stdGee {stdReg}

Regression standardization in conditional generalized estimating equations

Description

stdGee performs regression standardization in linear and log-linear fixed effects models, at specified values of the exposure, over the sample covariate distribution. Let Y, X, and Z be the outcome, the exposure, and a vector of covariates, respectively. It is assumed that data are clustered with a cluster indicator i. stdGee uses fitted fixed effects model, with cluster-specific intercept a_i (see details), to estimate the standardized mean \theta(x)=E\{E(Y|i,X=x,Z)\}, where x is a specific value of X, and the outer expectation is over the marginal distribution of (a_i,Z).

Usage

stdGee(fit, data, X, x, clusterid, subsetnew)

Arguments

fit

an object of class "gee", with argument cond = TRUE, as returned by the gee function in the drgee package. If arguments weights and/or subset are used when fitting the model, then the same weights and subset are used in stdGee.

data

a data frame containing the variables in the model. This should be the same data frame as was used to fit the model in fit.

X

a string containing the name of the exposure variable X in data.

x

an optional vector containing the specific values of X at which to estimate the standardized mean. If X is binary (0/1) or a factor, then x defaults to all values of X. If X is numeric, then x defaults to the mean of X. If x is set to NA, then X is not altered. This produces an estimate of the marginal mean E(Y)=E\{E(Y|X,Z)\}.

clusterid

an mandatory string containing the name of a cluster identification variable. Must be identical to the clusterid variable used in the gee call.

subsetnew

an optional logical statement specifying a subset of observations to be used in the standardization. This set is assumed to be a subset of the subset (if any) that was used to fit the regression model.

Details

stdGee assumes that a fixed effects model

\eta\{E(Y|i,X,Z)\}=a_i+h(X,Z;\beta)

has been fitted. The link function \eta is assumed to be the identity link or the log link. The conditional generalized estimating equation (CGGE) estimate of \beta is used to obtain estimates of the cluster-specific means:

\hat{a}_i=\sum_{j=1}^{n_i}r_{ij}/n_i,

where

r_{ij}=Y_{ij}-h(X_{ij},Z_{ij};\hat{\beta})

if \eta is the identity link, and

r_{ij}=Y_{ij}exp\{-h(X_{ij},Z_{ij};\hat{\beta})\}

if \eta is the log link, and (X_{ij},Z_{ij}) is the value of (X,Z) for subject j in cluster i, j=1,...,n_i, i=1,...,n. The CGEE estimate of \beta and the estimate of a_i are used to estimate the mean E(Y|i,X=x,Z):

\hat{E}(Y|i,X=x,Z)=\eta^{-1}\{\hat{a}_i+h(X=x,Z;\hat{\beta})\}.

For each x in the x argument, these estimates are averaged across all subjects (i.e. all observed values of Z and all estimated values of a_i) to produce estimates

\hat{\theta}(x)=\sum_{i=1}^n \sum_{j=1}^{n_i} \hat{E}(Y|i,X=x,Z_i)/N,

where N=\sum_{i=1}^n n_i. The variance for \hat{\theta}(x) is obtained by the sandwich formula.

Value

An object of class "stdGee" is a list containing

call

the matched call.

input

input is a list containing all input arguments.

est

a vector with length equal to length(x), where element j is equal to \hat{\theta}(x[j]).

vcov

a square matrix with length(x) rows, where the element on row i and column j is the (estimated) covariance of \hat{\theta}(x[i]) and \hat{\theta}(x[j]).

Note

The variance calculation performed by stdGee does not condition on the observed covariates \bar{Z}=(Z_{11},...,Z_{nn_i}). To see how this matters, note that

var\{\hat{\theta}(x)\}=E[var\{\hat{\theta}(x)|\bar{Z}\}]+var[E\{\hat{\theta}(x)|\bar{Z}\}].

The usual parameter \beta in a generalized linear model does not depend on \bar{Z}. Thus, E(\hat{\beta}|\bar{Z}) is independent of \bar{Z} as well (since E(\hat{\beta}|\bar{Z})=\beta), so that the term var[E\{\hat{\beta}|\bar{Z}\}] in the corresponding variance decomposition for var(\hat{\beta}) becomes equal to 0. However, \theta(x) depends on \bar{Z} through the average over the sample distribution for Z, and thus the term var[E\{\hat{\theta}(x)|\bar{Z}\}] is not 0, unless one conditions on \bar{Z}.

Author(s)

Arvid Sjolander.

References

Goetgeluk S. and Vansteelandt S. (2008). Conditional generalized estimating equations for the analysis of clustered and longitudinal data. Biometrics 64(3), 772-780.

Martin R.S. (2017). Estimation of average marginal effects in multiplicative unobserved effects panel models. Economics Letters 160, 16-19.

Sjolander A. (2019). Estimation of marginal causal effects in the presence of confounding by cluster. Biostatistics doi: 10.1093/biostatistics/kxz054

Examples

require(drgee)

n <- 1000
ni <- 2
id <- rep(1:n, each=ni)
ai <- rep(rnorm(n), each=ni)
Z <- rnorm(n*ni)
X <- rnorm(n*ni, mean=ai+Z)
Y <- rnorm(n*ni, mean=ai+X+Z+0.1*X^2)
dd <- data.frame(id, Z, X, Y)
fit <- gee(formula=Y~X+Z+I(X^2), data=dd, clusterid="id", link="identity",
  cond=TRUE)
fit.std <- stdGee(fit=fit, data=dd, X="X", x=seq(-3,3,0.5), clusterid="id")
print(summary(fit.std, contrast="difference", reference=2))
plot(fit.std)