Signature
x_bar = sum_(i=1)^n [x_i] / n; SE_inf = sqrt(sum_(i=1)^n [(x_i - x_bar)^2]) / n
| Inputs | Definition | Unit |
|---|---|---|
x_i | Value for observation i, for example one patient's cost | unit of the data |
n | Number of observations, for example patients in one arm | count |
x_bar | Mean of the n observations | unit of the data |
|---|---|---|
SE_inf | Ideal bootstrap standard error of the sample mean, the limit of the Monte Carlo estimate as B grows | unit of the data, for example pounds |
Function
Bootstrap standard error of a trial statistic from resampled data
Maps a statistic and the data that produced it to the standard deviation of the statistic across samples drawn with replacement from the data, each of the same size as the original sample. The empirical distribution of the observations, with mass 1/n on each, stands in for the unknown population distribution, so no normal shape is assumed. In trial-based economic evaluation the statistic is usually a mean cost, an incremental cost or incremental net benefit, and the resampling copies the trial design by drawing patients within each arm with their costs and effects kept together.
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Implementations
Excel
Ideal bootstrap standard error of a mean in one cell
DEVSQ returns the sum of squared deviations from the mean for the range named Costs, and COUNT returns n.
=SQRT(DEVSQ(Costs))/COUNT(Costs)
Assumptions
Independent observations from one arm
The observations are independent draws from one arm or one population, and each bootstrap sample has the same size n.
Ideal value without simulation noise
The formula gives the limit that the Monte Carlo estimate approaches as B grows, so it carries no simulation noise and serves as a check on a resampling routine.
Worked examples
Mean cost in arm T of the six-patient example
Arm T costs have a mean of 2,000 pounds and squared deviations summing to 10,000,000, so the ideal bootstrap standard error is about 527.05 pounds, against 577.35 from s divided by the square root of n.
x_i = [1000, 1200, 1400, 1600, 2000, 4800]; n = 6; x_bar = 2000; SE_inf = 527.05
Mean cost in arm U of the six-patient example
Arm U costs have a mean of 1,000 pounds and squared deviations summing to 580,000, giving an ideal bootstrap standard error of about 126.93 pounds, against 139.04 from the usual formula.
x_i = [600, 800, 900, 1000, 1100, 1600]; n = 6; x_bar = 1000; SE_inf = 126.93
Common errors
Reading the small-sample shortfall of the bootstrap as lower cost variability
With six patients the bootstrap standard error is about 9% below the usual value, 527.05 against 577.35 pounds in arm T, before skewness plays any part. The shortfall comes from the divisor n, not from the costs being less variable; with 50 patients the factor is about 0.990.
Sources
Efron and Tibshirani closed form for the bootstrap standard error of a mean
Efron B, Tibshirani R. Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science. 1986;1(1):54-75. Section 1, equations 1.3 and 1.5 to 1.6, which give the usual estimate with divisor n minus 1 and the bootstrap estimate with divisor n, and note that the difference is too small to matter in most applications.
Canonical Identity
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