Signature
E_C = exp(mu + sigma^2 / 2); M_C = exp(mu)
| Inputs | Definition | Unit |
|---|---|---|
mu | Mean of the natural log of cost, ln of the median | natural log of pounds |
sigma | Standard deviation of the natural log of cost, above zero | none, on the log scale |
E_C | Expected, or arithmetic mean, cost per patient | pounds per patient per year |
|---|---|---|
M_C | Median cost per patient, equal to the geometric mean for a log-normal distribution | pounds per patient per year |
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
Log-normal expected cost and median from named cells
With MeanLogCost and SDLogCost named, the formulas return the expected cost, held in ExpCost, and the median, held in MedianCost. A median of £2,000 gives MeanLogCost as =LN(2000).
=EXP(MeanLogCost+SDLogCost^2/2); =EXP(MeanLogCost)
Assumptions
Log cost normally distributed with no zero costs
The natural log of cost follows a normal distribution, which requires every cost to be positive; zero costs need a separate part of the model.
Log-normal parameters treated as known
mu and sigma are treated as known values. When they are estimated from a sample they carry sampling error, which this formula does not reflect.
Worked examples
Log-normal cost with a median of £2,000 and a standard deviation of log cost of 1.0
With mu equal to ln 2000, about 7.601, and sigma of 1.0, the expected cost is £2,000 times exp(0.5), about £3,297, about 65 per cent above the median, as in the article. For 5,000 patients the expected budget is about £16.49 million against £10.0 million from the median. The figures are illustrative.
mu = 7.600902; sigma = 1; E_C = 3297.44; M_C = 2000
Same median with a standard deviation of log cost of 1.2
With the same median and a standard deviation of log cost of 1.2, the expected cost rises to about £4,109, as in the article, so the mean depends on how the upper tail is modelled.
mu = 7.6009025; sigma = 1.2; E_C = 4108.87; M_C = 2000
Common errors
Reporting the exponentiated mean log cost as the expected cost
For 5,000 patients the median or back-transformed mean log cost gives a budget of £10.0 million against an expected £16.49 million, a shortfall of about £6.49 million or 39.3 per cent of the true total, as in the article.
Using the log-normal mean when costs are not log-normal
Briggs and colleagues found in simulations that the sample mean is always unbiased, while the log-normal mean estimator can perform very poorly when the true distribution is not log-normal; where the distribution is unknown the sample mean generally remains the estimator of choice.
Treating a log-normal mean cost as secure when it rests on the modelled tail
Raising the standard deviation of log cost from 1.0 to 1.2 with the same median moves the expected cost from about £3,297 to about £4,109. Thompson and Nixon found cost-effectiveness conclusions sensitive to how the upper tail of the cost distribution beyond the observed data is modelled.
Sources
NIST log-normal mean and median behind the expected cost
NIST/SEMATECH e-Handbook of Statistical Methods. Section 1.3.6.6.9, Lognormal distribution. Gaithersburg, MD: National Institute of Standards and Technology (page read). Common statistics: with scale parameter equal to 1 the mean is exp(0.5 sigma squared) and the median is the scale parameter m; the maximum likelihood estimate of m is exp of the estimated mu.
Arithmetic mean cost as the informative summary for budgets
Thompson SG, Barber JA. How should cost data in pragmatic randomised trials be analysed? BMJ. 2000;320(7243):1197-1200. doi:10.1136/bmj.320.7243.1197 (full text read). The total cost is estimated by multiplying the arithmetic mean cost by the number of patients; for positively skewed data the median and geometric mean are always less than the arithmetic mean; a t test on log-transformed data compares geometric means.
Simulation evidence on the log-normal mean estimator for costs
Briggs A, Nixon R, Dixon S, Thompson S. Parametric modelling of cost data: some simulation evidence. Health Economics. 2005;14(4):421-428. doi:10.1002/hec.941 (abstract read). The sample mean is always unbiased; the log-normal estimator can perform very badly when the true distribution is not log-normal; the sample mean generally remains the estimator of choice.
Sensitivity of cost-effectiveness to the modelled cost tail
Thompson SG, Nixon RM. How sensitive are cost-effectiveness analyses to choice of parametric distributions? Medical Decision Making. 2005;25(4):416-423. doi:10.1177/0272989X05276862 (abstract read). Conclusions are sensitive to the choice of distribution and in particular to how the upper tail of the cost distribution beyond the observed data is modelled.
Canonical Identity
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