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.