Signature
E_NU = E_N * E_U + Cov_NU
| Inputs | Definition | Unit |
|---|---|---|
E_N | Expected number of admissions per patient | admissions per patient |
E_U | Expected cost per admission | pounds per admission |
Cov_NU | Covariance of the number of admissions and the cost per admission, the expected product of their deviations from their means, equal to their correlation times the two standard deviations | pounds per patient |
E_NU | Expected value of the product of the two quantities, here the expected admission cost per patient | pounds per patient |
|---|
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
Expected product from summary cells and from paired draws
With MeanAdm, MeanUnitCost and CovAdmCost named, the first formula returns the expected product. With paired draws or patient records in the ranges AdmDraws and CostDraws, the second returns the mean of the products directly, which equals the first when the means and the population covariance COVARIANCE.P are taken from the same draws.
=MeanAdm*MeanUnitCost+CovAdmCost; =SUMPRODUCT(AdmDraws,CostDraws)/COUNT(AdmDraws)
Assumptions
Means and covariance taken from one joint distribution
E_N, E_U and Cov_NU describe the same population or the same set of paired simulation draws. Taking the two means from different sources and setting the covariance to zero assumes independence rather than establishing it.
Finite variances for the product identity
Both quantities have finite variances, so the covariance exists. No distributional form and no independence is needed; with independence the identity reduces to the product of the means.
Worked examples
Admissions and cost per admission with a covariance of 300
With a mean of 1.5 admissions, a mean of £3,000 per admission and a covariance of 300, the expected cost is 1.5 x 3,000 + 300 = 4,800 pounds per patient, against £4,500 from multiplying the means, as in the article. The figures are illustrative.
E_N = 1.5; E_U = 3000; Cov_NU = 300; E_NU = 4800
Same admission means with independent counts and unit costs
When admissions and cost per admission are independent the covariance is zero and the expected cost is the product of the means, £4,500 per patient, so the covariance of 300 accounts for the whole gap of £300 in the first example.
E_N = 1.5; E_U = 3000; Cov_NU = 0; E_NU = 4500
Same admission means with a negative covariance of 300
If patients with more admissions had cheaper admissions, a covariance of minus 300 would lower the expected cost to £4,200 per patient, below the product of the means; correlation can narrow or widen the gap, as the article notes. The figure is computed here for illustration.
E_N = 1.5; E_U = 3000; Cov_NU = -300; E_NU = 4200
Common errors
Multiplying mean admissions by mean cost per admission
Multiplying the two means drops the covariance: £4,500 in place of £4,800 per patient in the article's example, an understatement of 6.25 per cent of the expected cost that carries into every budget built on it.
Combining admission counts and unit costs from separate sources as if independent
When counts and unit costs come from different studies, the covariance cannot be estimated and is set to zero without being stated. The resulting expected cost rests on an independence assumption that patient-level data recording both quantities could check.
Sources
Covariance identity behind the expected admission cost
Casella G, Berger RL. Statistical Inference. 2nd ed. Pacific Grove, CA: Duxbury; 2002 (reprinted Boca Raton: Chapman and Hall/CRC; 2024). Section 4.5, covariance and correlation: the covariance of X and Y equals E[XY] minus E[X]E[Y], so the expected product is the product of the means plus the covariance; independent variables have zero covariance. Textbook result.
Canonical Identity
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