R: Small sample Adjustments for Wald-type tests using Sandwich...
saws
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Small sample Adjustments for Wald-type tests using Sandwich estimator of variance
Description
This function takes an object from a regression function and gives confidence intervals and p-values using
the sandwich estimator of variance corrected for small samples.
a list containing three elements: coefficients, u, omega (see details)
test
either a numeric vector giving elements of coefficient to test,
or an r by p matrix of constants for testing (see details)
beta0
null parameters for testing (see details)
conf.level
level for confidence intervals
method
one of "d3", "d5", "d1", "d2", "d4", or "dm" (see details)
bound
bound for bias correction, denoted b in Fay and Graubard, 2001
Details
Typically, the x object is created in a specialized function. Currently there are three such functions,
link{lmfitSaws},geeUOmega and clogistCalc. The function lmfitSaws
is a simple linear model function that creates all the output needed. The function geeUOmega takes output from the gee function of the gee package
and creates the 'u' matrix and the 'omega' array. The 'coefficients' is a vector with p parameter
estimates, and is a standard output from the regression. The matrix 'u' is K by p with u[i,] the ith
estimating equation, where there are K approximately independent estimating equations. The array 'omega' is K by p by p
where omega[i,,] is a p by p matrix estimating - du/dbeta (here beta=coefficients). See Fay and Graubard (2001) for details.
Suppose that the coefficient vector from the regression is beta. Then we test r hypotheses, based on the the matrix product,
TEST (beta-beta0)=0, where TEST is an r by p matrix. If the argument 'test' is an r by p matrix (where r is arbitrary), then TEST=test.
If 'test' is a vector, then each element of test corresponds to testing that row of beta is 0, i.e., TEST<-diag(p)[test,],
where p is the length of the coefficient vector. For example, test<-c(2,5), tests that beta[2]-beta0[2]=0 and that beta[5]-beta0[5]=0.
The alternatives are always two-sided.
There are several methods available. They are all discussed in Fay and Graubard (2001). The naming of the
methods follows that paper (see for example Table 1, where deltam corresponds to dm, etc.):
dm
the usual model based method which does not use the sandwich, uses a chi squared distribution
d1
the standard sandwich method which makes no corrections for small samples
d2
sandwich method, no bias correction, uses F distribution with df=dhat (see paper)
d3
(default method:sandwich method, no bias correction, uses F distribution with df=dtilde (see paper)
d4
sandwich method, with bias correction, uses F distribution with df=dhatH (see paper)
d5
sandwich method, with bias correction, uses F distribution with df=dtildeH (see paper)
Value
An object of class 'saws'. A list with elements:
originalCall
call from the original object
method
method used (see details)
test
test matrix (see details)
beta0
beta0 vector (see details)
coefficients
estimated coefficients
df
a vector of estimated degrees of freedom. This will have as many elements as there are coefficients
V
variance-covariance matrix
se
vector of standard errors of the coefficients
t.value
a vector of t-values: test (coef - beta0)/se
p.value
a vector of two-sided p-values
conf.int
p by 2 matrix of confidence intervals
Author(s)
M.P. Fay
References
Fay and Graubard (2001). Small-Sample Adjustments for Wald-Type Tests Using Sandwich Estimators.
Biometrics 57: 1198-1206. (for copy see /inst/doc/ directory)
See Also
For examples, see geeUOmega and clogistCalc. See also print.saws