Probabilistic sensitivity analysis
Assess how uncertainty in transition probabilities and state costs affects a cost-effectiveness comparison.
Concepts and reasoning
Treat uncertain inputs as uncertain
A deterministic economic model uses one transition probability, one cost, and one treatment effect for each input. Those numbers are estimates. Probabilistic sensitivity analysis, or PSA, repeatedly samples plausible parameter values from a joint uncertainty distribution and evaluates the model for each draw. The output is a distribution of modeled decision consequences conditional on the evidence and uncertainty model.
Each draw should represent a coherent possible set of parameters. Drawing every probability in a transition row independently can produce a row whose entries do not sum to 1. Similarly, treating two effects estimated from the same data as independent can generate implausible combinations. Correlations are part of the uncertainty specification, not a cosmetic refinement.
Probability of being best and expected benefit answer different questions
At a given willingness-to-pay threshold, each draw favors whichever strategy has higher net monetary benefit. The fraction favoring an intervention produces a point on its cost-effectiveness acceptability curve. The decision based on expected net benefit, however, depends on the sizes of gains and losses across draws, not just how often they are positive.
A strategy might have a small gain in many draws and a large loss in a few. It could then have a high probability of positive incremental net benefit but negative expected incremental net benefit. The curve describes decision uncertainty; it is not by itself the criterion for maximizing expected health value. The DARTH PSA tutorial provides reproducible modeling examples.
Mathematical foundations
Distributions must respect their support
For a row of
For a strictly positive cost with mean
Summarizing the draws
For
Use the same draw index for the cost and effectiveness result from one coherent scenario. Pairing unrelated cost and QALY draws would destroy their relationship. The Monte Carlo standard error of an acceptability estimate is approximately
Worked example
Work through a small uncertainty distribution
Consider four equally weighted hypothetical draws evaluated at 50,000 per QALY:
| Draw | Incremental cost | Incremental QALYs | Incremental net benefit |
|---|---|---|---|
| 1 | 3000 | 0.100 | 2000 |
| 2 | 7000 | 0.100 | −2000 |
| 3 | 2000 | 0.200 | 8000 |
| 4 | 9000 | 0.100 | −4000 |
Two draws have positive net benefit, so the acceptability estimate is 0.500. Mean incremental net benefit is
For a larger illustration, 720 positive draws out of 1000 give acceptability 0.720 and approximate MCSE
On the cost-effectiveness plane, a point with extra cost and extra QALYs must be compared with the threshold line. A point with lower cost and better health favors the intervention without needing a ratio. Correlation among draws changes the shape of the cloud and can materially alter the decision uncertainty.
Interpreting the results
Read the joint distribution of incremental outcomes
Each point on the cost-effectiveness plane represents one draw of uncertain model parameters followed by evaluation of both strategies. Its horizontal coordinate is incremental health and its vertical coordinate is incremental cost. Points in different quadrants represent different trade-offs; a ratio alone hides that information.
At a threshold λ, the acceptability curve reports the fraction of draws with positive incremental net monetary benefit. A probability of 0.70 means the intervention is preferred in 70% of draws from the specified uncertainty model. It is not a 70% probability of benefiting an individual patient.
The decision based on expected net benefit and the probability of being preferred answer different questions. One strategy can have higher expected benefit despite losing in more than half the draws if gains and losses differ in magnitude. Report the distributions and assumptions that generated the curve, not just its height at one threshold.
Assumptions and limitations
Specify uncertainty using evidence
Parameter distributions should describe uncertainty in estimates, not merely variation among patients. Transition probabilities in a row are constrained to sum to one, so independent sampling of each cell is generally incoherent. A Dirichlet distribution supplies dependence within a row, but its concentration still needs justification.
Shared evidence may induce correlation across strategies, outcomes, or transition rows. Ignoring that dependence can either overstate or understate incremental uncertainty. Structural zeros require particular care: a transition assigned probability zero may be treated as impossible rather than simply unobserved in a small study.
Gamma costs require nonnegative support and an appropriate mean/variance relationship. Fixed utilities omit utility uncertainty. Structural choices such as the health-state definition and extrapolation model are not automatically represented by distributions around numerical inputs. Their consequences should be assessed through explicit alternative scenarios.
Diagnostics and sensitivity analysis
Separate Monte Carlo noise from model uncertainty
Increase the number of parameter draws and examine whether acceptability probabilities and key distribution summaries stabilize. For a probability estimated from independent draws, its Monte Carlo standard error is approximately the square root of p(1−p) divided by the number of draws. More draws improve numerical precision without narrowing the underlying parameter uncertainty.
Inspect sampled values and their implied outcomes. Unexpected extreme costs, impossible transitions, or implausible survival tails may reveal inappropriate distributions or units. Check whether zero-probability transitions remain fixed and whether shared inputs should have been correlated.
Repeat analyses with defensible distributional and structural alternatives. A very stable curve under one poorly supported uncertainty model is not robust evidence. Record how parameters, distributions, dependence, and thresholds were selected so the probability statement has an interpretable meaning.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- DARTH: cohort modeling and probabilistic sensitivity analysis — A worked tutorial that also covers time-dependent models beyond this app’s current scope.