Etc In particular, effectiveness efficacy SISMEC

15 Economic modelling Health economists actually evaluate costeffectiveness using models An economic model Represents explicitly the disease and treatment processes Allows the synthesis of evidence from a variety of sources SISMEC

16 Vaccination example Cost Efficacy p 1 Disease V+mD Q-q 1 Vaccine 1-p 1 No Disease V Q Subject No Vaccine p 2 1-p 2 Disease No Disease md 0 Q-q 1 Q SISMEC

17 Vaccination model sources Parameter Meaning Source p1 Infection rate, vaccinated Trial p2 Infection rate, unvaccinated Reported cases V Vaccine cost Known m Mean GP visits if infected Registry D GP visit cost Standard unit cost Q General health utility Equals 1? q1 Loss of utility for infection Valuation study SISMEC

18 Parameter uncertainty Typically all of the inputs to an economic model are uncertain In addition to sourcing estimates of all the inputs, we need to describe uncertainty Formally, we need a joint probability distribution for all inputs We then need to determine the induced uncertainty in model outputs (cost-effectiveness measures) This is known as PSA (probabilistic sensitivity analysis) SISMEC

19 The challenge of data gaps Consider the uncertainty in a single parameter If the estimate comes from analysing data, the appropriate solution is the posterior distribution But there are always gaps Parameters for which there is no data source Data relate to something slightly different» Efficacy, not effectiveness» Observational data which may have bias Solution: elicit expert judgements And/or many possible sources Solution: evidence synthesis (e.g. meta-analysis) SISMEC

20 The challenge of PSA Usual approach is Monte Carlo Generate samples of input values Run model for each set of inputs Sample of outputs gives output uncertainty But may be impractical If model is complex and takes too long to run Stevenson, Oakley& Chilcott, Med Decis Making, 24, , 2004 O Hagan, Stevenson & Madan, manuscript on Monte Carlo PSA, 2005 ( SISMEC

21 Valuing health How do we derive QALYs? First we need a system to describe health state Usually multi-dimensional Next we need a measure of health-related quality of life (HRQoL) for each state Equals 1 for perfect health Equals 0 if dead May be less than 0 for some health states Then QALYs are calculated using HRQoL times length of time in each state SISMEC

22 The challenge of HRQoL A sample of subjects are asked to give values for different health states By asking them to trade off bets or time in competing states HRQoL then derived by statistical analysis of these data but several statistical challenges Not all possible states are valued in the sample Skewness and heteroscedasticity Respondent effects (repeated measures) Monotonicity constraints SISMEC

23 Example Analysis of SF-6D system 6 dimensions states Bayesian model fits better than standard regression analysis Kharroubi, O Hagan & Brazier, Appl Stat (in press), 2005 SISMEC

24 Conclusions There are many challenging statistical problems in health economics Costs Trial design Quantifying uncertainty (and bridging data gaps) Efficient computation of uncertainties Valuing health Bayesian methods are proving to be essential in tackling many of these SISMEC

25 Conclusions (continued) Health economics is growing in importance Constrained healthcare budgets Ever-increasing cost of new treatments Statisticians are in a unique position to contribute to this young field and to influence its development More information: SISMEC