standardize_gee: Regression standardization in conditional generalized...

In stdReg2: Regression Standardization for Causal Inference

Regression standardization in conditional generalized estimating equations

Description

standardize_gee 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. standardize_gee 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

standardize_gee(
  formula,
  link = "identity",
  data,
  values,
  clusterid,
  case_control = FALSE,
  ci_level = 0.95,
  ci_type = "plain",
  contrasts = NULL,
  family = "gaussian",
  reference = NULL,
  transforms = NULL
)

Arguments

formula	A formula to be used with "gee" in the drgee package.
link	The link function to be used with "gee".
data	The data.
values	A named list or data.frame specifying the variables and values at which marginal means of the outcome will be estimated.
clusterid	An optional string containing the name of a cluster identification variable when data are clustered.
case_control	Whether the data comes from a case-control study.
ci_level	Coverage probability of confidence intervals.
ci_type	A string, indicating the type of confidence intervals. Either "plain", which gives untransformed intervals, or "log", which gives log-transformed intervals.
contrasts	A vector of contrasts in the following format: If set to "difference" or "ratio", then \psi(x)-\psi(x_0) or \psi(x) / \psi(x_0) are constructed, where x_0 is a reference level specified by the reference argument. Has to be NULL if no references are specified.
family	The family argument which is used to fit the glm model for the outcome.
reference	A vector of reference levels in the following format: If contrasts is not NULL, the desired reference level(s). This must be a vector or list the same length as contrasts, and if not named, it is assumed that the order is as specified in contrasts.
transforms	A vector of transforms in the following format: If set to "log", "logit", or "odds", the standardized mean \theta(x) is transformed into \psi(x)=\log\{\theta(x)\}, \psi(x)=\log[\theta(x)/\{1-\theta(x)\}], or \psi(x)=\theta(x)/\{1-\theta(x)\}, respectively. If the vector is NULL, then \psi(x)=\theta(x).

Details

standardize_gee 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 (CGEE) 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 std_glm. Obtain numeric results in a data frame with the tidy.std_glm function. This is a list with the following components:

res_contrast

An unnamed list with one element for each of the requested contrasts. Each element is itself a list with the elements:

estimates

Estimated counterfactual means and standard errors for each exposure level

covariance

Estimated covariance matrix of counterfactual means

fit_outcome

The estimated regression model for the outcome

fit_exposure

The estimated exposure model

exposure_names

A character vector of the exposure variable names

est_table

Data.frame of the estimates of the contrast with inference

transform

The transform argument used for this contrast

contrast

The requested contrast type

reference

The reference level of the exposure

ci_type

Confidence interval type

ci_level

Confidence interval level

res

A named list with the elements:

estimates

Estimated counterfactual means and standard errors for each exposure level

covariance

Estimated covariance matrix of counterfactual means

fit_outcome

The estimated regression model for the outcome

fit_exposure

The estimated exposure model

exposure_names

A character vector of the exposure variable names

Note

The variance calculation performed by standardize_gee 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 Sjölander.

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.

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

Examples

require(drgee)

set.seed(4)
n <- 300
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.std <- standardize_gee(
  formula = Y ~ X + Z + I(X^2),
  link = "identity",
  data = dd,
  values = list(X = seq(-3, 3, 0.5)),
  clusterid = "id"
)
print(fit.std)
plot(fit.std)

stdReg2 documentation built on April 13, 2025, 5:12 p.m.