This function implements a GEE estimator. It implements classical GEE, IPW-GEE, augmented GEE and IPW-Augmented GEE (Doubly robust).
geeDREstimation(formula, id, data = parent.frame(), family = gaussian,
corstr = "independence", Mv = 1, weights = NULL, aug = NULL,
pi.a = 1/2, corr.mat = NULL, init.beta = NULL, init.alpha = NULL,
init.phi = 1, scale.fix = FALSE, sandwich = TRUE, maxit = 20,
tol = 1e-05, print.log = FALSE, typeweights = "VW", nameTRT = "TRT",
model.weights = NULL, model.augmentation.trt = NULL,
model.augmentation.ctrl = NULL, stepwise.augmentation = FALSE,
stepwise.weights = FALSE, nameMISS = "MISSING", nameY = "OUTCOME",
sandwich.nuisance = FALSE, fay.adjustment = FALSE, fay.bound = 0.75)An object of type 'CRTgeeDR'
$beta Final values for regressors estimates
$phi scale parameter estimate
$alpha Final values for association parameters in the working correlation structure when exchangeable
$coefnames Name of the regressors in the main regression
$niter Number of iteration done by the algorithm before convergence
$converged convergence status
$var.naiv Variance of the estimates model based (naive)
$var Variance of the estimates sandwich
$var.nuisance Variance of the estimates nuisance adjusted sandwich
$var.fay Variance of the estimates nuisance adjusted sandwich with Fay correction for small samples
$call Call function
$corr Correlation structure used
$clusz Number of unit in each cluster
$FunList List of function associated with the family
$X design matrix for the main regression
$offset Offset specified in the regression
$eta predicted values
$weights Weights vector used in the diagonal term for the IPW
$ps.model Summary of the regression fitted for the PS if computed internally
$om.model.trt Summary of the regression fitted for the OM for treated if computed internally
$om.model.ctrl Summary of the regression fitted for the OM for control if computed internally
an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.
a vector which identifies the clusters. The length of "id" should be the same as the number of observations. Data are assumed to be sorted so that observations on a cluster are contiguous rows for all entities in the formula.
an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which CRTgeeDR is called.
a description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. (See family for details of family functions.)
a character string specifying the correlation structure. The following are permitted: '"independence"', '"exchangeable"', '"ar1"', '"unstructured"' and '"userdefined"'
for "m-dependent", the value for m
A vector of weights for each observation. If an observation has weight 0, it is excluded from the calculations of any parameters. Observations with a NA anywhere (even in variables not included in the model) will be assigned a weight of 0.
A list of vector (one for A=1 treated, one for A=0 control) for each observation representing E(Y|X,A=a).
A number, Probability of treatment attribution P(A=1)
The correlation matrix for "fixed". Matrix should be symmetric with dimensions >= the maximum cluster size. If the correlation structure is "userdefined", then this is a matrix describing which correlations are the same.
an optional vector with the initial values of beta. If not specified, then the intercept will be set to InvLink(mean(response)). init.beta must be specified if not using an intercept.
an optional scalar or vector giving the initial values for the correlation. If provided along with Mv>1 or unstructured correlation, then the user must ensure that the vector is of the appropriate length.
an optional initial overdispersion parameter. If not supplied, initialized to 1.
if set to TRUE, then the scale parameter is fixed at the value of init.phi.
if set to TRUE, the sandwich variance is provided together with the naive estimator of variance.
maximum number of iterations.
tolerance in calculation of coefficients.
if set to TRUE, a report is printed.
a character string specifying the weights implementation. The following are permitted: "GENMOD" for \(W^{1/2}V^{-1}W^{1/2}\), "WV" for \(V^{-1}W\)
Name of the variable containing information for the treatment
an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted for the propensity score. Must model the probability of being observed.
an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted for the ouctome model for the treated group (A=1).
an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted for the ouctome model for the control group (A=0).
if set to TRUE, a stepwise for the augmentation model is performed during the fit of the augmentation model for the OM
if set to TRUE, a stepwise for the propensity score is performed during the fit of the augmentation model for the OM
Name of the variable containing information for the Missing indicator
Name of the variable containing information for the outcome
if set to TRUE, the nuisance adjusted sandwich variance is provided.
if set to TRUE, the small-sample nuisance adjusted sandwich variance with Fay's adjustement is provided.
if set to 0.75 by default, bound value used for Fay's adjustement.
Melanie Prague [based on R packages 'geeM' L. S. McDaniel, N. C. Henderson, and P. J. Rathouz. Fast Pure R Implementation of GEE: Application of the Matrix Package. The R Journal, 5(1):181-188, June 2013.]
The estimator is founds by solving: $$ 0= \sum_{i=1}^M \Bigg[ \boldsymbol D_i^T \boldsymbol V_i^{-1} \boldsymbol W_i(\boldsymbol X_i, A_i, \boldsymbol \eta_W) \left( \boldsymbol Y_i - \boldsymbol B(\boldsymbol X_i, A_i, \boldsymbol \eta_B) \right) $$ $$ \qquad + \sum_{a=0,1} p^a(1-p)^{1-a} \boldsymbol D_i^T \boldsymbol V_i^{-1} \Big( \boldsymbol B(\boldsymbol X_i,A_i=a, \boldsymbol \eta_B) -\boldsymbol \mu_i(\boldsymbol \beta,A_i=a)\Big) \Bigg]$$ where \(\boldsymbol D_i=\frac{\partial \boldsymbol \mu_i(\boldsymbol \beta,A_i)}{\partial \boldsymbol \beta^T}\) is the design matrix, \(\boldsymbol V_i\) is the covariance matrix equal to \(\boldsymbol U_i^{1/2} \boldsymbol C(\boldsymbol \alpha)\boldsymbol U_i^{1/2}\) with \(\boldsymbol U_i\) a diagonal matrix with elements \({\rm var}(y_{ij})\) and \(\boldsymbol C(\boldsymbol \alpha)\) is the working correlation structure with non-diagonal terms \(\boldsymbol \alpha\). Parameters \(\boldsymbol \alpha\) are estimated using simple moment estimators from the Pearson residuals. The matrix of weights \(\boldsymbol W_i(\boldsymbol X_i, A_i, \boldsymbol \eta_W)=diag\left[R_{ij}/\pi_{ij}(\boldsymbol X_i, A_i, \boldsymbol \eta_W)\right]_{j=1,\dots,n_{i}}\), where \(\pi_{ij}(\boldsymbol X_i, A_i, \boldsymbol \eta_W)=P(R_{ij}|\boldsymbol X_i, A_i)\) is the Propensity score (PS). The function \(\boldsymbol B(\boldsymbol X_i,A_i=a,\boldsymbol \eta_B)\), which is called the Outcome Model (OM), is a function linking \(Y_{ij}\) with \(\boldsymbol X_i\) and \(A_i\). The \(\boldsymbol \eta_B\) are nuisance parameters that are estimated. The estimator is most efficient if the OM is equal to \(E(\boldsymbol Y_i|\boldsymbol X_i,A_i=a)\) The estimator denoted \(\hat{\beta}_{aug}\) is found by solving the estimating equation. Although analytic solutions sometimes exist, coefficient estimates are generally obtained using an iterative procedure such as the Newton-Raphson method. Automatic implementation is such that, \(\hat{ \boldsymbol \eta}_W\) in \(\boldsymbol W_i(\boldsymbol X_i, A_i, \hat{ \boldsymbol \eta}_W)\) are obtained using a logistic regression and \(\hat{ \boldsymbol \eta}_B\) in \(\boldsymbol B(\boldsymbol X_i,A_i,\hat{ \boldsymbol \eta}_B)\) are obtained using a linear regression.
The variance of \(\hat{\boldsymbol \beta}_{aug}\) is estimated by the sandwich variance estimator. There are two external sources of variability that need to be accounted for: estimation of \(\boldsymbol \eta_W\) for the PS and of \(\boldsymbol \eta_B\) for the OM. We denote \(\boldsymbol \Omega=(\boldsymbol \beta, \boldsymbol \eta_W,\boldsymbol \eta_B)\) the estimated parameters of interest and nuisance parameters. We can stack estimating functions and score functions for \(\boldsymbol \Omega\): $$\small \boldsymbol U_i(\boldsymbol \Omega)= \left( \begin{array}{c} \boldsymbol \Phi_i(\boldsymbol Y_i,\boldsymbol X_i,A_i,\boldsymbol \beta, \boldsymbol \eta_W, \boldsymbol \eta_B) \\ \boldsymbol S^W_i(\boldsymbol X_i, A_i, \boldsymbol \eta_W)\\ \boldsymbol S^B_i(\boldsymbol X_i, A_i, \boldsymbol \eta_B)\\ \end{array} \right)$$ where \(\boldsymbol S^W_i\) and \(\boldsymbol S^B_i\) represent the score equations for patients in cluster \(i\) for the estimation of \(\boldsymbol \eta_W\) and \(\boldsymbol \eta_B\) in the PS and the OM. A standard Taylor expansion paired with Slutzky's theorem and the central limit theorem give the sandwich estimator adjusted for nuisance parameters estimation in the OM and PS: $$Var(\boldsymbol \Omega)={{E\left[\frac{\partial \boldsymbol U_i(\boldsymbol \Omega)}{\partial \boldsymbol \Omega}\right]}^{-1}}^{T} \underbrace{{E\left[ \boldsymbol U_i(\boldsymbol \Omega)\boldsymbol U_i^T(\boldsymbol \Omega) \right]}}_{\boldsymbol \Delta_{adj}} \underbrace{E\left[\frac{\partial \boldsymbol U_i(\boldsymbol \Omega)}{\partial \boldsymbol \Omega}\right]^{-1} }_{\boldsymbol \Gamma^{-1}_{adj}}.$$
Details regarding implementation can be found in
'Augmented GEE for improving efficiency and validity of estimation in cluster randomized trials by leveraging cluster-and individual-level covariates' - 2012 - Stephens A., Tchetgen Tchetgen E. and De Gruttola V. : Stat Med 31(10) - 915-930.
'Accounting for interactions and complex inter-subject dependency for estimating treatment effect in cluster randomized trials with missing at random outcomes' - 2015 - Prague M., Wang R., Stephens A., Tchetgen Tchetgen E. and De Gruttola V. : in revision.
'Fast Pure R Implementation of GEE: Application of the Matrix Package' - 2013 - McDaniel, Lee S and Henderson, Nicholas C and Rathouz, Paul J : The R Journal 5(1) - 181-197.
'Small-Sample Adjustments for Wald-Type Tests Using Sandwich Estimators' - 2001 - Fay, Michael P and Graubard, Barry I : Biometrics 57(4) - 1198-1206.
data(data.sim)
if (FALSE) {
#### STANDARD GEE
geeresults<-geeDREstimation(formula=OUTCOME~TRT,
id="CLUSTER" , data = data.sim,
family = "binomial", corstr = "independence")
summary(geeresults)
#### IPW GEE
ipwresults<-geeDREstimation(formula=OUTCOME~TRT,
id="CLUSTER" , data = data.sim,
family = "binomial", corstr = "independence",
model.weights=I(MISSING==0)~TRT*AGE)
summary(ipwresults)
#### AUGMENTED GEE
augresults<-geeDREstimation(formula=OUTCOME~TRT,
id="CLUSTER" , data = data.sim,
family = "binomial", corstr = "independence",
model.augmentation.trt=OUTCOME~AGE,
model.augmentation.ctrl=OUTCOME~AGE, stepwise.augmentation=FALSE)
summary(augresults)
}
#### DOUBLY ROBUST
drresults<-geeDREstimation(formula=OUTCOME~TRT,
id="CLUSTER" , data = data.sim,
family = "binomial", corstr = "independence",
model.weights=I(MISSING==0)~TRT*AGE,
model.augmentation.trt=OUTCOME~AGE,
model.augmentation.ctrl=OUTCOME~AGE, stepwise.augmentation=FALSE)
summary(drresults)
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