Concept Architecture
Concept
Theoretically, the Delta Method is an asymptotic statistical technique used to approximate the variance and sampling distribution of a function of one or more estimated parameters. It is founded on first-order Taylor series expansion and asymptotic theory, allowing uncertainty to be propagated through non-linear transformations of random variables. In health economics, the Delta Method is widely used to estimate standard errors and confidence intervals for derived measures such as incremental cost-effectiveness ratios, risk ratios, odds ratios and transformed regression parameters.
Mathematically, the Delta Method linearises a differentiable function around the expected value of an estimator using a first-order Taylor expansion. The variance of the transformed estimator is then approximated using the gradient of the function together with the variance-covariance matrix of the original estimator. Under regularity conditions, the transformed estimator is asymptotically normally distributed.
In practice, the Delta Method is implemented after estimating model parameters and their variance-covariance matrix. Statistical software routinely applies the method to obtain standard errors for transformed parameters in regression models, survival analyses, Bayesian approximations and health economic evaluations. It provides computationally efficient variance estimation when exact analytical solutions are unavailable.
Purpose
Used to approximate the variance and standard error of transformed estimators, construct confidence intervals for non-linear functions of estimated parameters, propagate uncertainty through mathematical models and support statistical inference in health economic analyses.
Mathematical Formulae
Primary Formula
Var(g(??)) � [g?(?)]� ? Var(??)
Supporting Formulae
For a vector of parameters:
Var(g(??)) � ?g(??)? � ?g(??)
where:
?g(??) = gradient vector of the transformation
� = variance-covariance matrix of ??
Asymptotic distribution:
g(??) ~ N(g(?), Var(g(??)))
Related Mathematical Methods
Taylor Series Expansion
Asymptotic Normality
Variance-Covariance Matrix
Maximum Likelihood Estimation
Generalised Linear Models
Bootstrap Method
Sandwich Variance Estimation
Example
A health economist estimates a logistic regression model in which the regression coefficient for treatment is:
?? = 0.405
SE(??) = 0.12
The odds ratio is obtained by exponentiation:
OR = exp(0.405) = 1.50
Using the Delta Method:
SE(OR) � exp(??) ? SE(??)
SE(OR) � 1.50 ? 0.12 = 0.18
The estimated standard error is then used to construct a confidence interval for the odds ratio.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(B2) | Transform regression coefficient to odds ratio |
| POWER | =POWER(B3,2) | Calculate variance components |
| MMULT | =MMULT(A2:C2,D2:F4) | Matrix multiplication for multivariate Delta Method calculations |
| TRANSPOSE | =TRANSPOSE(A2:C2) | Calculate gradient vector products |
| SUMPRODUCT | =SUMPRODUCT(A2:C2,D2:F2) | Evaluate variance approximation using gradient and covariance terms |
VBA (Optional)
Automate Delta Method variance calculations for transformed model parameters and generate standard errors and confidence intervals across health economic models.
Sources
Oehlert GW. A Note on the Delta Method. The American Statistician. 1992;46(1):27?29.
Casella G, Berger RL. Statistical Inference.
Van der Vaart AW. Asymptotic Statistics.
Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
Related Concepts (3)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is the delta method?
A statistical technique approximating the variance of a function of random variables using a Taylor series expansion, used for ratios such as the ICER.
Source: Oehlert 1992
How does the delta method find the uncertainty of a combined quantity?
The delta method finds the approximate variance of a quantity that is a function of estimated variables, such as a ratio, by using a Taylor series to linearise that function around the estimates. This turns an awkward nonlinear combination into a simple linear approximation whose variance can be computed from the variances of its parts. It is commonly used to attach a confidence interval to the incremental cost-effectiveness ratio, which combines uncertain costs and effects. Approximating the spread of a function of estimates is its role. Drummond and colleagues (2015) discuss this.
Source: Drummond et al. 2015
How does the delta method work?
The delta method works by taking a function of random variables and approximating it with a first-order Taylor series expansion around the means of the variables, which linearises the function; the variance of this linear approximation, computed from the variances and covariances of the variables and the function's derivatives, gives the approximate variance of the function. So the delta method works by linearising a function and propagating the uncertainty of its inputs through the derivatives, yielding an approximate variance for the output, which is why it is useful for obtaining standard errors of complex quantities built from estimated components, such as ratios and other transformations.
Source: Oehlert 1992
When is the delta method used?
The delta method is used when the standard error of a function of estimates is needed but not directly available, such as for ratios, products, or other non-linear transformations of estimated quantities. In health economics, it is applied to derive the variance of the incremental cost-effectiveness ratio from the variances of costs and effects. So the delta method is used to obtain approximate standard errors and confidence intervals for derived or combined estimates, particularly ratios, providing a way to quantify uncertainty for quantities that are functions of other estimates, though it is an approximation and alternatives such as bootstrapping may be preferred in some cases.
Source: Oehlert 1992
What are the limitations of the delta method?
The limitations of the delta method include that it is an approximation based on linearising the function, so it may be inaccurate when the function is strongly non-linear or the variability large, and when the approximation is poor the resulting standard errors and intervals can be misleading, particularly for ratios where the denominator may be near zero. So the delta method is used with awareness that its accuracy depends on the function being reasonably linear over the relevant range and on the variability not being too great, which is why for problematic cases, such as cost-effectiveness ratios with uncertain denominators, resampling methods like the bootstrap are often preferred as they avoid the linear approximation.
Source: Oehlert 1992
How is the delta method used for the ICER?
The delta method is used for the incremental cost-effectiveness ratio by treating the ratio of incremental cost to incremental effect as a function of these two estimated quantities and applying the Taylor series approximation to derive its variance from the variances and covariance of the incremental cost and effect. So the delta method provides an approximate standard error and confidence interval for the incremental cost-effectiveness ratio, allowing its uncertainty to be quantified, though because the ratio can behave awkwardly when the incremental effect is small or uncertain, the approximation may be unreliable there, which is why alternative approaches such as bootstrapping or cost-effectiveness planes are also used.
Source: Drummond et al. 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 15 Dec 2025
Content version: 1.0.0
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- Term code
- HE-ES-SA-046
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