Doubly Robust Generalized Estimating Equations package

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 Sjolander 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)

Arguments

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 Sjolander

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, ...)

Arguments

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 logit\{E(Y|A=0,L_1,L_2;\gamma_0, \gamma_1, \gamma_2)\}=\gamma_0 + \gamma_1 L_1 + \gamma_2 L_2 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:

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 Sjolander

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., Sjolander (2016), Doubly Robust Methods for Handling Confounding by Cluster, Biostatistics, 17, 2, 264–276

See Also

gee for O-estimation and robVcov for estimation of variance.

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)

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)

Arguments

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

Author(s)

Johan Zetterqvist, Arvid Sjolander

See Also

drgee and gee.

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, ...)

Arguments

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:

Author(s)

Johan Zetterqvist, Arvid Sjolander

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, ...)

Arguments

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:

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 Sj?lander

References

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

See Also

glm

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)

Arguments

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:

Author(s)

Johan Zetterqvist, Arvid Sj?lander

See Also

robVcov.

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)

Arguments

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.