Suppose \(x\) is a random vector of length \(p\) that is at least approximately
normally distributed with mean \(\beta\) and estimated covariance matrix
\(C\). Then any function \(g(\beta)\) of \(\beta\), is estimated by
\(g(x)\), which is in large samples normally distributed with mean \(g(\beta)\)
and estimated variance \(h'Ch\), where \(h\) is the first derivative of
\(g(\beta)\) with respect to \(\beta\) evaluated at \(x\). This function
returns both \(g(x)\) and its standard error, the square root of the estimated
variance.
The default method requires that you provide \(x\) in the argument
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 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 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)” used for the intercept in linear and generalized
linear models is an exception, and it will be correctly interpreted by
deltaMethod. Another option is to use back-ticks to quote nonstandard R names, as in deltaMethod(m2,"X1/`X1:X2`").
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 produced by the nls function, 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 models, such as those produced by lm and glm, the names of
the coefficient estimates are the corresponding regressor names, not
parameter names.
For mixed-effects models fit with lmer and glmer 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.
Note: Earlier versions of deltaMethod included an argument
parameterPrefix that implemented the same functionality as the
parameterNames argument, but which caused several problems that
were not easily fixed without the change in syntax.