Signature
mu_D = 1 - mu_U; k = mu_D^2 / s^2; theta = s^2 / mu_D
| Inputs | Definition | Unit |
|---|---|---|
mu_U | Mean utility of the health state, below one | utility, with full health equal to one |
s | Standard error of the mean utility, which is also the standard error of the decrement | utility |
mu_D | Mean decrement, one minus the mean utility, above zero | utility, on the scale where full health is one |
|---|---|---|
k | Shape of the gamma distribution for the decrement | none |
theta | Scale of the gamma distribution, so that the mean decrement is k times theta | utility |
Function
Beta distribution density, moments and fitting for an uncertain probability
Maps two positive shape parameters, alpha and beta, to a probability density on the unit interval that describes uncertainty about a probability, such as a transition probability in a decision tree or Markov model. The parameters are fitted from event counts, with alpha counting events and beta non-events, or from a reported mean and standard error by the method of moments, and the fitted distribution is sampled in probabilistic sensitivity analysis. The formulae below give the mean and variance, the two fitting routes, the gamma alternative for a utility decrement and the Dirichlet extension for states with three or more exits. Gamma(.) in the density is the gamma function.
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Implementations
Excel
Gamma decrement parameters and a sampled utility in Excel
With named cells MeanUtility and StdErr, the first three formulas return the decrement, the shape and the scale in cells named Decrement, GammaShape and GammaScale. The fourth draws a utility by passing a uniform random number through GAMMA.INV, whose third argument is a scale.
=1-MeanUtility; =Decrement^2/StdErr^2; =StdErr^2/Decrement; =1-GAMMA.INV(RAND(),GammaShape,GammaScale)
Assumptions
Decrement approach used when a normal or beta does not fit the mean utility
The NICE Decision Support Unit notes that a normal distribution usually describes uncertainty in a mean utility adequately and suggests the decrement approach where it does not. A gamma decrement allows sampled mean utilities below zero, so it suits states where such a mean is plausible given the evidence and the value set.
Scale parameterisation of the gamma decrement
theta is a scale, so the mean decrement is k times theta and the variance k times theta squared. Software that takes a rate or inverse scale needs 1/theta instead.
Worked examples
Gamma decrement for a severe state with mean utility 0.10
The article's illustrative severe state: a mean utility of 0.10 with a standard error of 0.08 gives a mean decrement of 0.90, a shape of 126.5625 and a scale of about 0.007111, shown as 126.6 and 0.0071 in the article. About 11% of sampled mean utilities then fall below zero.
mu_U = 0.10; s = 0.08; mu_D = 0.90; k = 126.5625; theta = 0.007111
Gamma decrement for a moderate state with mean utility 0.70
Illustrative figures: a mean utility of 0.70 with a standard error of 0.05 gives a decrement of 0.30, a shape of 36 and a scale of about 0.008333. Sampled utilities below zero are then vanishingly rare.
mu_U = 0.70; s = 0.05; mu_D = 0.30; k = 36; theta = 0.008333
Common errors
Gamma rate entered where Excel GAMMA.INV expects a scale
The third argument of GAMMA.INV is a scale. Entering the rate 1/theta, about 140.6 in the severe-state example, gives a mean decrement of about 17,800 instead of 0.90.
Sampled gamma decrement used as the utility itself
The gamma draw is the decrement. Using it directly as the utility turns a severe state with a mean utility of 0.10 into one with a mean of 0.90; the sampled utility is one minus the draw.
Sources
Utility decrements sampled from a gamma distribution in DSU TSD 12
Ara R, Wailoo A. NICE DSU Technical Support Document 12: The use of health state utility values in decision models. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; July 2011. Section 4.1, pp. 16-17, which states that uncertainty in a mean utility can in most cases be described by a normal distribution and otherwise suggests describing utilities as decrements from full health, 1 minus the utility, sampled from a log normal or gamma distribution.
Gamma distribution mean and variance in Bayesian Data Analysis
Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd edition. Boca Raton: Chapman & Hall/CRC; 2013. Appendix A, Table A.1, pp. 578-579, which gives the gamma distribution with shape alpha and inverse scale beta, mean alpha/beta and variance alpha/beta squared; matching these to a mean and standard error gives the shape, and the scale as the reciprocal of the inverse scale.
Excel GAMMA.INV with a scale parameter
Microsoft. GAMMA.INV function. Microsoft Support; accessed 1 October 2026. The function returns the inverse of the cumulative gamma distribution; its example, GAMMA.INV(0.068094, 9, 2) returning about 10, matches a gamma with shape 9 and scale 2. It returns the #NUM! error when alpha or beta is zero or below.
Canonical Identity
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