# WEIGHT MATRICES FOR THE ESTIMATING EQUATIONS Weight matrices for the estimating equations. ```r weightMat(b,gam,rh,xdat,ydat,id,tvec,margmodel,corstr,link) weightMat.ord(b,gam,rh,xdat,ydat,id,tvec,corstr,link) ``` ## Arguments - `b`: The regression coefficients. - `gam`: The uinivariate parameters that are not regression coefficients. That is the parameter $\gamma$ of negative binomial distribution or the $q$-dimensional vector of the univariate cutpoints of ordinal model. $\gamma$ is NULL for Poisson and binary regression. - `rh`: The vector of normal copula parameters. - `xdat`: c("$(\\mathbf{x}_1 , \\mathbf{x}_2 , \\ldots ,\n$", "$ \\mathbf{x}_n )^\\top$"), where the matrix $\mathbf{x}_i,\,i=1,\ldots,n$ for a given unit will depend on the times of observation for that unit ($j_i$) and will have number of rows $j_i$, each row corresponding to one of the $j_i$ elements of $y_i$ and $p$ columns where $p$ is the number of covariates including the unit first column to account for the intercept (except for ordinal regression where there is no intercept). This xdat matrix is of dimension $(N\times p),$ where $N =\sum_{i=1}^n j_i$ is the total number of observations from all units. - `ydat`: $(y_1 , y_2 , \ldots , y_n )^\top$, where the reponse data vectors $y_i, i=1,\ldots,n$ are of possibly different lengths for different units. In particular, we now have that $y_i$ is ($j_i \times 1$), where $j_i$ is the number of observations on unit $i$. The total number of observations from all units is $N =\sum_{i=1}^n j_i$. The ydat are the collection of data vectors $y_i, i = 1,\ldots,n$ one from each unit which summarize all the data together in a single, long vector of length $N$. - `id`: An index for individuals or clusters. - `tvec`: A vector with the time indicator of individuals or clusters. - `margmodel`: Indicates the marginal model. Choices are poisson for Poisson, bernoulli for Bernoulli, and nb1 , nb2 for the NB1 and NB2 parametrization of negative binomial in Cameron and Trivedi (1998). - `corstr`: Indicates the latent correlation structure of normal copula. Choices are exch , ar , and unstr for exchangeable, ar(1) and unstrucutred correlation structure, respectively. - `link`: The link function. Choices are log for the log link function, logit for the logit link function, and probit for the probit link function. ## Details The fixed weight matrices $W_{i,\rm working}$ based on a working discretized MVN, of the weighted scores equations in Nikoloulopoulos et al. (2011) $$ g_1= g_1(a)=\sum_{i=1}^n X_i^T\,W_{i,\rm working}^{-1}\, s_i(a)=0, $$ where c("$W_{i,\\rm working}^{-1}=\\Delta_i\\Omega_{i,\\rm working}^{-1}=\n$", "$\\Delta_i({\\tilde a})\\Omega_i({\\tilde a},{\\tilde R})^{-1}$") is based on the covariance matrix of $s_i(a)$ computed from the fitted discretized MVN model with estimated parameters c("${\\tilde a},\n$", "${\\tilde R}$"). Note that `weightMat.ord` is a variant of the code for ordinal (probit and logistic) regression. ## Returns A list containing the following components: - **omega**: The array with the $\Omega_i,\ i=1,\ldots,n$ matrices. - **delta**: The array with the $\Delta_i,\ i=1,\ldots,n$ matrices. - **X**: The array with the $X_i,\ i=1,\ldots,n$ matrices. ## See Also `wtsc`, `solvewtsc`, `godambe`, `wtsc.wrapper` ## References Nikoloulopoulos, A.K., Joe, H. and Chaganty, N.R. (2011) Weighted scores method for regression models with dependent data. **Biostatistics**, 12 , 653--665. tools:::Rd_expr_doi("10.1093/biostatistics/kxr005") . Nikoloulopoulos, A.K. (2016) Correlation structure and variable selection in generalized estimating equations via composite likelihood information criteria. **Statistics in Medicine**, 35 , 2377--2390. tools:::Rd_expr_doi("10.1002/sim.6871") . Nikoloulopoulos, A.K. (2017) Weighted scores method for longitudinal ordinal data. **Arxiv e-prints**, . [https://arxiv.org/abs/1510.07376](https://arxiv.org/abs/1510.07376). ## Author(s) Aristidis K. Nikoloulopoulos A.Nikoloulopoulos@uea.ac.uk Harry Joe harry.joe@ubc.ca ## Examples ```r ################################################################################ # binary regression ################################################################################ ################################################################################ # read and set up the data set ################################################################################ data(toenail) xdat<-cbind(1,toenail$treat,toenail$time,toenail$treat*toenail$time) # response ydat<-toenail$y #id id<-toenail$id #time tvec<-toenail$time ################################################################################ # select the marginal model ################################################################################ margmodel="bernoulli" link="probit" ################################################################################ # select the correlation structure ################################################################################ corstr="ar" ################################################################################ # perform CL1 estimation ################################################################################ i.est<-iee(xdat,ydat,margmodel,link) cat("\niest: IEE estimates\n") print(c(i.est$reg,i.est$gam)) # est.rho<-cl1(b=i.est$reg,gam=i.est$gam,xdat,ydat,id,tvec,margmodel,corstr,link) # cat("\nest.rho: CL1 estimates\n") # print(est.rho$e) # [1] 0.8941659 ################################################################################ # obtain the fixed weight matrices ################################################################################ WtScMat<-weightMat(b=i.est$reg,gam=i.est$gam,rh=0.8941659, xdat,ydat,id,tvec,margmodel,corstr,link) ################################################################################ # Ordinal regression ################################################################################ ################################################################################ # read and set up data set ################################################################################ data(arthritis) nn=nrow(arthritis) bas2<-bas3<-bas4<-bas5<-rep(0,nn) bas2[arthritis$b==2]<-1 bas3[arthritis$b==3]<-1 bas4[arthritis$b==4]<-1 bas5[arthritis$b==5]<-1 t2<-t3<-rep(0,nn) t2[arthritis$ti==3]<-1 t3[arthritis$ti==5]<-1 xdat=cbind(t2,t3,arthritis$trt,bas2,bas3,bas4,bas5,arthritis$age) ydat=arthritis$y id<-arthritis$id #time tvec<-arthritis$time ################################################################################ # select the link ################################################################################ link="probit" ################################################################################ # select the correlation structure ################################################################################ corstr="exch" ################################################################################ # perform CL1 estimation ################################################################################ i.est<-iee.ord(xdat,ydat,link) cat("\niest: IEE estimates\n") print(c(i.est$reg,i.est$gam)) est.rho<-cl1.ord(b=i.est$reg,gam=i.est$gam,xdat,ydat,id,tvec,corstr,link) WtScMat<-weightMat.ord(b=i.est$reg,gam=i.est$gam,rh=est.rho$e,xdat,ydat,id,tvec,corstr,link) ```

weightMat function