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Delta method for approximating the standard error of a transformation \(g(X)\) of a random variable \(X = (x_1, x_2, \ldots)\), given estimates of the mean and covariance matrix of \(X\).


deltamethod(g, mean, cov, ses = TRUE)



A formula representing the transformation. The variables must be labelled x1, x2,...{} For example,

~ 1 / (x1 + x2)

If the transformation returns a vector, then a list of formulae representing (\(g_1, g_2, \ldots\)) can be provided, for example

list( ~ x1 + x2, ~ x1 / (x1 + x2) )


The estimated mean of \(X\)


The estimated covariance matrix of \(X\)


If TRUE, then the standard errors of \(g_1(X), g_2(X),\ldots\) are returned. Otherwise the covariance matrix of \(g(X)\) is returned.


A vector containing the standard errors of \(g_1(X), g_2(X), \ldots\) or a matrix containing the covariance of \(g(X)\).


The delta method expands a differentiable function of a random variable about its mean, usually with a first-order Taylor approximation, and then takes the variance. For example, an approximation to the covariance matrix of \(g(X)\) is given by

$$ Cov(g(X)) = g'(\mu) Cov(X) [g'(\mu)]^T $$

where \(\mu\) is an estimate of the mean of \(X\). This function uses symbolic differentiation via deriv.

A limitation of this function is that variables created by the user are not visible within the formula g. To work around this, it is necessary to build the formula as a string, using functions such as sprintf, then to convert the string to a formula using as.formula. See the example below.

If you can spare the computational time, bootstrapping is a more accurate method of calculating confidence intervals or standard errors for transformations of parameters. See boot.msm. Simulation from the asymptotic distribution of the MLEs (see e.g. Mandel 2013) is also a convenient alternative.


Oehlert, G. W. (1992) A note on the delta method. American Statistician 46(1).

Mandel, M. (2013) Simulation based confidence intervals for functions with complicated derivatives. The American Statistician 67(2):76-81.


C. H. Jackson


## Simple linear regression, E(y) = alpha + beta x 
x <- 1:100
y <- rnorm(100, 4*x, 5)
toy.lm <- lm(y ~ x)
estmean <- coef(toy.lm)
estvar <- summary(toy.lm)$cov.unscaled * summary(toy.lm)$sigma^2

## Estimate of (1 / (alphahat + betahat))
1 / (estmean[1] + estmean[2])
#> (Intercept) 
#>   0.4072937 
## Approximate standard error
deltamethod (~ 1 / (x1 + x2), estmean, estvar) 
#> [1] 0.1699689

## We have a variable z we would like to use within the formula.
z <- 1
## deltamethod (~ z / (x1 + x2), estmean, estvar) will not work.
## Instead, build up the formula as a string, and convert to a formula.
form <- sprintf("~ %f / (x1 + x2)", z)
#> [1] "~ 1.000000 / (x1 + x2)"
deltamethod(as.formula(form), estmean, estvar)
#> [1] 0.1699689