deltaMethod is a generic function that uses the delta method to get a
standard error for a nonlinear function of a vector of random variables
with known or estimated covariance matrix.
deltaMethod(object, ...) ## Default S3 method: deltaMethod(object, g., vcov., func=g., constants, level=0.95, rhs, ..., envir=parent.frame()) ## S3 method for class 'lm' deltaMethod(object, g., vcov.=vcov(object, complete=FALSE), parameterNames=names(coef(object)), ..., envir=parent.frame()) ## S3 method for class 'nls' deltaMethod(object, g., vcov.=vcov(object, complete=FALSE), ..., envir=parent.frame()) ## S3 method for class '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 class 'polr' deltaMethod(object, g., vcov.=vcov(object, complete=FALSE), ..., envir=parent.frame()) ## S3 method for class 'survreg' deltaMethod(object, g., vcov. = vcov(object, complete=FALSE), parameterNames = names(coef(object)), ..., envir=parent.frame()) ## S3 method for class 'coxph' deltaMethod(object, g., vcov. = vcov(object, complete=FALSE), parameterNames = names(coef(object)), ..., envir=parent.frame()) ## S3 method for class 'mer' deltaMethod(object, g., vcov. = vcov(object, complete=FALSE), parameterNames = names(fixef(object)), ..., envir=parent.frame()) ## S3 method for class 'merMod' deltaMethod(object, g., vcov. = vcov(object, complete=FALSE), parameterNames = names(fixef(object)), ..., envir=parent.frame()) ## S3 method for class 'lme' deltaMethod(object, g., vcov. = vcov(object, complete=FALSE), parameterNames = names(fixef(object)), ..., envir=parent.frame()) ## S3 method for class 'lmList' deltaMethod(object, g., ..., envir=parent.frame())
For the default method,
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
A quoted string used to annotate output. The default of
A character vector of length
This argument is a named vector whose elements are constants that
are used in the
level for confidence interval, default
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
Suppose x is a random vector of length p that is at least approximately normally distributed with mean β and estimated covariance matrix C. Then any function g(β) of β, is estimated by g(x), which is in large samples normally distributed with mean g(β) and estimated variance h'Ch, where h is the first derivative of g(β) with respect to β 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
g. that when evaluated gives the function g. The
names(object) must return the names of the elements of
that are used in the expression
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
You can provide an alternative function for computing the sample variance
matrix, for example to use a sandwich estimator.
For mixed models using
nlme, the coefficient estimates
are returned by the
fixef function, while for
nlsList coefficient estimates are returned by
coef as a matrix. Methods for these models are provided to get the
correct estimates and variance matrix.
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
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
- 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
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
objects, the delta method is computed for each element of the list of
For nonlinear regression objects produced by the
nls function, the call
returns the estimated
coefficient vectors with names corresponding to parameter names.
m2 <- nls(y ~ theta/(1 + gamma * x), start = list(theta=2, gamma=3)) will
have parameters named
In many other familiar regression models, such as those produced by
glm, the names of
the coefficient estimates are the corresponding regressor names, not
For mixed-effects models fit with
glmer from the
lme4 package or
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
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.
An object of class
"deltaMethod", inheriting from
"data.frame", for which a
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.
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
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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