car (version 2.0-9)

deltaMethod: Estimate and Standard Error of a Nonlinear Function of Estimated Regression Coefficients

Description

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.

Usage

deltaMethod(object, ...)

## S3 method for class 'default':
deltaMethod(object, g, vcov., func=g, ...)
## S3 method for class 'lm':
deltaMethod(object, g, vcov.=vcov, parameterPrefix="b", ...)
## S3 method for class 'nls':
deltaMethod(object, g, vcov.=vcov, ...)
## S3 method for class 'multinom':
deltaMethod(object, g, vcov.=vcov, parameterPrefix="b", ...)
## S3 method for class 'polr':
deltaMethod(object, g, vcov.=vcov, ...)
## S3 method for class 'survreg':
deltaMethod(object, g, vcov.=vcov, ...)
## S3 method for class 'coxph':
deltaMethod(object, g, vcov.=vcov, ...)

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.

vcov.

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

func

A quoted string used to annotate output. The default of func = g is usually appropriate.

parameterPrefix

Typically a single letter with default "b" giving the prefix of the names of the parameter names used in the argument g for some regression models; see details.

...

Additional arguments; not currently used.

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 vcov., 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 several 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 deltaMethod function. In the argument g you must provide a quoted character string that gives the function of interest. For example, if you set m2 <- lm(Y ~ X1 + X2), then deltaMethod(m2,"X1/X2") applies the delta method to the ratio of the coefficient estimates for X1 and X2. For the product of the interecpt and the coefficient of X2, use deltaMethod(m2, "(Intercept)*X2"), since the name of the interecpt parameter estimate is (Intercept). For models of type lm, glm and polr, you can replace the names in the coefficient vector by b0, b1, ..., bp; you can also change the prefix b using the parameterPrefix argument. For multinom objects you can only use the parameter names starting with the prefix, not the names of the corresponding regressors. 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.

References

Fox, J. (2008) Applied Regression Analysis and Generalized Linear Models, Second Edition. Sage. Fox, J. and Weisberg, S. (2011) An R Companion to Applied Regression, Second Edition, Sage. S. Weisberg (2005) Applied Linear Regression, Third Edition, Wiley, Section 6.1.2.

See Also

First derivatives of g are computed using symbolic differentiation by the function D.

Examples

m1 <- lm(time ~ t1 + t2, data = Transact) 
deltaMethod(m1, "b1/b2") # ratio of coefficients
deltaMethod(m1, "t1/t2") # use names of preds. rather than coefs.
deltaMethod(m1, "t1/t2", vcov=hccm) # use hccm function to est. vars.
# 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))