deltaMethod is a generic function that uses the delta method to get a
first-order approximate
standard error for a nonlinear function of a vector of random variables
with known or estimated covariance matrix.deltaMethod(object, ...)
## S3 method for class 'default':
deltaMethod(object, g, vcov., func=g, constants, ...)
## S3 method for class 'lm':
deltaMethod(object, g, vcov.=vcov,
parameterNames=names(coef(object)), ...)
## S3 method for class 'nls':
deltaMethod(object, g, vcov.=vcov, ...)
## S3 method for class 'multinom':
deltaMethod(object, g, vcov. = vcov,
parameterNames = if (is.matrix(coef(object)))
colnames(coef(object)) else names(coef(object)), ...)
## S3 method for class 'polr':
deltaMethod(object, g, vcov.=vcov, ...)
## S3 method for class 'survreg':
deltaMethod(object, g, vcov. = vcov,
parameterNames = names(coef(object)), ...)
## S3 method for class 'coxph':
deltaMethod(object, g, vcov. = vcov,
parameterNames = names(coef(object)), ...)
## S3 method for class 'mer':
deltaMethod(object, g, vcov. = vcov,
parameterNames = names(fixef(object)), ...)
## S3 method for class 'merMod':
deltaMethod(object, g, vcov. = vcov,
parameterNames = names(fixef(object)), ...)
## S3 method for class 'lme':
deltaMethod(object, g, vcov. = vcov,
parameterNames = names(fixef(object)), ...)
## S3 method for class 'lmList':
deltaMethod(object, g, ...)object is either (1) a vector of p
named elements, so names(object) returns a list
of p character strings that are the names of the elements of
object; or (func = g is usually appropriate.p that gives the
names of the parameters in the same order as they appear in the vector of
estimates. This argument will be useful if some of the names in the
vector of estimates include special charactf argument. This is needed only when the function is
called from within another function to comply to R scoping rules. See
example below.Estimate for the estimate, SE for its standard error.
The value of g is given as a row label.object, $C$ in the argument vcov., and a text expression
in argument g that when evaluated gives the function $g$. The
call names(object) must return the names of the elements of x
that are used in the expression g.
Since
the delta method is often applied to functions of regression parameter
estimates, the argument object may be the name of a regression
object from which the the estimates and their estimated variance matrix can
be extracted. In most regression models, estimates are returned by the
coef(object) and the variance matrix from vcov(object).
You can provide an alternative function for computing the sample variance
matrix, for example to use a sandwich estimator.
For mixed models using lme4 or nlme, the coefficient estimates
are returned by the fixef function, while for multinom,
lmList and nlsList coefficient estimates are returned by
coef as a matrix. Methods for these models are provided to get the
correct estimates and variance matrix.
The argument g must be a quoted character string
that gives the function of interest. For example, if you set
m2 <- lm(Y ~ X1 + X2 + X1:X2), then deltaMethod(m2,"X1/X2") applies the
delta method to the ratio of the coefficient estimates for X1 and
X2. The argument g can consist of constants and names
associated with the elements of the vector of coefficient estimates.
In some cases the names may include characters including such as the colon
: used in interactions, or mathematical symbols like + or
- signs that would confuse the
function that computes numerical derivatives, and for this case you can replace
the names of the estimates with the parameterNames argument. For
example, the ratio of the
X2 main effect to the interaction term could be computed using
deltaMethod(m2, "b1/b3", parameterNames=c("b0", "b1", "b2", "b3")).
The name (Intercept)deltaMethod.
For multinom objects, the coef function returns a matrix of
coefficients, with each row giving the estimates for comparisons of one category
to the baseline. The deltaMethod function applies the delta method to
each row of this matrix. Similarly, for lmList and nlsList
objects, the delta method is computed for each element of the list of
models fit.
For nonlinear regression objects of type nls, the call coef(object)
returns the estimated
coefficient vectors with names corresponding to parameter names.
For example,
m2 <- nls(y ~ theta/(1 + gamma * x), start = list(theta=2, gamma=3)) will
have parameters named c("theta", "gamma").
In many other familiar regression methods, such as lm and glm, the names of
the coefficient estimates are the corresponding variable names, not
parameter names.
For mixed-effects models fit with lmer and nlmer from the
lme4 package or lme and nlme from the nlme package,
only fixed-effect coefficients are considered.
For regression models for which methods are not provided, you can extract
the named vector of coefficient estimates and and estimate of its covariance
matrix and then apply the default deltaMethod function.
Earlier versions of deltaMethod included an argument
parameterPrefix that implemented the same functionality as the
parameterNames argument, but it caused several unintended bugs that
were not easily fixed without the change in syntax.g are computed using symbolic differentiation
by the function D.m1 <- lm(time ~ t1 + t2, data = Transact)
deltaMethod(m1, "b1/b2", parameterNames= paste("b", 0:2, sep=""))
deltaMethod(m1, "t1/t2") # use names of preds. rather than coefs.
deltaMethod(m1, "t1/t2", vcov=hccm) # use hccm function to est. vars.
# to get the SE of 1/intercept, rename coefficients
deltaMethod(m1, "1/b0", parameterNames= paste("b", 0:2, sep=""))
# The next example calls the default method by extracting the
# vector of estimates and covariance matrix explicitly
deltaMethod(coef(m1), "t1/t2", vcov.=vcov(m1))
# the following works:
a <- 5
deltaMethod(m1, "(t1 + a)/t2")
# ...but embedded in a function this will fail
f1 <- function(mod, ...) {
z <- 3
deltaMethod(m1, "(t1+z)/t2", ...)
}
f1(m1)
# if z is defined globally f1 could even return the wrong answer.
# the following function works
f2 <- function(mod, ...) {
deltaMethod(m1, "(t1+z)/t2", ...)
}
f2(m1, constants=c(z=3))
# as does
f3 <- function(mod) {
a <- 3
deltaMethod(m1, "(t1+z)/t2", constants=c(z=a))
}
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