standardize_gee: Regression standardization in conditional generalized...
| standardize_gee | R Documentation |
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 |
link |
The link function to be used with |
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 |
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 |
transforms |
A vector of transforms in the following format:
If set to |
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)
