Signature
E_g = g_mu + 0.5 * d2g * Var_theta
| Inputs | Definition | Unit |
|---|---|---|
g_mu | Output evaluated at the mean of the parameter, the result of a deterministic run | units of the output |
d2g | Second derivative of the output with respect to the parameter, evaluated at its mean; for the probability 1 minus exp(minus h t) it is minus t squared times exp(minus h t) | output units per squared parameter unit |
Var_theta | Variance of the uncertain parameter, the square of its standard deviation | squared parameter units |
E_g | Approximate expected value of the output over the distribution of the parameter | units of the output, for example a probability |
|---|
Function
Expected value of an uncertain cost, health outcome or model output
Maps the probability distribution of an uncertain quantity, such as a cost per patient, a QALY total or a model output that depends on uncertain parameters, to its probability-weighted mean, which carries the units of the quantity. The discrete form applied at chance nodes is HE-FM-CHN-001 on the chance node page, the Monte Carlo mean over probabilistic simulations is HE-FM-ENB-001 and the constant-hazard event probability used below is HE-FM-TP-001. The records here cover the cases in which the mean of a function differs from the function of the means: a product of correlated quantities, a curved output with an uncertain parameter, an event probability under a gamma-distributed hazard and the mean of a log-normal cost. Notation follows the Expected Value article.
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Implementations
Excel
Second-order expected output from named cells
With OutputAtMean, SecondDeriv and ParamVar named, the first formula returns the approximation, held in ApproxOutput. For the fracture example the second formula computes it directly from MeanHazard, Horizon and HazardVar.
=OutputAtMean+0.5*SecondDeriv*ParamVar; =(1-EXP(-MeanHazard*Horizon))-0.5*Horizon^2*EXP(-MeanHazard*Horizon)*HazardVar
Assumptions
Smooth output and a concentrated parameter distribution
The output is twice differentiable near the mean and the parameter distribution is concentrated enough for third and higher-order terms to be small. With a wide distribution or strong curvature the approximation can miss by a large margin, as for the log-normal cost in HE-EX-EV-006.
Single uncertain parameter in the second-order expansion
The form is written for one parameter. With several parameters each pair adds a cross term with its covariance; leaving the cross terms out assumes the parameters are uncorrelated.
Worked examples
Five-year fracture probability at an uncertain hazard
At a mean hazard of 0.10 a year and a horizon of 5 years the deterministic probability is about 0.3935 and its second derivative about minus 15.16. With a hazard variance of 0.005 the correction is about minus 0.038 and the approximate expectation about 0.356, close to the exact 0.36 from HE-FM-EV-003, as in the article.
g_mu = 0.39347; d2g = -15.163; Var_theta = 0.005; E_g = 0.35556
Five-year fracture-free survival at the same uncertain hazard
Fracture-free survival exp(minus 5h) is convex, so the same terms with the opposite sign raise its approximate expectation from about 0.6065 to about 0.6444, above the exact 0.64: the sign of the correction follows the curvature. The approximation is computed here for illustration.
g_mu = 0.60653; d2g = 15.163; Var_theta = 0.005; E_g = 0.64444
Exponential of a log cost with a standard deviation of 1.0
For a log-normal cost with a median of £2,000 and a standard deviation of log cost of 1.0, the output is the exponential of log cost, whose second derivative equals the output itself, £2,000 at the mean log cost. The approximation gives £3,000 against the exact mean of about £3,297 from HE-FM-EV-004, so with a large variance the expansion understates the gap. The figures are computed here for illustration.
g_mu = 2000; d2g = 2000; Var_theta = 1; E_g = 3000
Common errors
Using the standard deviation of the parameter in place of its variance
Multiplying by the standard deviation of the hazard, about 0.0707, rather than its variance of 0.005 gives a correction of about minus 0.536 and an approximate five-year probability of about minus 0.14, which is impossible.
Leaving out the factor of one half in the second-order correction
Without the half, the approximate fracture probability becomes about 0.318, further from the exact 0.36 than the deterministic 0.3935 is.
Reporting the second-order approximation as the expected value
The expansion is a guide to sign and rough size. For the log-normal cost it gives £3,000 against the exact £3,297; where no closed form exists the simulated mean of a probabilistic analysis is the estimate, as the article states.
Sources
Jensen 1906 for the gap between mean output and output at the mean
Jensen JLWV. Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Mathematica. 1906;30:175-193. doi:10.1007/BF02418571 (bibliographic record checked). The original statement of the inequality between the mean of a convex function and the function of the mean.
Textbook basis for the second-order expected output
Casella G, Berger RL. Statistical Inference. 2nd ed. Pacific Grove, CA: Duxbury; 2002 (reprinted Boca Raton: Chapman and Hall/CRC; 2024). Section 4.7, inequalities (Jensen's inequality), and section 5.5.4, the delta method, which expands a smooth function of a random variable in a Taylor series about its mean. Textbook result.
Canonical Identity
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