Expected Net Benefit

Expected Net Benefit as a Decision Rule in Cost-Effectiveness Analysis

A probabilistic cost-effectiveness model produces thousands of plausible results for each option, yet a health system has to make one adoption decision. Expected net benefit reduces that spread to a single number per option, and decision theory shows that the option with the highest value is the one to adopt on current evidence, whatever the level of statistical significance. The same quantity then reappears as the baseline in value of information analysis, so a clear grasp of it links the adoption decision to the question of whether more research is worthwhile.

In this sense the term belongs to cost-effectiveness analysis, where health effects are valued at a cost-effectiveness threshold. It is distinct from the net benefit statistic used to assess clinical prediction models in decision curve analysis. Expected net benefit can be expressed in monetary or health units, and in practice the expectation is estimated from probabilistic sensitivity analysis. A nonlinear model run at mean inputs gives the wrong answer. Expected net benefit and the probability that an option is cost-effective can point to different options, and the expected value of perfect information is calculated from the same simulations.

From net benefit to its expectation

Net benefit places costs and health effects on one scale by valuing health at the threshold $\lambda$. In its monetary form it is net monetary benefit; dividing through by the threshold gives net health benefit, measured in QALYs. For option $j$ the two forms are

$$\text{NMB}_j(\theta) = \lambda E_j(\theta) - C_j(\theta), \qquad \text{NHB}_j(\theta) = E_j(\theta) - \frac{C_j(\theta)}{\lambda}$$

where $\text{NMB}_j(\theta)$ is net monetary benefit in pounds, $\text{NHB}_j(\theta)$ is net health benefit in QALYs, $E_j(\theta)$ is the health effect of option $j$ (usually QALYs), $C_j(\theta)$ is its cost, $\lambda$ is the cost-effectiveness threshold in pounds per QALY and $\theta$ is the vector of uncertain model parameters. Both forms rank options identically at any positive $\lambda$, because one is the other multiplied by a constant.

Because $\theta$ is uncertain, net benefit is a random quantity. Expected net benefit is its mean over the joint distribution of the parameters, which is expected value analysis applied to net benefit, and it can be written as

$$\text{ENB}_j = \mathbb{E}_\theta\left[\text{NMB}_j(\theta)\right] = \lambda\,\mathbb{E}_\theta\left[E_j(\theta)\right] - \mathbb{E}_\theta\left[C_j(\theta)\right]$$

where $\text{ENB}_j$ is the expected net benefit of option $j$ and $\mathbb{E}_\theta$ denotes the expectation taken over the distribution of $\theta$, which in a Bayesian analysis is the posterior distribution given current evidence. The second equality holds because net benefit is linear in costs and effects, so expected net benefit can be built from expected costs and expected QALYs. The level of net benefit for a single option has no absolute meaning, since it depends on which costs are counted; only differences between options inform a decision.

Why the mean decides adoption

The decision rule is to adopt the option with the maximum expected net benefit at the chosen threshold. All relevant options are compared at once, not in a sequence of pairs. Formally, the chosen option is

$$j^{*} = \arg\max_{j} \; \mathbb{E}_\theta\left[\text{NMB}_j(\theta)\right]$$

where $j^{*}$ is the option to adopt and the maximum is taken over every mutually exclusive option in the decision problem. With several options, this gives the same choice as a full incremental analysis in which dominated and extendedly dominated options are removed and the most effective remaining option whose ICER does not exceed $\lambda$ is selected. Net benefit avoids the difficulties of ratios, which is why Stinnett and Mullahy proposed the net health benefit framework for analysing uncertainty in cost-effectiveness analysis.

Claxton argued in 1999 that rules of statistical inference are arbitrary and irrelevant to the adoption decision. On this view, decisions should rest only on mean net benefit, whether or not the difference between options is statistically significant, and a decision maker who waits for significance forgoes health or resources whenever the option with the higher mean is delayed. The distribution of net benefit still matters, but for a separate question, namely whether more information should be acquired before or alongside adoption. This separation of the adoption decision from the research decision is the basis of value of information analysis.

The NICE manual (PMG36) asks for expected net health benefits to be presented as well as ICERs, using values of £25,000 and £35,000 per QALY gained, and notes that they may be particularly informative in cases including when there are several technologies or comparators, when differences in costs or QALYs are small, or when a technology provides less health benefit at lower cost.

Estimating expected net benefit from probabilistic sensitivity analysis

In practice the expectation is estimated by Monte Carlo simulation. Probabilistic sensitivity analysis assigns a distribution to each uncertain parameter, draws a set of values, runs the model for every option with that same set, and repeats. The estimate of expected net benefit is the mean across the simulations,

$$\widehat{\text{ENB}}_j = \frac{1}{N} \sum_{n=1}^{N} \text{NMB}_j\left(\theta^{(n)}\right)$$

where $\widehat{\text{ENB}}_j$ is the estimated expected net benefit of option $j$, $N$ is the number of simulations and $\theta^{(n)}$ is the parameter set drawn in simulation $n$. Every option is evaluated with the same draw of $\theta$ in a given simulation, because the options share parameters such as baseline risk and costs of care. This preserves the correlation between the options' net benefits, on which the probability of cost-effectiveness and value of information depend, and it also reduces the Monte Carlo error in the estimated difference in expected net benefit between options.

The estimate carries Monte Carlo error, which falls as $N$ rises. PMG36 states that enough simulations should be used to minimise its effect, and that the variance of probabilistic outputs such as net benefits can be reviewed as the number of simulations increases to judge whether more runs are needed. When two options have close expected net benefits, a small number of simulations can reverse their ranking by chance alone.

Expected values from nonlinear models

Many decision models are nonlinear functions of their parameters. Survival depends on hazards through exponentials, rates are converted to probabilities, and in a Markov model transition probabilities compound over repeated cycles. For such a model, the mean of the outputs is generally different from the output at the mean inputs, so a deterministic run at point estimates does not give expected costs, expected QALYs or expected net benefit.

A small illustration shows the size of the effect. Under a constant hazard $h$, mean survival is the reciprocal of the hazard, $\frac{1}{h}$. Suppose, as a hypothetical case, that $h$ is equally likely to be 0.1 or 0.3 per year. The model run at the mean hazard gives

$$\frac{1}{\mathbb{E}[h]} = \frac{1}{0.2} = 5$$

where $\mathbb{E}[h]$ is the mean hazard and the result is mean survival in years. Averaging the model output over the two equally likely values gives

$$\mathbb{E}\left[\frac{1}{h}\right] = \frac{1}{2}\left(\frac{1}{0.1} + \frac{1}{0.3}\right) \approx 6.67$$

where $\mathbb{E}\left[\frac{1}{h}\right]$ is expected survival in years. The deterministic run understates expected survival by about 1.67 years, and QALYs and ongoing costs that accrue over survival would be understated with it. PMG36 accordingly states that the committee's preferred cost-effectiveness estimate should be derived from a probabilistic analysis when possible, unless the model is linear.

A worked example with five simulations of a two-option model

This example is illustrative and uses hypothetical numbers. A new treatment B is compared with standard care A at a threshold of £25,000 per QALY. A probabilistic sensitivity analysis is reduced to five equally likely simulations so that every step can be checked by hand; a real analysis would use thousands.

Each net benefit is 25,000 times the QALYs minus the cost; in simulation 1, for example, option A has 150,000 minus 12,000, which is £138,000.

Expected net benefit of each option (step 1)

The expected net benefit of each option is the mean of its five net benefits. Both calculations are shown in full.

$$\mathbb{E}[\text{NMB}_A] = \frac{138{,}000 + 134{,}000 + 142{,}000 + 128{,}000 + 135{,}500}{5} = 135{,}500$$

where $\mathbb{E}[\text{NMB}_A]$ is the expected net benefit of standard care A in pounds. For the new treatment the corresponding figure is

$$\mathbb{E}[\text{NMB}_B] = \frac{134{,}000 + 131{,}000 + 140{,}000 + 136{,}000 + 147{,}500}{5} = 137{,}700$$

where $\mathbb{E}[\text{NMB}_B]$ is the expected net benefit of treatment B in pounds. Option B has the higher expected net benefit by £2,200 per patient, so the decision rule selects B. In health units the difference is

$$\mathbb{E}[\text{NHB}_B] - \mathbb{E}[\text{NHB}_A] = \frac{2{,}200}{25{,}000} = 0.088$$

where $\mathbb{E}[\text{NHB}_B] - \mathbb{E}[\text{NHB}_A]$ is the difference in expected net health benefit in QALYs per patient. The same conclusion follows from the ICER built on expected values,

$$\text{ICER} = \frac{20{,}400 - 12{,}000}{6.324 - 5.90} = \frac{8{,}400}{0.424} \approx 19{,}811$$

where ICER is the incremental cost per QALY gained of B over A in pounds. Since it lies below £25,000, B is preferred, and with only two options B remains the choice at any threshold above about £19,811 per QALY.

Probability that each option is cost-effective (step 2)

The probability that an option is cost-effective is the share of simulations in which it has the highest net benefit. This is the quantity plotted on a cost-effectiveness acceptability curve.

Option A has the highest net benefit in three of five simulations, a probability of 60 per cent, and B in two of five, a probability of 40 per cent. The two summaries disagree. A is more often the better option, but B is better on average. The reason is skewness. B loses by small amounts in three simulations and wins by large amounts in two, so the median incremental net benefit is minus £2,000 while the mean is plus £2,200.

Expected value of perfect information per patient (step 3)

With perfect information the decision maker would pick the best option in each simulation, earning the highest net benefit in the last column. The expected value of that choice is its mean, £139,500. The expected value of perfect information is the difference between this and the expected net benefit of the option chosen now,

$$\text{EVPI} = 139{,}500 - 137{,}700 = 1{,}800$$

where EVPI is the expected value of perfect information in pounds per patient. The same figure is the mean loss from choosing B in the simulations where A would have been better,

$$\text{EVPI} = \frac{4{,}000 + 3{,}000 + 2{,}000 + 0 + 0}{5} = 1{,}800$$

where each term is the net benefit forgone in one simulation by adopting B. In health units this is 0.072 QALYs per patient, which is £1,800 divided by £25,000.

Scaling to the population (step 4)

The per-patient figures become decision-relevant when multiplied by the number of patients affected. In this hypothetical case 2,000 patients a year are treated over a 10-year decision horizon, and discounting is ignored for simplicity, so 20,000 patients are affected. The population gain from adopting B is

$$20{,}000 \times 2{,}200 = 44{,}000{,}000$$

where the result is the gain in population expected net benefit from adopting B instead of A, £44 million. The population EVPI is

$$20{,}000 \times 1{,}800 = 36{,}000{,}000$$

where the result is the population expected value of perfect information, £36 million. This figure is an upper bound on the value of further research that could resolve the uncertainty for this population.

Probability of cost-effectiveness versus expected net benefit

The worked example illustrates a general result. The option with the highest probability of being cost-effective need not be the option with the highest expected net benefit. For a comparison of two options, the probability of cost-effectiveness exceeds one half when the median incremental net benefit is positive, while the decision rule depends on the mean. The two can point to different options when the distribution of incremental net benefit is skewed and its mean lies close to zero, so that the mean and the median fall on opposite sides of zero.

Fenwick, Claxton and Sculpher proposed the cost-effectiveness acceptability frontier to address this. At each threshold the frontier plots the probability of cost-effectiveness of the option with the highest expected net benefit, so it follows the decision rule and switches between options at the threshold where the expected net benefits cross, which need not be where the acceptability curves cross. In the worked example the frontier at £25,000 sits at 40 per cent, the value for B, even though A's curve lies higher at that threshold.

Barton, Briggs and Fenwick showed that with multiple options, an option subject to extended dominance can have the highest probability of being cost-effective at some thresholds, the optimal option may never have the highest probability, and the EVPI can rise even as the probability of making the wrong decision falls. They concluded that acceptability curves can represent decision uncertainty but should not be used to determine the optimal decision. PMG36 asks that the presentation of acceptability curves include a representation and explanation of the acceptability frontier, described as showing the probability that the technology with the highest expected net benefit is cost effective.

Expected net benefit inside value of information analysis

Value of information analysis compares the expected net benefit available now with what could be achieved if uncertainty were resolved. Its simplest measure considers perfect information about all parameters at once. The comparison is

$$\text{EVPI} = \mathbb{E}_\theta\left[\max_j \text{NMB}_j(\theta)\right] - \max_j \mathbb{E}_\theta\left[\text{NMB}_j(\theta)\right]$$

where the first term is the expected net benefit when the best option can be chosen for each realisation of $\theta$, and the second term is the maximum expected net benefit, the value of the decision made on current information. Expected net benefit is therefore the reference point against which every value of information measure is set. The first term can never be smaller than the second, so EVPI is zero or positive, and it is zero only when the same option is best in every realisation of $\theta$.

Population EVPI and other population values follow by scaling per-patient results by the discounted number of patients who will face the decision over its expected lifetime,

$$\text{Population EVPI} = \text{EVPI} \times \sum_{t=1}^{T} \frac{I_t}{(1+r)^t}$$

where $I_t$ is the number of patients affected in period $t$, $r$ is the discount rate and $T$ is the time horizon over which the decision is expected to apply. The gain in population expected net benefit from adopting the best option is scaled in the same way. The joint ISPOR-SMDM task force report on model parameter estimation and uncertainty argued that EVPI, alongside acceptability curves, is the most appropriate way to present decision uncertainty from a probabilistic analysis. Partial EVPI applies the same comparison to particular parameters, and the expected value of sample information applies it to a particular study design and sample size. The expected net benefit of sampling then subtracts the cost of that study from its population value, and further research of that design is worthwhile on these grounds only when the result is positive.

Choices that change expected net benefit

Expected net benefit is conditional on a set of analytic choices, and a change in any of them can change the option selected. Reporting it therefore needs the choices stated alongside the result.

• Threshold. The value of $\lambda$ scales the health term, so the preferred option can switch as the threshold moves; results are commonly shown across a range, or at the values a decision maker specifies, such as the £25,000 and £35,000 per QALY named in PMG36.
• Comparators. The maximum is taken over the options included, and omitting a relevant comparator can make an option look best when it would not be in a complete comparison.
• Parameter distributions. The expectation is only as good as the distributions assigned to the inputs; PMG36 asks that they represent the available evidence and are justified, not chosen arbitrarily.
• Structural assumptions. Alternative model structures can shift expected net benefit; PMG36 notes that structural uncertainty can sometimes be parameterised within a probabilistic model, for example by model averaging.
• Perspective, time horizon and discounting. These determine which costs and effects enter $E_j$ and $C_j$ and so shift every option's expected net benefit, often by different amounts.

Misreadings of expected net benefit

Several misreadings of expected net benefit lead to poorer decisions. One is to adopt the option with the highest probability of being cost-effective; as the worked example shows, that option can have lower expected net benefit, and choosing it forgoes net benefit on average.

A second error is to report the deterministic result at mean inputs as though it were expected net benefit. The two coincide for a model that is linear in its parameters, or multilinear in independent parameters as a simple decision tree is, but not for Markov or survival models in which probabilities compound over cycles or enter through exponentials. A third is to treat a small or statistically uncertain difference in expected net benefit as a reason to reject the new option; under the decision-theoretic view the uncertainty affects the value of further research, not which option is adopted now.

A fourth is to compare options only in pairs against a common comparator and pick the largest pairwise gain, which can ignore dominance among the other options; the rule applies to all options at once. Finally, expected net benefit treats the decision maker as neutral to risk in net benefit and says nothing about affordability, the distribution of health gains across groups or whether a decision can be reversed later. Those matters are handled through budget impact analysis, equity analysis and the value of information and managed access considerations that sit alongside the adoption decision.

Sources

• 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
• Stinnett AA, Mullahy J. Net health benefits: a new framework for the analysis of uncertainty in cost-effectiveness analysis. Medical Decision Making. 1998;18(2 Suppl):S68-S80. doi:10.1177/0272989X98018002S09
• Fenwick E, Claxton K, Sculpher M. Representing uncertainty: the role of cost-effectiveness acceptability curves. Health Economics. 2001;10(8):779-787. doi:10.1002/hec.635
• Barton GR, Briggs AH, Fenwick EA. Optimal cost-effectiveness decisions: the role of the cost-effectiveness acceptability curve (CEAC), the cost-effectiveness acceptability frontier (CEAF), and the expected value of perfection information (EVPI). Value in Health. 2008;11(5):886-897. doi:10.1111/j.1524-4733.2008.00358.x
• Briggs AH, Weinstein MC, Fenwick EA, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6. Value in Health. 2012;15(6):835-842. doi:10.1016/j.jval.2012.04.014
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford: Oxford University Press; 2006.
• 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.
• 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, 4.7.5, 4.7.10, 4.7.11, 4.7.12, 4.7.15 and 4.10.8.