Doubly Robust Inverse Probability Weighted Augmented GEE estimator

Description

The CRTgeeDR package allows you to estimates parameters in a regression model (with possibly a link function). It allows treatment augmentation and IPW for missing data alone.

Details

The only function you're likely to need from CRTgeeDR is geeDREstimation. Otherwise refer to the help documentation.

The data.sim Dataset.

Description

HIV risk of infection after STI/HIV intervention in a cluster randomized trial.

Format

A data frame with 10000 rows and 8 variables

Details

A dataset containing the HIV risk scores and presence of risky behaviors (yes/no) and other covarites of 10000 subjects among 100 communities. The variables are as follows:

• IDPAT subject id

• CLUSTER cluster id

• TRT treatment status, 1 is received STI/HIV intervention

• X1 A covariate following a N(0,1)

• JOB employement status

• MARRIED marital status

• AGE age

• HIV.KNOW Score for HIV knowlege

• RELIGION religiosity score

• OUTCOME Binary outcome - 1 if the subject is at high risk of HIV infection, 0 otherwise. NA if missing.

• MISSING 1 if the ouctome is missing - 0 otherwise.

Fit CRTgeeDR object.

Description

Fit CRTgeeDR object to a dataset

Usage

## S3 method for class 'CRTgeeDR'
fitted(object, ...)

Arguments

Doubly Robust Inverse Probability Weighted Augmented GEE Estimator

Description

This function implements a GEE estimator. It implements classical GEE, IPW-GEE, augmented GEE and IPW-Augmented GEE (Doubly robust).

Usage

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)

Arguments

Details

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}}.

Value

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

Author(s)

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.]

References

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.

Examples

data(data.sim)
 ## Not run: 
 #### 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)
 
## End(Not run)
 #### 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)

Get Mean, Sd and CI for estimates from CRTgeeDR object.

Description

Get the estimates, standard deviations and confidence intervals from an CRTgeeDR object associated with a regressor given in argument.

Usage

getCI(object, nameTRT = "TRT", quantile = 1.96)

Arguments

Get the observed vs fitted residuals

Description

Get the histogram and some basic statistics for the weights used in the IPW part.

Usage

getOMPlot(object, save = FALSE, name = "plotOM", typeplot = 0)

Arguments

Get the histogram of weights for IPW and adequation for the glm weights model

Description

Get the histogram and some basic statistics for the weights used in the IPW part.

Usage

getPSPlot(object, save = FALSE, name = "plotPS", typeplot = NULL)

Arguments

Predict CRTgeeDR object.

Description

Predict CRTgeeDR object to a dataset

Usage

## S3 method for class 'CRTgeeDR'
predict(object, newdata = NULL, ...)

Arguments

Prints CRTgeeDR object.

Description

Prints CRTgeeDR object

Usage

## S3 method for class 'CRTgeeDR'
print(x, ...)

Arguments

Print the summarizing CRTgeeDR object.

Description

Print Summary CRTgeeDR object

Usage

## S3 method for class 'summary.CRTgeeDR'
print(x, ...)

Arguments

Summarizing CRTgeeDR object.

Description

Summary CRTgeeDR object

Usage

## S3 method for class 'CRTgeeDR'
summary(object, ...)

Arguments