# JM Imputation of clustered data with continuous variables only with cluster-specific covariance matrices Impute a clustered dataset with continuous outcomes only. A joint multivariate model for partially observed data is assumed and imputations are generated through the use of a Gibbs sampler. A different covariance matrix is estimated within each cluster. Categorical covariates may be considered, but they have to be included with dummy variables. ```r jomo1ranconhr(Y, X=NULL, Z=NULL, clus, beta.start=NULL, u.start=NULL, l1cov.start=NULL, l2cov.start=NULL, l1cov.prior=NULL, l2cov.prior=NULL, nburn=1000, nbetween=1000, nimp=5, a=(ncol(Y)+50),a.prior=NULL, meth="random", output=1, out.iter=10) ``` ## Arguments - `Y`: A data frame, or matrix, with responses of the joint imputation model. Rows correspond to different observations, while columns are different variables. Missing values are coded as NA. - `X`: A data frame, or matrix, with covariates of the joint imputation model. Rows correspond to different observations, while columns are different variables. Missing values are not allowed in these variables. In case we want an intercept, a column of 1 is needed. The default is a column of 1. - `Z`: A data frame, or matrix, for covariates associated to random effects in the joint imputation model. Rows correspond to different observations, while columns are different variables. Missing values are not allowed in these variables. In case we want an intercept, a column of 1 is needed. The default is a column of 1. - `clus`: A data frame, or matrix, containing the cluster indicator for each observation. - `beta.start`: Starting value for beta, the vector(s) of fixed effects. Rows index different covariates and columns index different outcomes. The default is a matrix of zeros. - `u.start`: A matrix where different rows are the starting values within each cluster for the random effects estimates u. The default is a matrix of zeros. - `l1cov.start`: Starting value for the covariance matrices, stacked one above the other. Dimension of each square matrix is equal to the number of outcomes in the imputation model. The default is the identity matrix for each cluster. - `l2cov.start`: Starting value for the level 2 covariance matrix. Dimension of this square matrix is equal to the number of outcomes in the imputation model times the number of random effects. The default is an identity matrix. - `l1cov.prior`: Scale matrix for the inverse-Wishart prior for the covariance matrices. The default is the identity matrix. - `l2cov.prior`: Scale matrix for the inverse-Wishart prior for the level 2 covariance matrix. The default is the identity matrix. - `nburn`: Number of burn in iterations. Default is 1000. - `nbetween`: Number of iterations between two successive imputations. Default is 1000. - `nimp`: Number of Imputations. Default is 5. - `a`: Starting value for the degrees of freedom of the inverse Wishart distribution of the cluster-specific covariance matrices. Default is 50+D, with D being the dimension of the covariance matrices. - `a.prior`: Hyperparameter (Degrees of freedom) of the chi square prior distribution for the degrees of freedom of the inverse Wishart distribution for the cluster-specific covariance matrices. Default is D, with D being the dimension of the covariance matrices. - `meth`: This can be set to "Fixed" or "Random". In the first case the function will consider fixed study-specific covariance matrices, in the second, random study-specific distributed according to an inverse-Wishart distribution. - `output`: When set to any value different from 1 (default), no output is shown on screen at the end of the process. - `out.iter`: When set to K, every K iterations a dot is printed on screen. Default is 10. ## Details The Gibbs sampler algorithm used is similar to the one described in detail in Chapter 9 of Carpenter and Kenward (2013), where we exclude the presence of level 2 variables and we estimate separetely different covariance matrices within each study. When option meth="random" is specified, all the covariance matrices ae assumed to be random draws from the same underlying inverse Wishart distributions. Details of this algorithm may be found in (Yucel, 2011). Regarding the choice of the priors, a flat prior is considered for beta, while an inverse-Wishart prior is given to the covariance matrices, with p-1 degrees of freedom, aka the minimum possible, to guarantee the greatest uncertainty. Binary or continuous covariates in the imputation model may be considered without any problem, but when considering a categorical covariate it has to be included with dummy variables (binary indicators) only. ## Returns On screen, the posterior mean of the fixed effects estimates and of the covariance matrix are shown. The only argument returned is the imputed dataset in long format. Column "Imputation" indexes the imputations. Imputation number 0 are the original data. ## References Carpenter J.R., Kenward M.G., (2013), Multiple Imputation and its Application. Chapter 9, Wiley, ISBN: 978-0-470-74052-1. Yucel R.M., (2011), Random-covariances and mixed-effects models for imputing multivariate multilevel continuous data, Statistical Modelling, 11 (4), 351-370, DOI: 10.1177/1471082X100110040. ## Examples ```r # we define the inputs # nimp, nburn and nbetween are smaller than they should. This is #just because of CRAN policies on the examples. Y<-cldata[,c("measure","age")] clus<-cldata[,c("city")] X=data.frame(rep(1,1000),cldata[,c("sex")]) colnames(X)<-c("const", "sex") Z<-data.frame(rep(1,1000)) beta.start<-matrix(0,2,2) u.start<-matrix(0,10,2) l1cov.start<-matrix(diag(1,2),20,2,2) l2cov.start<-diag(1,2) l1cov.prior=diag(1,2); nburn=as.integer(50); nbetween=as.integer(20); nimp=as.integer(5); l2cov.prior=diag(1,5); a=3 # Finally we run either the model with fixed or random cluster-specific covariance matrices: imp<-jomo1ranconhr(Y,X,Z,clus,beta.start,u.start,l1cov.start, l2cov.start, l1cov.prior,l2cov.prior,nburn,nbetween,nimp,meth="fixed") cat("Original value was missing(",imp[4,1],"), imputed value:", imp[1004,1]) #or: #imp<-jomo1ranconhr(Y,X,Z,clus,beta.start,u.start,l1cov.start, l2cov.start, # l1cov.prior,l2cov.prior,nburn,nbetween,nimp,a,meth="random") # Check help page for function jomo to see how to fit the model and # combine estimates with Rubin's rules ```

jomo1ranconhr function