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 default
deltaMethod(object, g., vcov., func=g., constants, level=0.95,
rhs, ..., envir=parent.frame())
# S3 method for lm
deltaMethod (object, g., vcov.=vcov(object, complete=FALSE),
parameterNames=names(coef(object)), ..., envir=parent.frame())
# S3 method for nls
deltaMethod(object, g., vcov.=vcov(object, complete=FALSE), ..., envir=parent.frame())
# S3 method for multinom
deltaMethod (object, g., vcov. = vcov(object, complete=FALSE),
parameterNames = if (is.matrix(coef(object)))
colnames(coef(object)) else names(coef(object)), ..., envir=parent.frame())
# S3 method for polr
deltaMethod (object, g., vcov.=vcov(object, complete=FALSE), ..., envir=parent.frame())
# S3 method for survreg
deltaMethod (object, g., vcov. = vcov(object, complete=FALSE),
parameterNames = names(coef(object)), ..., envir=parent.frame())
# S3 method for coxph
deltaMethod (object, g., vcov. = vcov(object, complete=FALSE),
parameterNames = names(coef(object)), ..., envir=parent.frame())
# S3 method for mer
deltaMethod (object, g., vcov. = vcov(object, complete=FALSE),
parameterNames = names(fixef(object)), ..., envir=parent.frame())
# S3 method for merMod
deltaMethod (object, g., vcov. = vcov(object, complete=FALSE),
parameterNames = names(fixef(object)), ..., envir=parent.frame())
# S3 method for lme
deltaMethod (object, g., vcov. = vcov(object, complete=FALSE),
parameterNames = names(fixef(object)), ..., envir=parent.frame())
# S3 method for lmList
deltaMethod (object, g., ..., envir=parent.frame())
For the default method, 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 (2) a model object for which there are coef and vcov methods,
and for which the named coefficient vector returned by coef is asymptotically normally distributed with asymptotic
covariance matrix returned by vcov.
For the other methods, object is a
regression object for which coef(object) or fixef(object) returns a vector of parameter
estimates.
A quoted string that is the function of the parameter estimates to be evaluated; see the details below.
The (estimated) covariance matrix of the coefficient
estimates. For the default method, this argument is required. For all
other methods, this argument must either provide the estimated covariance
matrix or a function that when applied
to object returns a covariance matrix. The default is
to use the function vcov.
A quoted string used to annotate output. The default of
func = g. is usually appropriate.
A character vector of length 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 characters, like I(x2^2), or
x1:x2 that will confuse the numerical differentiation function. See
details below.
This argument is a named vector whose elements are constants that
are used in the f argument. It isn't generally necessary to specify this argument but it may be convenient to do so.
level for confidence interval, default 0.95.
hypothesized value for the specified function of parameters; if absent no hypothesis test is performed.
Used to pass arguments to the generic method.
Environment in which g. is evaluated; not normally specified by the user.
An object of class "deltaMethod", inheriting from "data.frame", for which a print method is provided. The object contains columns
named Estimate for the estimate, SE for its standard error, and columns for confidence limits and possibly a hypothesis test.
The value of g. is given as a row label.
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.
Fox, J. (2016) Applied Regression Analysis and Generalized Linear Models, Third Edition. Sage.
Fox, J. and Weisberg, S. (2019) An R Companion to Applied Regression, Third Edition, Sage.
Weisberg, S. (2014) Applied Linear Regression, Fourth Edition, Wiley, Section 6.1.2.
First derivatives of g. are computed using symbolic differentiation
by the function D.
# NOT RUN {
m1 <- lm(time ~ t1 + t2, data = Transact)
deltaMethod(m1, "b1/b2", parameterNames= paste("b", 0:2, sep=""))
deltaMethod(m1, "t1/t2", rhs=1) # use names of preds. rather than coefs.
deltaMethod(m1, "t1/t2", vcov=hccm) # use hccm function to est. vars.
deltaMethod(m1, "1/(Intercept)")
# The next example invokes the default method by extracting the
# vector of estimates and covariance matrix explicitly
deltaMethod(coef(m1), "t1/t2", vcov.=vcov(m1))
# }
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