Expected Value Analysis

Expected Value Analysis in Health Care Decisions

Most decision models in health economics end in the same step. Each strategy is summarised by its expected cost and its expected health outcome, and the choice is made on those averages. Expected value analysis turns uncertain pathways into expected costs and quality-adjusted life years (QALYs). Public decision makers are advised to choose on expected values rather than on statistical significance or the chance of a bad outcome, although that advice has limits, and a probabilistic analysis can reverse the result of a deterministic run when the model is non-linear.

The mechanics of trees are covered on the decision tree, chance node and folding back pages, and the mathematical expectation itself on the expected value page. Expected value analysis is the decision principle that sits on top of them, concerned with what it means to choose by expected value, what it assumes about attitudes to risk, and how it differs from choosing by expected utility.

From pathways to expected costs and QALYs

A decision model describes each strategy as a set of mutually exclusive pathways, each with a probability, a cost and a health outcome. Expected value analysis weights every pathway by its probability and adds the results, separately for costs and for health effects. For strategy $j$ the two expectations are as follows.

$$\mathbb{E}[C_j] = \sum_{k=1}^{K_j} \pi_{jk} C_{jk}, \qquad \mathbb{E}[Q_j] = \sum_{k=1}^{K_j} \pi_{jk} Q_{jk}$$

where $\mathbb{E}[C_j]$ and $\mathbb{E}[Q_j]$ are the expected cost and expected QALYs of strategy $j$, $K_j$ is the number of pathways in that strategy, $\pi_{jk}$ is the probability of pathway $k$ (the product of the conditional branch probabilities along it, summing to one across the pathways of the strategy), and $C_{jk}$ and $Q_{jk}$ are the cost and QALYs at the end of pathway $k$. In a tree, the same totals are reached by rolling back from the terminal nodes, averaging at each chance node; in a Markov model the cohort proportions in each state play the part of the pathway probabilities.

The comparison between strategies is then made on these expectations, never on pathway values or on the most likely pathway. The NICE manual (PMG36, section 4.2.16) requires that reported ICERs be the ratio of expected additional total cost to expected additional QALYs, and section 4.10.6 asks for the expected value of each component of cost, expected total costs and expected QALYs for each option. The rule applied to the expectations, an incremental cost-effectiveness ratio against a threshold or net monetary benefit, is set out on the decision rule and expected net benefit pages.

Why public decisions rest on expected values

Choosing by expected value treats an option with a wide spread of possible results in the same way as one with a narrow spread and the same mean. The main argument for doing so in public decisions comes from the economics of public investment, while a second argument, from decision theory, concerns the role of statistical inference rather than attitudes to risk.

Arrow and Lind addressed the question in 1970 for public investment generally. They showed that when the returns of a public project are spread across a large number of taxpayers, each bears a small share of the risk, and the total cost of risk-bearing goes to zero as that number becomes large, so the value of the investment approaches its expected value. They stressed that the result comes from the government spreading risk among many people, not from pooling projects. Their conclusion carried two conditions that matter in health. The returns must be independent of other components of national income, and risks actually borne by private individuals should be discounted as those individuals would discount them.

Claxton's 1999 argument, that adoption should rest on mean net benefit whatever the statistical significance and that the distribution of net benefit matters only for whether more information is needed, takes the expected-value objective, and so risk neutrality, as given; it is set out on the expected net benefit and decision pages.

Expected value, expected utility and who bears the risk

Expected value averages outcomes in their own units, such as pounds or QALYs. Expected utility averages a utility of those outcomes, and a curved utility function encodes an attitude to risk. A concave function gives risk aversion, so a certain outcome can be preferred to a gamble with a higher mean. Choosing on expected QALYs is therefore the same as assuming risk neutrality over QALYs, and Lakdawalla and Phelps note that standard cost-effectiveness models assume consumers are risk neutral in health.

The distinction matters most at the level of the patient, who lives through one pathway and does not experience the average. McNeil, Weichselbaum and Pauker interviewed patients with operable lung cancer and found them quite averse to risks involving the possibility of immediate death; when these attitudes were combined with survival data, radiotherapy appeared to be the preferred plan for several patients, although surgery may offer longer five-year survival and life expectancy. For a shared clinical decision, the patient's own risk attitude is a legitimate input, and an expected utility analysis can represent it.

At the level of the payer, the case for risk neutrality is stronger but not complete. Ben-Zion and Gafni argued that the use of expected values as outcome measures in risky public health programmes is inappropriate and proposed certainty equivalents measured by the standard gamble. Sendi summarises their reasoning as the fact that health is not a transferable good, a point close to Arrow and Lind's own exception for risks borne by private individuals, and adds that a payer may be averse to cost risk because budget constraints must be met.

Worked example comparing surgery with medical management

The numbers in this example are illustrative. A patient group can be offered surgery (S) or medical management (M), and outcomes are measured in discounted lifetime QALYs and costs at a threshold of £25,000 per QALY.

Medical management costs £4,000 and a relapse adds £6,000 of treatment. The relapse risk $r$ applies in each of two periods, and its best estimate is 0.25. For simplicity, a relapse carries the same discounted cost and QALYs whichever period it occurs in.

1. Roll back medical management. The chance node for period 2, reached after no relapse in period 1, has expected QALYs of 0.25 × 6 + 0.75 × 9 = 8.25 and expected cost of 0.25 × £10,000 + 0.75 × £4,000 = £5,500. At the period 1 node, expected QALYs are 0.25 × 6 + 0.75 × 8.25 = 7.6875 and expected cost is 0.25 × £10,000 + 0.75 × £5,500 = £6,625.

2. Roll back surgery. Expected QALYs are 0.95 × 8.3 + 0.05 × 0 = 7.885 and expected cost is £9,000, since every patient incurs the cost of the operation.

3. Compare on expected values. Surgery adds £2,375 (£9,000 − £6,625) and 0.1975 QALYs (7.885 − 7.6875), an ICER of about £12,025 per QALY, below the threshold. In net monetary benefit terms, surgery gives 25,000 × 7.885 − 9,000 = £188,125 and medical management 25,000 × 7.6875 − 6,625 = £185,562.50, so expected value analysis selects surgery by £2,562.50 per patient.

4. Read the result at the right level. Surgery wins on the population average, yet one patient in twenty dies at operation, while the worst outcome under medical management is 6 QALYs. A patient who is strongly averse to immediate death could rationally prefer medical management, as in the lung cancer study above, without contradicting the payer's analysis.

When the deterministic run is not the expected value

The rollback above used the best estimate of $r$. If $r$ is itself uncertain, the expected outcome is the average of the model's results over the distribution of $r$, which equals the result at the average $r$ only when the outputs are linear in it. When every parameter appears at most once on each pathway and the parameters are independent, a tree's result at the mean inputs equals its expected result, but here $r$ enters twice on the same pathway, through the term $(1-r)^2$. For that term the expectation is as follows.

$$\mathbb{E}\left[(1-r)^2\right] = \left(1 - \mathbb{E}[r]\right)^2 + \operatorname{Var}(r)$$

where $\mathbb{E}[r]$ is the mean relapse risk per period and $\operatorname{Var}(r)$ is its variance across the uncertainty in the evidence. Suppose the evidence leaves $r$ equally likely to be 0.05 or 0.45, so its mean is 0.25 and its variance is (0.05² + 0.45²) ÷ 2 − 0.25² = 0.1025 − 0.0625 = 0.04. The deterministic run uses 0.75² = 0.5625 as the probability of avoiding relapse, while the expected value of that probability is 0.5625 + 0.04 = 0.6025.

Carrying the two values of $r$ through the model gives 8.7075 QALYs at £4,585 when $r$ is 0.05, and 6.9075 QALYs at £8,185 when $r$ is 0.45. The probabilistic means are 7.8075 QALYs and £6,385, against 7.6875 QALYs and £6,625 in the deterministic run. Surgery's inputs enter linearly, so its expectation is unchanged. Net monetary benefit for medical management rises to 25,000 × 7.8075 − 6,385 = £188,802.50, now £677.50 above surgery, and the ICER of surgery rises to about £33,742 per QALY (£2,615 ÷ 0.0775). The deterministic run and the expected value analysis recommend different strategies.

The NICE manual (PMG36, section 4.7.12) states that the committee's preferred cost-effectiveness estimate should be derived from a probabilistic analysis when possible unless the model is linear, and that use of deterministic results should be clearly justified. A survival example of the same effect, where mean survival depends on the reciprocal of a hazard, is given on the expected net benefit page.

Where choosing on expected values needs qualification

The expected value rule rests on assumptions that hold well for many routine adoption decisions and poorly for some. The main qualifications concern risk attitudes, irreversibility and correlation.

• On risk aversion, patients bear their own health outcomes, and payers bear budget risk. Lakdawalla and Phelps note that technologies that reduce the variance of health outcomes generate value to risk-averse consumers even when average outcomes are unchanged, value that a comparison of expected QALYs omits.
• On irreversibility, Palmer and Smith argued that uncertainty, together with the degree of irreversibility of a decision, can give flexibility in the timing of a decision an economic value, which a one-off comparison of expected values does not capture, a point taken up on the decision page.
• Outcomes can also be correlated. In a cost-effectiveness model, parameter uncertainty is shared by every patient who receives the treatment, so it does not average out across a larger patient population in the way patient-level variation does. If the true effect is worse than expected, every treated patient bears the shortfall in health, and the argument of Ben-Zion and Gafni, as Sendi summarises it, applies, since health is not a transferable good, so patients cannot share that loss with others in the way taxpayers share a financial one.

The gap between the expected value of the decision made now and the expected value with uncertainty resolved before choosing is the expected value of perfect information, the starting point of value of information analysis.

Common misreadings

The commonest error is to read an expected value as the outcome a typical patient will have. In the example, no patient receives 7.885 QALYs from surgery, since each either dies at operation or gains 8.3 QALYs. A second is to report a deterministic run as the expected result of a non-linear model, which the example shows can select the wrong strategy. A third is to read choice on the expectation as ignoring uncertainty; uncertainty instead informs the separate decision about further evidence. A last error runs the other way, applying a payer's risk-neutral rule to an individual clinical choice in which the patient's own attitude to risk ought to count.

Sources

• Arrow KJ, Lind RC. Uncertainty and the evaluation of public investment decisions. American Economic Review. 1970;60(3):364-378.
• Ben-Zion U, Gafni A. Evaluation of public investment in health care: is the risk irrelevant? Journal of Health Economics. 1983;2(2):161-165. doi:10.1016/0167-6296(83)90005-X
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford: Oxford University Press; 2006.
• Claxton K. The irrelevance of inference: a decision-making approach to the stochastic evaluation of health care technologies. Journal of Health Economics. 1999;18(3):341-364. doi:10.1016/S0167-6296(98)00039-3
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford: Oxford University Press; 2015.
• Lakdawalla DN, Phelps CE. Health technology assessment with risk aversion in health. Journal of Health Economics. 2020;72:102346. doi:10.1016/j.jhealeco.2020.102346
• McNeil BJ, Weichselbaum R, Pauker SG. Fallacy of the five-year survival in lung cancer. New England Journal of Medicine. 1978;299(25):1397-1401. doi:10.1056/NEJM197812212992506
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; published 31 January 2022, last updated 13 October 2026. Sections 4.2.16, 4.7.12 and 4.10.6.
• Palmer S, Smith PC. Incorporating option values into the economic evaluation of health care technologies. Journal of Health Economics. 2000;19(5):755-766. doi:10.1016/S0167-6296(00)00048-5
• Sendi P. Dealing with bad risk in cost-effectiveness analysis: the cost-effectiveness risk-aversion curve. PharmacoEconomics. 2021;39(2):161-169. doi:10.1007/s40273-020-00969-5