Package 'MIIPW'

Fit a geeglm model using AIPW

Description

provides augmented inverse probability weighted estimates of parameters for GEE model of response variable using different covariance structure

Usage

AIPW(
  data,
  formula,
  id,
  visit,
  family,
  init.beta = NULL,
  init.alpha = NULL,
  init.phi = NULL,
  tol = 0.001,
  weights = NULL,
  corstr = "independent",
  maxit = 50,
  m = 2,
  pMat,
  method = NULL
)

Arguments

data	longitudinal data set where each subject's outcome has been measured at same time points and number of visits for each patient is similar. Covariance structure of the outcome variable like "unstructured","independent","AR-1" ,"exchangeable"
formula	formula for the response model
id	column name of id of subjects in the dataset
visit	column name of timepoints of visit in the dataset
family	name of the distribution for the response variable, For more information on how to use family objects, see family
init.beta	initial values for the regression coefficient of GEE model
init.alpha	initial values for the correlation structure
init.phi	initial values for the csale parameter for
tol	tolerance in calculation of coefficients
weights	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. Weights are updated as the mentioned the details.
corstr	a character string specifying the correlation structure. It could "independence", "exchangeable", "AR-1", "unstructured"
maxit	maximum number iteration for newton-raphson
m	number of imputation used to update the missing score function value due incomplete data.
pMat	predictor matrix as obtained in mice
method	method option for mice model,for information see mice

Details

AIPW

It uses the inverse probability weighted method to reduce the bias due to missing values in GEE model for longitudinal data. The response variable $\mathbf{Y}$ is related to the coariates as $g(\mu)=\mathbf{X}\beta$, where g is the link function for the glm. The estimating equation is

$\sum_{i=1}^{k}\sum_{j=1}^{n}(\frac{\delta_{ij}}{\pi_{ij}}S(Y_{ij},\mathbf{X}_{ij},\mathbf{X}'_{ij})+(1-\frac{\delta_{ij}}{\pi_{ij}})\phi(\mathbf{V}=\mathbf{v}))=0$

where $\delta_{ij}=1$ if there is missing value in covariates and 0 otherwise, $\mathbf{X}$ is fully observed all subjects and $\mathbf{X}'$ is partially missing, where $\mathbf{V}=(Y,\mathbf{X})$. The missing score function values due to incomplete data are estimated using an imputation model through mice which we have considered as $\phi(\mathbf{V}=\mathbf{v}))$.

Value

A list of objects containing the following objects

call

details about arguments passed in the function

beta

estimated regression coeffictient value for the response model

niter

number of iteration required

betalist

list of beta values at different iteration

weight

estimated weights for the observations

mu

mu values according glm

phi

etsimated phi value for the glm model

hessian

estimated hessian matrix obtained from the last iteration

betaSand

sandwich estimator value for the variance covariance matrix of the beta

Author(s)

Atanu Bhattacharjee, Bhrigu Kumar Rajbongshi and Gajendra Kumar Vishwakarma

References

Wang, C. Y., Shen-Ming Lee, and Edward C. Chao. "Numerical equivalence of imputing scores and weighted estimators in regression analysis with missing covariates." Biostatistics 8.2 (2007): 468-473.

Seaman, Shaun R., and Stijn Vansteelandt. "Introduction to double robust methods for incomplete data." Statistical science: a review journal of the Institute of Mathematical Statistics 33.2 (2018): 184.

Vansteelandt, Stijn, James Carpenter, and Michael G. Kenward. "Analysis of incomplete data using inverse probability weighting and doubly robust estimators." Methodology: European Journal of Research Methods for the Behavioral and Social Sciences 6.1 (2010): 37.

See Also

SIPW,miSIPW,miAIPW

Examples

## Not run: 
##
formula<-C6kine~ActivinRIB+ActivinRIIA+ActivinRIIAB+Adiponectin+AgRP+ALCAM
pMat<-mice::make.predictorMatrix(srdata1[names(srdata1)%in%all.vars(formula)])
m1<-AIPW(data=srdata1,
formula<-formula,id='ID',
visit='Visit',family='gaussian',init.beta = NULL,
init.alpha=NULL,init.phi=1,tol=.00001,weights = NULL,
corstr = 'exchangeable',maxit=50,m=3,pMat=pMat)
##

## End(Not run)

---

Fit a geeglm model using meanScore

Description

provides mean score estimates of parameters for GEE model of response variable using different covariance structure

Usage

MeanScore(
  data,
  formula,
  id,
  visit,
  family,
  init.beta = NULL,
  init.alpha = NULL,
  init.phi = NULL,
  tol = 0.001,
  weights = NULL,
  corstr = "independent",
  maxit = 50,
  m = 2,
  pMat,
  method = NULL
)

Arguments

data	longitudinal data set where each subject's outcome has been measured at same time points and number of visits for each patient is similar. Covariance structure of the outcome variable like "unstructured","independent" ,"AR-1","exchnageable"
formula	formula for the response model
id	column name of id of subjects in the dataset
visit	column name of timepoints of visit in the dataset
family	name of the distribution for the response variable, For more information on how to use family objects, see family
init.beta	initial values for the regression coefficient of GEE model
init.alpha	initial values for the correlation structure
init.phi	initial values for the scale parameter
tol	tolerance in calculation of coefficients
weights	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. Weights are updated as the mentioned the details.
corstr	a character string specifying the correlation structure. It could "independence", "exchangeable", "AR-1", "unstructured"
maxit	maximum number iteration for newton-raphson
m	number of imputation used to update the missing score function value due incomplete data.
pMat	predictor matrix as obtained in mice
method	method option for mice model,for information see mice

Details

meanScore

It uses the mean score method to reduce the bias due to missing covariate in GEE model.The response variable $\mathbf{Y}$ is related to the coariates as $g(\mu)=\mathbf{X}\beta$, where g is the link function for the glm. The estimating equation is

$\sum_{i=1}^{k}\sum_{j=1}^{n}(\delta_{ij}S(Y_{ij},\mathbf{X}_{ij},\mathbf{X}'_{ij})+(1-\delta_{ij})\phi(\mathbf{V}=\mathbf{v}))=0$

where $\delta_{ij}=1$ if there is missing value in covariates and 0 otherwise, $\mathbf{X}$ is fully observed all subjects and $\mathbf{X}'$ is partially missing, where $\mathbf{V}=(Y,\mathbf{X})$. The missing score function values due to incomplete data are estimated using an imputation model through mice which we have considered as $\phi(\mathbf{V}=\mathbf{v}))$. The estimated value $\phi(\mathbf{V}=\mathbf{v}))$ is obtained through multiple imputation.

Value

A list of objects containing the following objects

call

details about arguments passed in the function

beta

estimated regression coeffictient value for the response model

niter

number of iteration required

betalist

list of beta values at different iteration

weight

estimated weights for the observations

mu

mu values according glm

phi

etsimated phi value for the glm model

hessian

estimated hessian matrix obtained from the last iteration

betaSand

sandwich estimator value for the variance covariance matrix of the beta

Author(s)

Atanu Bhattacharjee, Bhrigu Kumar Rajbongshi and Gajendra Kumar Vishwakarma

References

Wang, C. Y., Shen-Ming Lee, and Edward C. Chao. "Numerical equivalence of imputing scores and weighted estimators in regression analysis with missing covariates." Biostatistics 8.2 (2007): 468-473.

Seaman, Shaun R., and Stijn Vansteelandt. "Introduction to double robust methods for incomplete data." Statistical science: a review journal of the Institute of Mathematical Statistics 33.2 (2018): 184.

Vansteelandt, Stijn, James Carpenter, and Michael G. Kenward. "Analysis of incomplete data using inverse probability weighting and doubly robust estimators." Methodology: European Journal of Research Methods for the Behavioral and Social Sciences 6.1 (2010): 37.

See Also

SIPW,miSIPW,miAIPW

Examples

## Not run: 
##
formula<-C6kine~ActivinRIB+ActivinRIIA+ActivinRIIAB+Adiponectin+AgRP+ALCAM
pMat<-mice::make.predictorMatrix(srdata1[names(srdata1)%in%all.vars(formula)])
m1<-MeanScore(data=srdata1,
formula<-formula,id='ID',
visit='Visit',family='gaussian',init.beta = NULL,
init.alpha=NULL,init.phi=1,tol=.00001,weights = NULL,
corstr = 'exchangeable',maxit=50,m=2,pMat=pMat)
##

## End(Not run)

---

Fit a geeglm model using miAIPW

Description

provides augmented inverse probability weighted estimates of parameters for GEE model of response variable using different covariance structure. The augmented terms are estimated by using multiple imputation model.

Usage

miAIPW(
  data,
  formula,
  id,
  visit,
  family,
  init.beta = NULL,
  init.alpha = NULL,
  init.phi = NULL,
  tol = 0.001,
  weights = NULL,
  corstr = "independent",
  maxit = 50,
  m = 2,
  pMat,
  method = NULL
)

Arguments

data	longitudinal data set where each subject's outcome has been measured at same time points and number of visits for each patient is similar. Covariance structure of the outcome variable like "unstuctured","independent","AR1" ,"Exchageable"
formula	formula for the response model
id	column name of id of subjects in the dataset
visit	column name of timepoints of visit in the dataset
family	name of the distribution for the response variable, For more information on how to use family objects, see family
init.beta	initial values for the regression coefficient of GEE model
init.alpha	initial values for the correlation structure
init.phi	initial values for the csale parameter for
tol	tolerance in calculation of coefficients
weights	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. Weights are updated as the mentioned the details.
corstr	a character string specifying the correlation structure. It could "independent", "exchangeable", "AR-1", "unstructured"
maxit	maximum number iteration for newton-raphson
m	number of imputation used to update the missing score function value due incomplete data.
pMat	predictor matrix as obtained in mice
method	method option for mice model,for information see mice

Details

miAIPW

It uses the augmented inverse probability weighted method to reduce the bias due to missing values in GEE model for longitudinal data. The response variable $\mathbf{Y}$ is related to the coariates as $g(\mu)=\mathbf{X}\beta$, where g is the link function for the glm. The estimating equation is

$\sum_{i=1}^{k}\sum_{j=1}^{n}(\frac{\delta_{ij}}{\pi_{ij}}S(Y_{ij},\mathbf{X}_{ij},\mathbf{X}'_{ij})+(1-\frac{\delta_{ij}}{\pi_{ij}})\phi(\mathbf{V}=\mathbf{v}))=0$

where $\delta_{ij}=1$ if there is missing value in covariates and 0 otherwise, $\mathbf{X}$ is fully observed all subjects and $\mathbf{X}'$ is partially missing, where $\mathbf{V}=(Y,\mathbf{X})$. The missing score function values due to incomplete data are estimated using an imputation model through mice which we have considered as $\phi(\mathbf{V}=\mathbf{v}))$. The estimated value $\phi(\mathbf{V}=\mathbf{v}))$ is obtained through multiple imputation.

Value

A list of objects containing the following objects

call

details about arguments passed in the function

beta

estimated regression coeffictient value for the response model

niter

number of iteration required

betalist

list of beta values at different iteration

weight

estimated weights for the observations

mu

mu values according glm

phi

etsimated phi value for the glm model

hessian

estimated hessian matrix obtained from the last iteration

betaSand

sandwich estimator value for the variance covariance matrix of the beta

Author(s)

Atanu Bhattacharjee, Bhrigu Kumar Rajbongshi and Gajendra Kumar Vishwakarma

References

Wang, C. Y., Shen-Ming Lee, and Edward C. Chao. "Numerical equivalence of imputing scores and weighted estimators in regression analysis with missing covariates." Biostatistics 8.2 (2007): 468-473.

Seaman, Shaun R., and Stijn Vansteelandt. "Introduction to double robust methods for incomplete data." Statistical science: a review journal of the Institute of Mathematical Statistics 33.2 (2018): 184.

Vansteelandt, Stijn, James Carpenter, and Michael G. Kenward. "Analysis of incomplete data using inverse probability weighting and doubly robust estimators." Methodology: European Journal of Research Methods for the Behavioral and Social Sciences 6.1 (2010): 37.

See Also

SIPW,miSIPW,miAIPW

Examples

## Not run: 
##
formula<-C6kine~ActivinRIB+ActivinRIIA+ActivinRIIAB+Adiponectin+AgRP+ALCAM
pMat<-mice::make.predictorMatrix(srdata1[names(srdata1)%in%all.vars(formula)])
m1<-miAIPW(data=srdata1,
formula<-formula,id='ID',
 visit='Visit',family='gaussian',init.beta = NULL,
init.alpha=NULL,init.phi=1,tol=.00001,weights = NULL,
corstr = 'exchangeable',maxit=4,m=2,pMat=pMat)
##

## End(Not run)

---

Fit a geeglm model using miSIPW

Description

provides simple inverse probability weighted estimates of parameters for GEE model of response variable using different covariance structure, missing values in covariates are multiply imputed for those subjects whose response is observed.

Usage

miSIPW(
  data,
  formula,
  id,
  visit,
  family,
  init.beta = NULL,
  init.alpha = NULL,
  init.phi = NULL,
  tol = 0.001,
  weights = NULL,
  corstr = "independent",
  maxit = 50,
  m = 2,
  pMat,
  method = NULL
)

Arguments

data	longitudinal data set where each subject's outcome has been measured at same time points and number of visits for each patient is similar. Covariance structure of the outcome variable like "unstuctured","independent","AR-1" ,"exchageable"
formula	formula for the response model
id	column name of id of subjects in the dataset
visit	column name of timepoints of visit in the dataset
family	name of the distribution for the response variable, For more information on how to use family objects, see family
init.beta	initial values for the regression coefficient of GEE model
init.alpha	initial values for the correlation structure
init.phi	initial values for the scale parameter
tol	tolerance in calculation of coefficients
weights	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. Weights are updated as the mentioned the details.
corstr	a character string specifying the correlation structure. It could "independence", "exchangeable", "AR-1", "unstructured"
maxit	maximum number iteration for newton-raphson
m	number of imputation used to update the missing score function value due incomplete data.
pMat	pMat predictor matrix as obtained in mice
method	method option for mice model,for information see mice

Details

miSIPW

It uses the simple inverse probability weighted method to reduce the bias due to missing values in GEE model for longitudinal data. The response variable $\mathbf{Y}$ is related to the coariates as $g(\mu)=\mathbf{X}\beta$, where g is the link function for the glm. The estimating equation is

$\sum_{i=1}^{k}\sum_{j=1}^{n}\frac{\delta_{ij}}{\pi_{ij}}S(Y_{ij},\mathbf{X}_{ij},\mathbf{X}'_{ij})$

=0 where $\delta_{ij}=1$ if there is missing no value in covariates and 0 otherwise. $\mathbf{X}$ is fully observed all subjects and $\mathbf{X}'$ is partially missing.

Value

A list of objects containing the following objects

call

details about arguments passed in the function

beta

estimated regression coeffictient value for the response model

niter

number of iteration required

betalist

list of beta values at different iteration

weight

estimated weights for the observations

mu

mu values according glm

phi

etsimated phi value for the glm model

hessian

estimated hessian matrix obtained from the last iteration

betaSand

sandwich estimator value for the variance covariance matrix of the beta

Author(s)

Atanu Bhattacharjee, Bhrigu Kumar Rajbongshi and Gajendra Kumar Vishwakarma

References

Wang, C. Y., Shen-Ming Lee, and Edward C. Chao. "Numerical equivalence of imputing scores and weighted estimators in regression analysis with missing covariates." Biostatistics 8.2 (2007): 468-473.

Seaman, Shaun R., and Stijn Vansteelandt. "Introduction to double robust methods for incomplete data." Statistical science: a review journal of the Institute of Mathematical Statistics 33.2 (2018): 184.

Vansteelandt, Stijn, James Carpenter, and Michael G. Kenward. "Analysis of incomplete data using inverse probability weighting and doubly robust estimators." Methodology: European Journal of Research Methods for the Behavioral and Social Sciences 6.1 (2010): 37.

See Also

SIPW,AIPW,miAIPW

Examples

## Not run: 
##
formula<-C6kine~ActivinRIB+ActivinRIIA+ActivinRIIAB+Adiponectin+AgRP+ALCAM
pMat<-mice::make.predictorMatrix(srdata1[names(srdata1)%in%all.vars(formula)])
m1<-miSIPW(data=srdata1,
formula=formula,id='ID',
visit='Visit',family='gaussian',init.beta = NULL,
init.alpha=NULL,init.phi=1,tol=0.001,weights = NULL,
corstr = 'exchangeable',maxit=50,m=2,pMat=pMat)
##

## End(Not run)

---

print method for ipw

Description

print method for ipw

Usage

print_ipw(x, ...)

Arguments

x	ipw object
...	further argument can be passed

Value

print result for ipw object

---

print method for meanscore

Description

print method for meanscore

Usage

print_meanscore(x, ...)

Arguments

x	meanscore object
...	further argument can be passed

Value

print result for meanscore object

---

Model Selection criteria QIC

Description

It provides model selection criteria such as quasi-likelihood under the independence model criterion (QIC), an approximation to QIC under large sample i.e QICu and quasi likelihood

Usage

QICmiipw(model.R, model.indep, family)

Arguments

model.R	fitted object obtained from GEE model MeanScore,SIPW,AIPW,miSIPW,miAIPW with correlation struture other than "independent"
model.indep	same fitted object as in model.indep with "independent" correlation struture
family	currently we have inlcuded "poisson","binomial","gaussian"

Details

QICmiipw

Value

returns a list containing QIC,QICu,Quasi likelihood

References

Pan, Wei. "Akaike's information criterion in generalized estimating equations." Biometrics 57.1 (2001): 120-125.

Examples

## Not run: 
 ##
 formula<-C6kine~ActivinRIB+ActivinRIIA+ActivinRIIAB+Adiponectin+AgRP+ALCAM
 pMat<-mice::make.predictorMatrix(srdata1[names(srdata1)%in%all.vars(formula)])
 m1<-MeanScore(data=srdata1,
             formula<-formula,id='ID',
             visit='Visit',family='gaussian',init.beta = NULL,
             init.alpha=NULL,init.phi=1,tol=.00001,weights = NULL,
             corstr = 'exchangeable',maxit=50,m=2,pMat=pMat)
 m11<-MeanScore(data=srdata1,
             formula<-formula,id='ID',
             visit='Visit',family='gaussian',init.beta = NULL,
             init.alpha=NULL,init.phi=1,tol=.00001,weights = NULL,
            corstr = 'independent',maxit=50,m=2,pMat=pMat)
QICmiipw(model.R=m1,model.indep=m11,family="gaussian")
##

## End(Not run)

---

Fit a geeglm model using SIPW

Description

provides simple inverse probability weighted estimates of parameters for GEE model of response variable using different covariance structure

Usage

SIPW(
  data,
  formula,
  id,
  visit,
  family,
  init.beta = NULL,
  init.alpha = NULL,
  init.phi = NULL,
  tol = 0.001,
  weights = NULL,
  corstr = "independent",
  maxit = 10,
  maxvisit = NULL
)

Arguments

data	longitudinal data set where each subject's outcome has been measured at same time points and number of visits for each patient is similar. Covariance structure of the outcome variable like "unstructured","independent" ,"exchangeable"
formula	formula for the response model
id	column name of id of subjects in the dataset
visit	column name of timepoints of visit in the dataset
family	name of the distribution for the response variable, For more information on how to use family objects, see family
init.beta	initial values for the regression coefficient of GEE model
init.alpha	initial values for the correlation structure
init.phi	initial values for the scale parameter
tol	tolerance in calculation of coefficients
weights	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. Weights are updated as the mentioned the details.
corstr	a character string specifying the correlation structure. It could "independence", "exchangeable", "AR-1", "unstructured"
maxit	maximum number of iteration
maxvisit	maximum number of visit

Details

SIPW

It uses the simple inverse probability weighted method to reduce the bias due to missing values in GEE model for longitudinal data.The response variable $\mathbf{Y}$ is related to the coariates as $g(\mu)=\mathbf{X}\beta$, where g is the link function for the glm. The estimating equation is

$\sum_{i=1}^{k}\sum_{j=1}^{n}\frac{\delta_{ij}}{\pi_{ij}}S(Y_{ij},\mathbf{X}_{ij},\mathbf{X}'_{ij})$

=0 where $\delta_{ij}=1$ if there is missing no value in covariates and 0 otherwise. $\mathbf{X}$ is fully observed all subjects and $\mathbf{X}'$ is partially missing.

Value

A list of objects containing the following objects

call

details about arguments passed in the function

beta

estimated regression coeffictient value for the response model

niter

number of iteration required

betalist

list of beta values at different iteration

weight

estimated weights for the observations

mu

mu values according glm

phi

etsimated phi value for the glm model

hessian

estimated hessian matrix obtained from the last iteration

betaSand

sandwich estimator value for the variance covariance matrix of the beta

Author(s)

Atanu Bhattacharjee, Bhrigu Kumar Rajbongshi and Gajendra Kumar Vishwakarma

References

Wang, C. Y., Shen-Ming Lee, and Edward C. Chao. "Numerical equivalence of imputing scores and weighted estimators in regression analysis with missing covariates." Biostatistics 8.2 (2007): 468-473.

Seaman, Shaun R., and Stijn Vansteelandt. "Introduction to double robust methods for incomplete data." Statistical science: a review journal of the Institute of Mathematical Statistics 33.2 (2018): 184.

Vansteelandt, Stijn, James Carpenter, and Michael G. Kenward. "Analysis of incomplete data using inverse probability weighting and doubly robust estimators." Methodology: European Journal of Research Methods for the Behavioral and Social Sciences 6.1 (2010): 37.

See Also

AIPW,miSIPW,miAIPW

Examples

## Not run: 
##
formula<-C6kine~ActivinRIB+ActivinRIIA+ActivinRIIAB+Adiponectin+AgRP+ALCAM
m1<-SIPW(data=srdata1,formula<-formula,id='ID',
visit='Visit',family='gaussian',corstr = 'exchangeable',maxit=5)
##

## End(Not run)

---

protein data

Description

Repeated measurement dataset, for each id we have four visit observations

Usage

data(srdata1)

Format

A dataframe with 164 rows and 9 columns

ID

ID of subjects

Visit

Number of times observations recorded

C6kine,.....,GFRalpha4

These are covariates

Examples

data(srdata1)

---

summary method for ipw

Description

summary method for ipw

Usage

summary_ipw(object, ...)

Arguments

object	ipw object
...	further argument can be passed

Value

summary of ipw object

---

summary method for meanscore

Description

summary method for meanscore

Usage

summary_meanscore(object, ...)

Arguments

object	meanscore object
...	further argument can be passed

Value

summary of meanscore object

---

internal function for updating alpha

Description

internal function for updating alpha

Usage

updateALpha(y, x, vfun, mu, w, phi, corstr, ni, mv = NULL, id, visit)

Arguments

y	response value for GEE model
x	model matrix for the GEE model
vfun	variance function for the GLM
mu	mu vector for the GLM
w	weight matrix
phi	scale parameter
corstr	correlation structure
ni	list of visits per subject
mv	NULL
id	id column
visit	visit column

Details

arguments are from Fisher Scoring Algorithm

---

internal function for updating beta through Fisher Scoring

Description

internal function for updating beta through Fisher Scoring

Usage

updateBeta(y, x, vfun, mu, w, D, Ralpha, beta)

Arguments

y	response value for GEE model
x	model matrix for the GEE model
vfun	variance function for the GLM
mu	mu vector for the GLM
w	weight matrix
D	derivation of the inverse link function
Ralpha	correlation matrix
beta	vector of beta value for GEE model

---

internal function for updating scale parameter

Description

internal function for updating scale parameter

Usage

UpdatePhi(y, x, vfun, mu, w)

Arguments

y	response value for GEE model
x	model matrix for the GEE model
vfun	variance function for the GLM
mu	mu vector for the GLM
w	weight matrix

---

internal function for sandwich estimator

Description

internal function for sandwich estimator

Usage

updateSandW(y, x, vfun, mu, w, D, Ralpha, beta, hessmat, blockdiag)

Arguments

y	response value for GEE model
x	model matrix for the GEE model
vfun	variance function for the GLM
mu	mu vector for the GLM
w	weight matrix
D	derivation of the inverse link function
Ralpha	correlation matrix
beta	vector of beta value for GEE model
hessmat	hessian matrix
blockdiag	vector containing the dim of block matrix for block diagonal matrix

Details

arguments are required for obtaining Sandwich Estimator for variance matrix of regression coefficient of GEE model

---