drgee is used to estimate an exposure-outcome effect adjusted
for additional covariates. The estimation is based on regression
models for the outcome, exposure or a combination of both.
For clustered data the models may
have cluster-specific intercepts.
drgee(outcome, exposure,
oformula, eformula, iaformula = formula(~1),
olink = c("identity", "log", "logit"),
elink = c("identity", "log", "logit"),
data, subset = NULL, estimation.method = c("dr", "o", "e"),
cond = FALSE, clusterid, clusterid.vcov, rootFinder = findRoots,
intercept = TRUE, ...)The outcome as variable or as a character string naming a variable
in the data argument. If missing, the outcome is assumed
to be the response of oformula.
The exposure as variable or as a character string naming a variable
in the data argument. If missing, the exposure is assumed
to be the response of eformula.
An expression or formula for the outcome nuisance model.
An expression or formula for the exposure nuisance model.
An expression or formula where the RHS should contain the variables
that "interact" (i.e. are supposed to be multiplied with) with the
exposure in the main model. "1" will always added. Default value is no
interactions, i.e. iaformula = formula(~1).
A character string naming the link function in the outcome nuisance
model. Has to be "identity", "log" or
"logit".Default is "identity".
A character string naming the link function in the exposure nuisance
model. Has to be "identity", "log" or
"logit". Default is "identity".
A data frame or environment containing the variables used. If missing, variables are expected to be found in the calling environment of the calling environment.
An optional vector defining a subset of the data to be used.
A character string naming the desired estimation method. Choose
"o" for O-estimation,
"e" for E-estimation or
"dr" for DR-estimation. Default is "dr".
A logical value indicating whether the nuisance models should have
cluster-specific intercepts. Requires a clusterid argument.
A function to solve a system of non-linear equations. Default
is findRoots.
A cluster-defining variable or a character string naming a
cluster-defining variable in the data argument. If it is not
found in the data argument, it will be searched for in the
calling frame. If missing, each observation will be considered to be
a separate cluster. This argument is required when cond = TRUE.
A cluster-defining variable or a character string naming a
cluster-defining variable in the data argument to be used for
adding contributions from the same cluster. These clusters can be
different from the clusters defined by clusterid. However,
each cluster defined by clusterid needs to be contained in
exactly one cluster defined by clusterid.vcov. This variable
is useful when the clusters are hierarchical.
A boolean to choose whether the nuisance parameters in doubly robust conditional logistic regression should be fitted with a model with an intercept. Only used for doubly robust condtional logistic regression.
Further arguments to be passed to the function rootFinder.
drgee returns an object of class drgee containing:
Estimates of the parameters in the main model.
Robust variance for all main model parameters.
Estimates of all estimated parameters.
Robust variance of the all parameter estimates.
An estimation object returned from the function specified
in the rootFinder, if this function is called for the
estimation of the main model parameters.
An estimation object returned from the function specified
in the rootFinder, if this function is called for the
estimation of the outcome nuisance parameters.
An estimation object returned from the function specified
in the rootFinder, if this function is called for the
estimation of the outcome nuisance parameters.
The matched call.
The value of the input argument estimation.method.
The original data object, if given as an input argument
The original oformula object, if given as an input argument
The original eformula object, if given as an input argument
The original iaformula object, if given as an input argument
The class methods coef and vcov can be used to extract the estimated parameters and their covariance matrix from a drgee object. summary.drgee produces a summary of the calculations.
drgee estimates the parameter \(\beta\) in a main
model \(g\{E(Y|A,L)\}-g\{E(Y|A=0,L)\}=\beta^T \{A\cdot X(L)\}\),
where \(Y\) is the outcome of interest, \(A\) is the exposure of
interest, and \(L\) is a vector of covariates that we wish to
adjust for. \(X(L)\) is a vector valued function of \(L\). Note that \(A
\cdot X(L)\) should be interpreted as a columnwise
multiplication and that \(X(L)\) will always contain a column of 1's.
Given a specification of an outcome nuisance model \(g\{E(Y|A=0,L)=\gamma^T
V(L)\) (where \(V(L)\) is a function of \(L\))
O-estimation is performed. Alternatively, leaving \(g\{E(Y|A=0,L)\)
unspecified and using an exposure nuisance model \(h\{E(A|L)\}=\alpha^T
Z(L)\) (where \(h\) is a link
function and \(Z(L)\) is a function of \(L\)) E-estimation
is performed. When \(g\) is logit, the exposure nuisance
model is required be of the form
\(logit\{E(A|Y=0,L)\}=\alpha^T Z(L)\).
In this case the exposure needs to binary.
Given both an outcome and an exposure nuisance model, DR-estimation can be performed. DR-estimation gives a consistent estimate of the parameter \(\beta\) when either the outcome nuisance model or the exposure nuisance model is correctly specified, not necessarily both.
Usage is best explained through an example. Suppose that we are
interested in the parameter vector \((\beta_0,
\beta_1)\) in a main model
\(logit\{E(Y|A,L_1,L_2)\}-logit\{E(Y|A=0,L_1,L_2)\}=\beta_0 A + \beta_1
A \cdot L_1\) where \(L_1\) and \(L_2\) are the covariates that we wish
to adjust for. To adjust for \(L_1\) and \(L_2\), we can use an outcome
nuisance model \(E(Y|A=0,L_1,L_2;\gamma_0, \gamma_1)=\gamma_0 + \gamma_1
L_1\) or an
exposure nuisance model \(logit\{E(A|Y=0,L_1,L_2)\}=\alpha_0+\alpha_1
L_1+\alpha_2 L_2\) to calculate estimates of \(\beta_0\) and \(\beta_1\)
in the main model. We specify the outcome nuisance model as oformula=Y~L_1
and olink = "logit". The exposure nuisance model is specified as
eformula = A~L_1+L_2 and elink = "logit".
Since the outcome \(Y\) and the exposure \(A\) are
identified as the LHS of oformula and eformla
respectively and since the outcome link is specified in the
olink argument,
the only thing left to specify for the main model is the
(multiplicative) interactions \(A\cdot X(L)=A\cdot
(1,L_1)^T\). This
is done by specifying \(X(L)\) as
iaformula = ~L_1, since \(1\) is always included in \(X(L)\).
We can then perform O-estimation, E-estimation or DR-estimation by
setting estimation.method to "o",
"e" or "dr" respectively. O-estimation uses only the
outcome nuisance model, and E-estimation uses only the exposure
nuisance model. DR-estimation uses both nuisance models, and gives a
consistent estimate of \((\beta_0,\beta_1)\) if either nuisance model is correct, not necessarily both.
When estimation.method = "o", the RHS of eformula will be
ignored. The eformula argument can also be replaced by an exposure
argument specifying what the exposure of interest is.
When estimation.method = "e", the RHS of oformula will be
ignored. The oformula argument can also be replaced by an outcome
argument specifying what the outcome of interest is.
When cond = TRUE the nuisance models will be assumed to have
cluster-specific intercept. These intercepts will not estimated.
When E-estimation or DR-estimation is chosen with
olink = "logit", the exposure link will be
changed to "logit". Note that this choice
of outcome link does not work for DR-estimation
when cond = TRUE.
Robust variance for the estimated parameter is calculated
using the function robVcov. A cluster robust variance is calculated when
a character string naming a cluster variable is
supplied in the clusterid argument.
For E-estimation when cond = FALSE and \(g\) is the identity
or log link, see Robins et al. (1992).
For DR-estimation when cond = TRUE and \(g\) is the identity
or log link, see Robins (1999). For DR-estimation when
\(g\) is the logit link, see Tchetgen et al. (2010).
O-estimation can also be performed using the gee function.
Orsini N., Belocco R., Sj<U+001AC86E>der A. (2013), Doubly Robust Estimation in Generalized Linear Models, Stata Journal, 13, 1, pp. 185--205
Robins J.M., Mark S.D., Newey W.K. (1992), Estimating Exposure Effects by Modelling the Expectation of Exposure Conditional on Confounders, Biometrics, 48, pp. 479--495
Robins JM (1999), Robust Estimation in Sequentially Ignorable Missing Data and Causal Inference Models, Proceedings of the American Statistical Association Section on Bayesian Statistical Science, pp. 6--10
Tchetgen E.J.T., Robins J.M., Rotnitzky A. (2010), On Doubly Robust Estimation in a Semiparametric Odds Ratio Model, Biometrika, 97, 1, 171--180
Zetterqvist J., Vansteelandt S., Pawitan Y., Sj<U+001AC86E>der (2016), Doubly Robust Methods for Handling Confounding by Cluster, Biostatistics, 17, 2, 264--276
gee for O-estimation, findRoots for
nonlinear equation solving and robVcov for
estimation of variance.
# NOT RUN {
## DR-estimation when
## the main model is
## E(Y|A,L1,L2)-E(Y|A=0,L1,L2)=beta0*A+beta1*A*L1
## and the outcome nuisance model is
## E(Y|A=0,L1,L2)=gamma0+gamma1*L1+gamma2*L2
## and the exposure nuisance model is
## E(A|Y=0,L1,L2)=expit(alpha0+alpha1*L1+alpha2*l2)
library(drgee)
expit<-function(x) exp(x)/(1+exp(x))
n<-5000
## nuisance
l1<-rnorm(n, mean = 0, sd = 1)
l2<-rnorm(n, mean = 0, sd = 1)
beta0<-1.5
beta1<-1
gamma0<--1
gamma1<--2
gamma2<-2
alpha0<-1
alpha1<-5
alpha2<-3
## Exposure generated from the exposure nuisance model
a<-rbinom(n,1,expit(alpha0 + alpha1*l1 + alpha2*l2))
## Outcome generated from the main model and the
## outcome nuisance model
y<-rnorm(n,
mean = beta0 * a + beta1 * a * l1 + gamma0 + gamma1 * l1 + gamma2 * l2,
sd = 1)
simdata<-data.frame(y,a,l1,l2)
## outcome nuisance model misspecified and
## exposure nuisance model correctly specified
## DR-estimation
dr.est <- drgee(oformula = formula(y~l1),
eformula = formula(a~l1+l2),
iaformula = formula(~l1),
olink = "identity", elink = "logit",
data = simdata, estimation.method = "dr")
summary(dr.est)
## O-estimation
o.est <- drgee(exposure = "a", oformula = formula(y~l1),
iaformula = formula(~l1), olink = "identity",
data = simdata, estimation.method = "o")
summary(o.est)
## E-estimation
e.est <- drgee(outcome = "y", eformula = formula(a~l1+l2),
iaformula = formula(~l1), elink="logit",
data = simdata, estimation.method = "e")
summary(e.est)
# }
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