Fits a marginal model using AIPW

provides augmented inverse probability weighted estimates of parameters for semiparametric marginal model of response variable of interest. The augmented terms are estimated by using multiple imputation model.

Usage

lmeaipw(
  data,
  M = 5,
  id,
  analysis.model,
  wgt.model,
  imp.model,
  qpoints = 4,
  psiCov,
  nu,
  psi,
  sigma = NULL,
  sigmaMiss,
  sigmaR,
  dist,
  link,
  conv = 1e-04,
  maxiter,
  maxpiinv = -1,
  se = TRUE,
  verbose = FALSE
)

Arguments

data: longitudinal data with each subject specified discretely
M: number of imputation to be used in the estimation of augmentation term
id: cloumn names which shows identification number for each subject
analysis.model: A formula to be used as analysis model
wgt.model: Formula for weight model, which consider subject specific random intercept
imp.model: For for missing response imputation, which consider subject specific random intercept
qpoints: Number of quadrature points to be used while evaluating the numerical integration
psiCov: working model parameter
nu: working model parameter
psi: working model parameter
sigma: working model parameter
sigmaMiss: working model parameter
sigmaR: working model parameter
dist: distribution for imputation model. Currently available options are Gaussian, Binomial
link: Link function for the mean
conv: convergence tolerance
maxiter: maximum number of iteration
maxpiinv: maximum value pi can take
se: Logical for Asymptotic SE for regression coefficient of the regression model.
verbose: logical argument

Value

A list of objects containing the following objects

Call: details about arguments passed in the function
nr.conv: logical for checking convergence in Newton Raphson algorithm
nr.iter: number of iteration required
nr.diff: absolute difference for roots of Newton Raphson algorithm
beta: estimated regression coefficient for the analysis model
var.beta: Asymptotic SE for beta

Details

lmeaipw

It uses the augmented inverse probability weighted method to reduce the bias due to missing values in response 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 $$\sum_{i=1}^{n}\sum_{j=t_1}^{t_k}\int_{a_i}\int_{b_i}(\frac{\delta_{ij}}{\hat\pi_{ij}(a_i)}S(Y_{ij},\mathbf{X}_{ij};\beta)+(1-\frac{\delta_{ij}}{\hat\pi_{ij}(a_i)})\phi(\mathbf{V}_{ij},b_i;\psi))da_idb_i=0$$ where $\delta_{ij}=1$ if there is missing value in the response and 0 otherwise, $\mathbf{X}$ is fully observed all subjects, where $\mathbf{V}_{ij}=(\mathbf{X}_{ij},A_{ij})$. The missing score function values due to incomplete data are estimated using an imputation model through FCS (here we have considered a mixed effect model) which we have considered as $\phi(\mathbf{V}_{ij}=\mathbf{v}_{ij}))$. The estimated value $\hat{\phi}(\mathbf{V}_{ij}=\mathbf{v}_{ij}))$ is obtained through multiple imputation. The working model for imputation of missing response is $$Y_{ij}|b_i\sim N(\mathbf{V}_{ij}\psi+b_i,\sigma)\; ; b_i\sim N(0,\sigma_{miss})$$ and for the missing data probability $$Logit(P(\delta_{ij}=1|\mathbf{V}_{ij}\nu+a_i))\;;a_i\sim N(0,\sigma_R)$$

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.

Author

Atanu Bhattacharjee, Bhrigu Kumar Rajbongshi and Gajendra Kumar Vishwakarma

Examples

 if (FALSE) {
##
library(JMbayes2)
library(lme4)
library(insight)
library(numDeriv)
library(stats)
lmer(log(alkaline)~drug+age+year+(1|id),data=na.omit(pbc2))
data1<-pbc2
data1$alkaline<-log(data1$alkaline)
names(pbc2)
apply(pbc2,2,function(x){sum(is.na(x))})
r.ij<-ifelse(is.na(data1$alkaline)==T,0,1)
data1<-cbind.data.frame(data1,r.ij)
data1$drug<-factor(data1$drug,levels=c("placebo","D-penicil"),labels = c(0,1))
data1$sex<-factor(data1$sex,levels=c('male','female'),labels=c(1,0))
data1$drug<-as.numeric(as.character(data1$drug))
data1$sex<-as.numeric(as.character(data1$sex))
r.ij~year+age+sex+drug+serBilir+(1|id)
model.r<-glmer(r.ij~year+age+sex+drug+serBilir+(1|id),family=binomial(link='logit'),data=data1)
model.y<-lmer(alkaline~year+age+sex+drug+serBilir+(1|id),data=na.omit(data1))
nu<-model.r@beta
psi<-model.y@beta
sigma<-get_variance_residual(model.y)
sigmaR<-get_variance(model.r)$var.random
sigmaMiss<-get_variance(model.y)$var.random
m11<-lmeaipw(data=data1,id='id',
analysis.model = alkaline~year,
wgt.model=~year+age+sex+drug+serBilir+(1|id),
imp.model = ~year+age+sex+drug+serBilir+(1|id),
psiCov = vcov(model.y),nu=nu,psi=psi,
sigma=sigma,sigmaMiss=sigmaMiss,sigmaR=sigmaR,dist='gaussian',link='identity',
maxiter = 200)
m11
##
}