mmmgee.test {mmmgee}

Hypothesis Tests for Linear Contrasts in Multiple Marginal GEE Models

Description

Global hypothesis tests, multiple testing procedures and simultaneous confidence intervals for multiple linear contrasts of regression coefficients in a single generalized estimating equation (GEE) model or across multiple GEE models.

Usage

mmmgee.test(x, L, r = NULL, statistic = c("wald", "score"),
  type = c("maximum", "quadratic"), asymptotic = TRUE,
  biascorr = FALSE, closed.test = FALSE, conf.int = FALSE,
  conf.level = 0.95, alternative = c("undirected", "greater", "less"),
  denomDF = NULL, scaled.F = FALSE, maxit = 20, tol = 10^(-8), ...)

Arguments

x

a geem2 object fitted with geem2 or a list of geem2. In the latter case, the geem2 objects must be different models calculated with data from the same subjects. In particular, the parameter id in the call to geem2 must refer to the same subjects in each model.

L

a contrast matrix defining a contrast for the stacked vector of regression coefficients of the marginal models, or a list of contrast matrices. In the latter case, the list must contain one matrix for each model listed in x, in the same order as the models. When using the the score test and x is a list, L must be a list.

r

right hand side vector of the null hypothesis or a list of vectors resembling the right hand side of the null hypothesis. If not specified, r is assumed to be a null vector of appropriate length. See details.

statistic

either "wald" or "score", see details. The default is "wald".

type

either "maximum" or "quadratic", see details. The default is "maximum".

asymptotic

logical, if TRUE the reference distribution for the maximum-type Wald test statistic is a multivariate normal distribution and the reference distribution for the quadratic form Wald test statistic is a chi-squared distribution. If FALSE, a multivariate t-distribution or an F-distribution is used instead. Ignored for the Score test, see details.

biascorr

logical indicating whether the Mancl and DeRouen Bias correction should be used when estimating the joint covariance matrix via mmmgee.

closed.test

logical, if TRUE, multiplicity adjusted p-values based on a closed test procedure using the selected type of test are calculated. With k hypotheses this involves the computation of 2^k tests, which may require considerable computation time.

conf.int

logical. If TRUE simultaneous confidence intervals corresponding to a single step maximum-type test are calculated using a multivariate normal or t approximation, depending on asymptotic.

conf.level

the nominal simultaneous coverage probability of the confidence intervals.

alternative

one of "undirected", "greater", or "less". Determines the direction of maximum-type tests and of confidence intervals. The default is "undirected".

denomDF

Defaults to NULL. In that case, denominator degrees of freedom for the multavariate t-distribution or F-distribution are calculated as min(n-p), where n and p are vectors of the number of independent clusters and the number of regression coefficients in the models in x. Alternatively, a numeric value may be entered to be used as denominator degrees of freedom.

scaled.F

logical. If TRUE and type="quadratic" and asymptotic=FALSE a scaled F distribution similar as for Hotelling's test is used. Ignored otherwise.

maxit

maximal number of iterations to be passed to geem2. Only required when using the score test, where the models are refitted under the restriction of the null hyptothesis.

tol

tolerance limit for the convergence criterion to be passed to geem2. Only required when using the score test, where the models are refitted under the restriction of the null hyptothesis.

...

additional arguments that are passed to pmvnorm, qmvnorm, pmvt and qmvt. In particular the algorithm to solve the multivariate normal or t-distribution integrals may be selected.

Details

The null hypothesis is H0:L\beta=r where L is a contrast matrix, \beta the stacked vector of regression coefficients rom the marginal models and r a real values right hand side vector. L can be specified as matrix or, if it is a block diagonal matrix with each block corresponding to a contrast for one marginal GEE model, as list of the matrices on the diagonal. The right hand side r can be speficied as vector or as list of vectors each corresponding to the part of the right hand side vector for one model.

When choosing statistic="wald" and type="maximum", the maximum of the standardized entries of L\hat{\beta} is used as test statistic and the p-value is calculated from a multivariate normal or t-distribution (depending on asymptotic being TRUE or FALSE) with correlation matrix estimated for L\hat{\beta}. For the t-distribution, denominator degrees of freedom are used as specified in denomDF. When choosing statistic="wald" and type="quadratic", a quadratic form of L\hat{\beta} and the inverse of the estimated covariance matrix of L\hat{\beta} is used as test statistic and the p-value is calculated from a chi-squared distribution or an F-distribution (depending on asymptotic being TRUE or FALSE).

With statistic="score", generalized score tests are calculated by replacing L\hat{\beta} by the first order approximation LAU where U is the stacked estimating equation (the score) and A is the negative inverse of the matrix of first derivatives of U, both evaluated at the location of constrained estimates for \beta under the null hypothesis. Analogous to the Wald statistic, a maximum-type and a quadratic form score test are available. For the score test the option asymptotic is ignored and the reference distribution is multivariate normal or chi-squared.

Value

A list with class mmmgeetest containing the following components, if required:

test

Contains a data frame with the test statistic, degrees of freedom (depending in the type of test) and the p-value. If closed test was required, a further data frame is reported with estimates, right hand side vector, unadjusted p-values and adjusted p-values for each line of H0:L\beta-r=0.

hypothesis

A list containing the contrast matrix L and the right hand side vector r.

conf.int

The simultaneous confidence intervals.

denomDF

The type and value of the denominator degrees of freedom used in the procedure.

mmmgee

The mmmgee object containing in particular the estimated covariance matrix for the coefficents of the models in x. See mmmgee.

Note

A single value for the denominator degrees of freedom is calculated for the covariance matrix estimate across all contrasts. In the closed testing procedure, this value is used for the degrees of freedom associated with the covariance matrix of any subset of contrasts.

Usual linear models or generalized linear models can be regarded as special case of GEE models and can be included in the analysis framework. Note however that mmmgee.test always uses the robust sandwich covariance matrix estimate, even if the calculation of the sandwich covariance was suppressed in the model objects in x.

Author(s)

Robin Ristl, [e-mail]

References

Dennis D. Boos. On generalized score tests. The American Statistician, 1992, 46(4):327-333.

Lloyd A. Mancl, Timothy A. DeRouen. A covariance estimator for GEE with improved small sample properties. Biometrics, 2001, 57(1):126-134.

See Also

geem2, mmmgee

Examples

data(keratosis)
m1<-geem2(clearance~trt,id=id,data=keratosis,family=binomial,corstr="independence")
m2<-geem2(pain~trt,id=id,data=keratosis[keratosis$lesion==1,],family=gaussian,corstr="independence")
L1<-L2<-diag(1,4)[-1,]
mmmgee.test(x=m1,L=list(L1),statistic="wald",type="maximum")
mmmgee.test(x=m1,L=list(L1),statistic="score",type="maximum")
mmmgee.test(x=list(m1,m2),L=list(L1,L2),type="maximum",asymptotic=FALSE,biascorr=TRUE)
mmmgee.test(x=list(m1,m2),L=list(L1,L2),type="maximum",closed.test=TRUE)
mmmgee.test(x=list(m1,m2),L=list(L1,L2),type="maximum",asymptotic=FALSE,
	alternative="less",conf.int=TRUE,denomDF=40)
mmmgee.test(x=list(m1,m2),L=list(L1,L2),type="quadratic",asymptotic=TRUE)
mmmgee.test(x=list(m1,m2),L=list(L1,L2),statistic="score",type="quadratic")
mmmgee.test(x=list(m1,m2),L=list(L1,L2),statistic="score",type="maximum")
#
## Not run: 
data(datasim)
mod1<-geem2(Y.lin~gr.lang+x1,id=id,data=datasim,family="gaussian",corstr="exchangeable")
mod2<-geem2(Y.poi~gr.lang+x2,id=id,data=datasim,family="poisson",corstr="exchangeable")
mod3<-geem2(Y.bin~gr.lang+x3,id=id,data=datasim,family="binomial",corstr="exchangeable")
Li<-matrix(c(0,1,0),nrow=1)
mmmgee.test(list(mod1,mod2,mod3),L=list(Li,Li,Li),statistic="Wald",type="maximum",
	biascorr=TRUE,asymptotic=FALSE,closed.test=TRUE)
mmmgee.test(list(mod1,mod2,mod3),L=list(Li,Li,Li),statistic="score",closed.test=TRUE)

## End(Not run)