# deltamethod: The delta method In msm: Multi-State Markov and Hidden Markov Models in Continuous Time

## Description

Delta method for approximating the standard error of a transformation g(X) of a random variable X = (x1, x2, …), given estimates of the mean and covariance matrix of X.

## Usage

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

## Arguments

 `g` 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 (g1, g2, …) can be provided, for example `list( ~ x1 + x2, ~ x1 / (x1 + x2) )` `mean` The estimated mean of X `cov` The estimated covariance matrix of X `ses` If `TRUE`, then the standard errors of g1(X), g2(X),… are returned. Otherwise the covariance matrix of g(X) is returned.

## Details

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.

## Value

A vector containing the standard errors of g1(X), g2(X), … or a matrix containing the covariance of g(X).

## Author(s)

C. H. Jackson [email protected]

## References

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.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```## 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]) ## Approximate standard error deltamethod (~ 1 / (x1 + x2), estmean, estvar) ## 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) form deltamethod(as.formula(form), estmean, estvar) ```

msm documentation built on May 31, 2017, 5:10 a.m.