alr3 (version 1.1.12)

# delta.method: Estimate and standard error of a nonlinear function of estimated regression coefficients

## Description

delta.method 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.

## Usage

delta.method(object, ...)

## S3 method for class 'default':
delta.method(object,g,var,...)
## S3 method for class 'lm':
delta.method(object, g, var=vcov,parameterPrefix="b",...)
## S3 method for class 'nls':
delta.method(object, g, var=vcov,...)
## S3 method for class 'lmList':
delta.method(object, g, var=vcov,parameterPrefix="b",...)
## S3 method for class 'nlsList':
delta.method(object, g, var=vcov,...)
## S3 method for class 'lme':
delta.method(object, g, var=vcov,parameterPrefix="b",...)
## S3 method for class 'nlme':
delta.method(object, g, var=vcov,...)
## S3 method for class 'multinom':
delta.method(object, g, var=vcov,parameterPrefix="b",...)
## S3 method for class 'polr':
delta.method(object, g, var=vcov,parameterPrefix="b",...)

## Arguments

object
For the default method, object is a named vector of p elements. This means that the call names(object) would return a list of p character strings that are the names of the elements of ob
g
A quoted string that is the function of the parameter estimates to be evaluated; see the details below.
var
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
parameterPrefix
Typically a single letter with default "b" giving the prefix of the names of the parameter names used in the argument g; see details.
...

## Value

• A data.frame with two components named Estimate for the estimate, SE for its standard error. The value of g is given as a row label.

## Details

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 var, and a text expression in argument g that when evaluated gives the function $g$. 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 vector $x$ will be taken from coef(object), and $C$ will be taken from vcov(object) unless you provide some other estimate of variance, for example, using a sandwich estimator. Methods have been provided for many common regression models. For regression models for which methods are not provided, you must extract the named vector of coefficient estimates and and estimate of its covariance matrix and then apply the default delta.method function. In the argument g you must provide a quoted character string that gives the function of interest, for example g="b1/b2", where b1 and b2 are names of two of the coefficient estimates. 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 example, in m2 <- lm(Y~X1+X2), names(coef(m2)) returns c("(Intercept)","X1","X2"). For models where the coefficient names are variable names, delta.method will provide names for the parameter estimates, given by b0, b1,...,bp, if parameterPrefix is left at its default value of b. In this case, b0 is the intercept (if the model has no intercept, then the numbering of the parameters starts with 1, not 0), b1 is the first estimated parameter after the intercept, and so on. Special characters should be avoided in the names of the elements of $x$ as these can cause problems, and the D function used to compute derivatives may get confused. However, embedded spaces or : are permitted.

## References

S. Weisberg (2005), Applied Linear Regression, third edition, Wiley, Section 6.1.2

First derivatives of g are computed using symbolic differentiation by the function D.
# cakes is a data frame with response Y, predictors X1 X2
delta.method(m2,"(b1+b5*350)/(-2*b3)")