deltamethod: Apply the (Multivariate) Delta Method

In wviechtb/metafor: Meta-Analysis Package for R

Apply the (Multivariate) Delta Method

Description

Function to apply the (multivariate) delta method to a set of estimates. \loadmathjax

Usage

deltamethod(x, vcov, fun, level, H0=0, digits)

Arguments

x	either a vector of estimates or a model object from which model coefficients can be extracted via coef(x).
vcov	when x is a vector of estimates, the corresponding variance-covariance matrix (ignored when x is a model object, in which case vcov(x) is used to extract the variance-covariance matrix).
fun	a function to apply to the estimates.
level	numeric value between 0 and 100 to specify the confidence interval level (see here for details). If unspecified, this either defaults to 95 or, if possible, the corresponding value from the model object.
H0	numeric value to specify the value under the null hypothesis for the Wald-type test(s) (the default is 0). Can also be a vector.
digits	optional integer to specify the number of decimal places to which the printed results should be rounded.

Details

Let \mjeqn\hat\theta\theta denote a vector of \mjseqnp estimates which can be specified via the x argument and let \mjseqn\Sigma denote the corresponding \mjeqnp \times ppxp variance-covariance matrix, which can be specified via the vcov argument. If x is not an vector with estimates, then the function assumes that x is a model object and will try to use coef(x) and vcov(x) to extract the model coefficients and the corresponding variance-covariance matrix (in this case, the vcov argument is ignored).

Let \mjeqnf(\cdot)f(.) be a function, specified via the fun argument, with \mjseqnp inputs/arguments (or with a single argument that is assumed to be a vector of length \mjseqnp), which returns a numeric (and atomic) vector of \mjseqnq transformed estimates. Then the function computes \mjeqnf(\hat\theta)f(\theta) and the corresponding variance-covariance matrix of the transformed estimates using the multivariate delta method (e.g., van der Vaart, 1998) with \mjdeqn\textVar[f(\hat\theta)] = \nabla f(\hat\theta)' \cdot \Sigma \cdot \nabla f(\hat\theta)Var[f(\theta)] = ▽f(\theta)' \Sigma ▽f(\theta) where \mjeqn\nabla f(\hat\theta)▽f(\theta) denotes the gradient of \mjeqnf(\cdot)f(.) evaluated at \mjeqn\hat\theta\theta. The function computes the gradient numerically using the derivative function from the calculus package.

The function also computes Wald-type tests and confidence intervals for the \mjseqnq transformed estimates. The level argument can be used to control the confidence interval level.

Value

An object of class "deltamethod". The object is a list containing the following components:

tab	a data frame with the transformed estimates, standard errors, test statistics, p-values, and lower/upper confidence interval bounds.
vcov	the variance-covariance matrix of the transformed estimates.
...	some additional elements/values.

The results are formatted and printed with the print function. Extractor functions include coef and vcov.

Author(s)

Wolfgang Viechtbauer ([e-mail], https://www.metafor-project.org).

References

van der Vaart, A. W. (1998). Asymptotic statistics. Cambridge, UK: Cambridge University Press.

Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1–48. ⁠https://doi.org/10.18637/jss.v036.i03⁠

See Also

conv.delta for a function to apply the (univariate) delta method to observed effect sizes or outcomes and their sampling variances.

Examples

############################################################################

### copy data into 'dat'
dat <- dat.craft2003

### construct dataset and var-cov matrix of the correlations
tmp <- rcalc(ri ~ var1 + var2 | study, ni=ni, data=dat)
V <- tmp$V
dat <- tmp$dat

### turn var1.var2 into a factor with the desired order of levels
dat$var1.var2 <- factor(dat$var1.var2,
   levels=c("acog.perf", "asom.perf", "conf.perf", "acog.asom", "acog.conf", "asom.conf"))

### multivariate random-effects model
res <- rma.mv(yi, V, mods = ~ 0 + var1.var2, random = ~ var1.var2 | study, struct="UN", data=dat)
res

### restructure estimated mean correlations into a 4x4 matrix
R <- vec2mat(coef(res))
rownames(R) <- colnames(R) <- c("perf", "acog", "asom", "conf")
round(R, digits=3)

### check that order in vcov(res) corresponds to order in R
round(vcov(res), digits=4)

### fit regression model with 'perf' as outcome and 'acog', 'asom', and 'conf' as predictors
matreg(1, 2:4, R=R, V=vcov(res))

### same analysis but using the deltamethod() function
deltamethod(coef(res), vcov(res), fun=function(r1,r2,r3,r4,r5,r6) {
   R <- vec2mat(c(r1,r2,r3,r4,r5,r6))
   setNames(c(solve(R[-1,-1]) %*% R[2:4,1]), c("acog","asom","conf"))
})

### using a function that takes a vector as input
deltamethod(coef(res), vcov(res), fun=function(r) {
   R <- vec2mat(r)
   setNames(c(solve(R[-1,-1]) %*% R[2:4,1]), c("acog","asom","conf"))
})

############################################################################

### construct dataset and var-cov matrix of the r-to-z transformed correlations
dat <- dat.craft2003
tmp <- rcalc(ri ~ var1 + var2 | study, ni=ni, data=dat, rtoz=TRUE)
V <- tmp$V
dat <- tmp$dat

### turn var1.var2 into a factor with the desired order of levels
dat$var1.var2 <- factor(dat$var1.var2,
   levels=c("acog.perf", "asom.perf", "conf.perf", "acog.asom", "acog.conf", "asom.conf"))

### multivariate random-effects model
res <- rma.mv(yi, V, mods = ~ 0 + var1.var2, random = ~ var1.var2 | study, struct="UN", data=dat)
res

### estimate the difference between r(acog,perf) and r(asom,perf)
deltamethod(res, fun=function(z1,z2,z3,z4,z5,z6) {
   transf.ztor(z1) - transf.ztor(z2)
})

### using a function that takes a vector as input
deltamethod(res, fun=function(z) {
   transf.ztor(z[1]) - transf.ztor(z[2])
})

############################################################################

wviechtb/metafor documentation built on Dec. 4, 2024, 1:03 a.m.