Package 'drgee' reference manual

Title:	Doubly Robust Generalized Estimating Equations
Description:	Fit restricted mean models for the conditional association between an exposure and an outcome, given covariates. Three methods are implemented: O-estimation, where a nuisance model for the association between the covariates and the outcome is used; E-estimation where a nuisance model for the association between the covariates and the exposure is used, and doubly robust (DR) estimation where both nuisance models are used. In DR-estimation, the estimates will be consistent when at least one of the nuisance models is correctly specified, not necessarily both. For more information, see Zetterqvist and Sjölander (2015) <doi:10.1515/em-2014-0021>.
Authors:	Johan Zetterqvist <drjohanzetterqvist at gmail.com> , Arvid Sjölander <arvid.sjolander at ki.se> with contributions from Alexander Ploner.
Maintainer:	Johan Zetterqvist <[email protected]>
License:	GPL-2 \| GPL-3
Version:	1.1.10
Built:	2024-11-26 06:33:08 UTC
Source:	CRAN

Doubly Robust Generalized Estimating Equations

Description

The main function is drgee, which estimates a parameter $\beta$ in a model defined as $g\{E(Y|A,L)\}-g\{E(Y|A=0,L)\}=\beta^T \{A\cdot X(L)\}$ . By supplying (nuisance) models for $g\{E(Y|A=0,L)$ and $E(A|L)$ , a consistent estimate of $\beta$ is obtained when at least one of these models is correctly specified.

In the function drgee, three estimation methods are implemented: O-estimation, where a model for $g\{E(Y|A=0,L)$ is used; E-estimation, where a model for $E(A|L)$ is used; and DR-estimation where models for both $g\{E(Y|A=0,L)$ and $E(A|L)$ are used.

The function gee is an implementation of standard GEE with independent working correlation matrix.

For conditional methods with clustered data, cluster-specific intercepts are assumed in all models.

The function drgeeData is used for extraction and manipulation of data, and is called by drgee and gee.

The function RobVcov is used to calculate standard errors of the estimates, given a vector of the residuals from estimating equations, the Jacobian, and a cluster-identifying variable.

The function findRoots solves a system of non linear equations.

Author(s)

Johan Zetterqvist, Arvid Sjölander with contributions from Alexander Ploner.

Cluster-center a design matrix around cluster means

Description

Given a design matrix and a cluster identification variable. A new design matrix is returned, where each column is centered around its cluster mean. Using C++ code for speed.

Usage

centerX(x, id)
centerX(x, id)

Arguments

`x`	A design matrix
`id`	A cluster identifying variable.

Details

centerX is written as a help function with the aim of being fast.

Value

A matrix with cluster centered columns.

Author(s)

Johan Zetterqvist, Arvid Sjölander

Doubly Robust Generalized Estimating Equations

Description

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.

Usage

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, ...)
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, ...)

Arguments

`outcome`	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`.
`exposure`	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`.
`oformula`	An expression or formula for the outcome nuisance model.
`eformula`	An expression or formula for the exposure nuisance model.
`iaformula`	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)`.
`olink`	A character string naming the link function in the outcome nuisance model. Has to be `"identity"`, `"log"` or `"logit"`.Default is `"identity"`.
`elink`	A character string naming the link function in the exposure nuisance model. Has to be `"identity"`, `"log"` or `"logit"`. Default is `"identity"`.
`data`	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.
`subset`	An optional vector defining a subset of the data to be used.
`estimation.method`	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"`.
`cond`	A logical value indicating whether the nuisance models should have cluster-specific intercepts. Requires a `clusterid` argument.
`rootFinder`	A function to solve a system of non-linear equations. Default is `findRoots`.
`clusterid`	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`.
`clusterid.vcov`	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.
`intercept`	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`.

Details

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.

Value

drgee returns an object of class drgee containing:

`coefficients`	Estimates of the parameters in the main model.
`vcov`	Robust variance for all main model parameters.
`coefficients.all`	Estimates of all estimated parameters.
`vcov.all`	Robust variance of the all parameter estimates.
`optim.object`	An estimation object returned from the function specified in the `rootFinder`, if this function is called for the estimation of the main model parameters.
`optim.object.o`	An estimation object returned from the function specified in the `rootFinder`, if this function is called for the estimation of the outcome nuisance parameters.
`optim.object.e`	An estimation object returned from the function specified in the `rootFinder`, if this function is called for the estimation of the outcome nuisance parameters.
`call`	The matched call.
`estimation.method`	The value of the input argument `estimation.method`.
`data`	The original data object, if given as an input argument
`oformula`	The original oformula object, if given as an input argument
`eformula`	The original eformula object, if given as an input argument
`iaformula`	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.

Author(s)

Johan Zetterqvist, Arvid Sjölander

References

Orsini N., Belocco R., Sjölander 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., Sjölander (2016), Doubly Robust Methods for Handling Confounding by Cluster, Biostatistics, 17, 2, 264–276

Examples


## 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)
## 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)

Extracting Variables and Model Matrices for Generalized Estimating equations

Description

Given a main model, an outcome nuisance model and an exposure nuisance model drgeeData extracts the model variables and matrices from a data.frame or an environment object. It also performs some data cleaning and error checking.

Usage

drgeeData(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)
drgeeData(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)

Arguments

`outcome`	The outcome as a variable or as a character string naming a 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, the outcome is assumed to be the response of `oformula`.
`exposure`	The exposure as a variable or as a character string naming a 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, the outcome is assumed to be the response of `eformula`.
`oformula`	An expression or formula for the outcome nuisance model. The outcome is identified as the response in this formula.
`eformula`	An expression or formula for the exposure nuisance model. The exposure is identified as the response in this formula.
`iaformula`	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 to create the terms associated with the parameters of interest. "1" will always added. Default value is no interactions, i.e. `formula(~1)`.
`olink`	A character string naming the link function in the outcome nuisance model. Have to be `"identity"`, `"log"` or `"logit"`. Default is `"identity"`.
`elink`	A character string naming the link function in the exposure nuisance model. Have to be `"identity"`, `"log"` or `"logit"`. Default is `"identity"`. When `olink="logit"` this is replaced by `"logit"`.
`data`	A data frame or environment containing the variables in `iaformula`, `oformula` and `eformula`. If missing, variables are expected to be found in the calling frame.
`subset`	An optional vector defining a subset of the data to be used.
`estimation.method`	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"`.
`cond`	A logical value indicating whether the nuisance models should have cluster-specific intercepts. If `cond=TRUE` the design matrices for the nuisance models do not have an intercept. Requires a `clusterid` argument.
`clusterid`	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`.
`clusterid.vcov`	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.

Details

drgeeData is called by drgee and gee to extract data from a data.frame or environment object. The data can then be used to for O-estimation, E-estimation or DR-estimation. drgeeData uses model.frame and model.matrix to remove incomplete observations and to convert factors to dummy variables. It also performs check the supplied data for errors or inconsistencies.

The class method summary.drgeeData produces strings for the formulas with terms referring to the columns in the produced design matrices.

Value

drgee.data returns an object of class drgeeData containing

`used.rows`	The rows numbers in the original data for the used rows (after subset selection and exlusions).
`orig.order`	The original order of the observations.
`y`	The outcome matrix.
`a`	The exposure matrix.
`x`	The matrix of of interactions defined in `iaformula`. This matrix will always contain a column with 1's.
`ax`	The matrix of elementwise product(s) of `a` and each column in `x`.
`v`	The matrix of terms in the outcome nuisance model.
`z`	The matrix of terms in the exposure nuisance model.
`yx`	The matrix of elementwise product(s) of `y` and each column in `x`.
`id`	A factor defining clusters. For independent observations, the number of levels equals the number of complete observations.
`clustname`	A string for the name of the cluster defining variable.
`y.names`	A string for the name of the outcome.
`a.names`	A string for the name of the exposure.
`x.names`	A string vector for the variable names in `x`.
`ax.names`	A string vector for the variable names in `ax`.
`v.names`	A string vector for the variable names in `v`.
`z.names`	A string vector for the variable names in `z`.
`yx.names`	A string vector for the variable names in `yx`.
`olink`	A character string naming the link function in the outcome nuisance model.
`elink`	A character string naming the link function in the outcome nuisance model.
`cond`	A logical value indicating whether cluster-specific intercepts should be assumed. If `TRUE`, the is no column for the intercept in `v` and `z`. Outcome concordant will also be removed.
`oterms`	The `terms` object corresponding to the outcome nuisance model.
`eterms`	The `terms` object corresponding to the exposure nuisance model.

Author(s)

Johan Zetterqvist, Arvid Sjölander

Non Linear Equation System Solving

Description

A wrapper around nleqslv from the nleqslv package to solve a non linear system of equations.

Usage

findRoots(beta.init, eq.func, d.eq.func = NULL, arg.list, ...)
findRoots(beta.init, eq.func, d.eq.func = NULL, arg.list, ...)

Arguments

`beta.init`	An initial guess for the zero.
`eq.func`	A function of two variables for which the zero are sought. Its first argument `beta` should be a vector over which the zeros are sought and the second argument `arg.list` a list of additional arguments.
`d.eq.func`	A function to return the Jacobian of `eq.func` taking the same arguments as `eq.func`. Supplying this function can speed up calculations. Default is `NULL`.
`arg.list`	The second argument to `eq.func` and `d.eq.func`
`...`	A list of additional arguments to be passed to `nleqslv`

Details

findRoots calculates zeros fo the function eq.func and is the default equation solving function in drgee. It is supplied as a separate function in order to allow users to use other equation solvers by writing their own wrapper with the same interface as findRoots.

Value

The value is a list containing the following arguments:

`roots`	The zero(s) of the function `eq.func`.
`optim.object`	The optimization object returned from `nleqslv`.

Author(s)

Johan Zetterqvist, Arvid Sjölander

Generalized Estimating Equations

Description

gee performs estimation of parameters in a restricted mean model using standard GEEs with independent working correlation matrix. For clustered data, cluster-robust standard errors are calculated. When cond=TRUE, cluster-specific intercepts are assumed.

Usage

gee(formula, link = c("identity", "log", "logit"), data, subset, cond = FALSE,
clusterid, clusterid.vcov, rootFinder = findRoots, ...)
gee(formula, link = c("identity", "log", "logit"), data, subset, cond = FALSE,
clusterid, clusterid.vcov, rootFinder = findRoots, ...)

Arguments

`formula`	An expression or formula representing the expected outcome conditional on covariates.
`link`	A character string naming the link function to use. Has to be `"identity"`, `"log"` or `"logit"`. Default is `"identity"`.
`data`	A data frame or environment containing the variables appearing in `formula`. If missing, the variables are expected to be found in the environment of the `formula` argument.
`subset`	An optional vector defining a subset of the data to be used.
`cond`	A logical value indicating whether cluster-specific intercepts should be assumed. Requires a `clusterid` argument.
`clusterid`	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`.
`clusterid.vcov`	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.
`rootFinder`	A function to solve a system of non linear equations. Default is `findRoots`.
`...`	Further arguments to be passed to the function `rootFinder`.

Details

Estimates parameters in a regression model, defined by formula. When cond=FALSE, the estimated coefficients are identical to those obtained with glm, but since no distributional assumptions are made, a robust variance is calculated. When cond=TRUE and link is "identity" or "log", the coefficients will be calculated using conditional estimating equations as described in Goetgeluk and Vansteelandt (2008). When cond=TRUE and link="logit", the coefficients will be calculated by conditional logistic regression (with robust standard errors).

Value

gee return an object of class gee containing:

`coefficients`	Estimates of the parameters.
`vcov`	Robust variance of the estimates.
`call`	The matched call.
`y`	The outcome vector.
`x`	The design matrix. For conditional methods there is no column corresponding to the intercept.
`optim.object`	An estimation object returned from the function specified in the `rootFinder`, if this function is called.
`res`	The residuals from the estimating equations.
`d.res`	The derivative of the residuals from the estimating equations.
`data`	The original data object, if given as an input argument
`formula`	The original formula 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 gee object. To obtain the 'naive' variance, i.e. the variance obtained from maximum likelihood estimation assuming correct parameteric model and no clustering, use the class method naiveVcov. The class method summary.drgee produces a summary of the calculations.

Author(s)

Johan Zetterqvist, Arvid Sjölander

References

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

Get the (conditional) score residuals from conditional logistic regression

Description

A function to obtain the conditional score residuals from conditional logistic regression as well as the sum of derivatives of these residuals with respect to the parameters evaluated at the estimated parameters.

Usage

getScoreResidualsFromClogit(fit, coefs, y, x, id)
getScoreResidualsFromClogit(fit, coefs, y, x, id)

Arguments

`fit`	A fitted object. Could be any object that has a class.function `coefficients`, but is intended to be used with `clogit` in the `survival` package.
`coefs`	An estimate of the parameters. If the argument `fit` is supplied, this will be replaced by `coefficients(fit)`.
`y`	The observed outcomes.
`x`	The design matrix.
`id`	The cluster identification variable.

Details

getScoreResidualsFromClogit calculates the residuals from the conditional score equations used in conditional logistic regression as we as the sums of their derivatives with respect to the paramaters.

This is useful if one wants to use a calculate the sandwich estimate of a variance where the uncertainty in the estimation of a conditional odds ratio needs to be taken into account.

Value

getScoreResidualsFromClogit return a list containing:

`U`	An $n$ x $p$ matrix, where $n$ is the number of observations and $p$ is the length of the estimated parameter.
`dU.sum`	An $p$ x $p$ matrix containing the sums of the derivatives. The rows correspond to the estimating equations and the columns correspond to the partial derivatives.
`dU.mean`	The mean cluster sum of the derivatives of the score residuals from the conditional likelihood.

Author(s)

Johan Zetterqvist, Arvid Sjölander

Robust Variance Calculation

Description

robVcov and robustVcov calculates the asymptotic variance for Z-estimators.

Usage

robustVcov(U, d.U.sum, id = NULL)
robVcov(U, d.U, id = NULL)
robustVcov(U, d.U.sum, id = NULL)
robVcov(U, d.U, id = NULL)

Arguments

`U`	A n x q matrix of the estimating equations evaluated at the estimated model parameters, where n is the number of observations and q is the number of estimating equations.
`d.U.sum`	The sum of the jacobian of `U` evaluated at the solution to the estimating equations, with rows corresponding to the estimating equations and columns corresponding to the model parameters. The number of model parameters is assumed to equal the number of estimating equations such that `d.U.sum` is a q x q square matrix.
`d.U`	The mean of `d.U.sum` taken over all observations.
`id`	A factor with levels corresponding the clusters in the data. Default is `NULL` in which case all observations are considered to be independent.

Details

For robust variance estimation, see van der Vaart (2000).

For clustered data, the rows in U are added clusterwise resulting in a cluster robust estimate of the variance.

Value

The estimated covariance matrix.

Author(s)

Johan Zetterqvist, Arvid Sjölander

References

van der Vaart, A.W. (2000), Asymptotic Statistics, Cambridge University Press, pp. 52–53.

Package 'drgee'

Help Index

Doubly Robust Generalized Estimating Equations

Description

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Cluster-center a design matrix around cluster means

Description

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Doubly Robust Generalized Estimating Equations

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Examples

Extracting Variables and Model Matrices for Generalized Estimating equations

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Non Linear Equation System Solving

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Generalized Estimating Equations

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Get the (conditional) score residuals from conditional logistic regression

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Author(s)

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Robust Variance Calculation

Description

Usage

Arguments

Details

Value

Author(s)

References