# Generate two/three-level simulation data This function generates a two/three-level dataset with given parameters. ```r sim.data.multi(Z.list, N, K = 1, Theta, Sigma, Psi = diag(rep(1, 3)), Lambda = diag(rep(1, 3))) ``` ## Arguments - `Z.list`: a list of data. When `K = 1` (a two-level dataset), each list is a vector containing the treatment/exposure assignment of the trials for each subject; When `K > 1` (a three-level dataset), each list is a list of length `K`, and each contains a vector of treatment/exposure assignment. - `N`: an integer, indicates the number of subjects. - `K`: an integer, indicates the number of sessions of each subject. - `Theta`: a 2 by 2 matrix, containing the population level model coefficients. - `Sigma`: a 2 by 2 matrix, is the covariance matrix of the model errors in the single-level model. - `Psi`: the covariance matrix of the random effects in the mixed effects model of the model coefficients. This is used only when `K > 1`. Default is a 3-dimensional identity matrix. - `Lambda`: the covariance matrix of the model errors in the mixed effects model if `K > 1` or the linear model if `K = 1` of the model coefficients. ## Details When `K > 1` (three-level data), for the $n_{ik}$ trials in each session of each subject, the single level mediation model is $$ M_{ik}=Z_{ik}A_{ik}+E_{1ik}, $$ $$ R_{ik}=Z_{ik}C_{ik}+M_{ik}B_{ik}+E_{2ik}, $$ where $Z_{ik}$, $M_{ik}$, $R_{ik}$, $E_{1ik}$, and $E_{2ik}$ are vectors of length $n_{ik}$. `Sigma` is the covariance matrix of $(E_{1ik},E_{2ik})$ (for simplicity, `Sigma` is the same across sessions and subjects). For coefficients $A_{ik}$, $B_{ik}$ and $C_{ik}$, we assume a mixed effects model. The random effects are from a trivariate normal distribution with mean zero and covariance `Psi`; and the model errors are from a trivariate normal distribution with mean zero and covariance `Lambda`. For the fixed effects $A$, $B$ and $C$, the values are specified in `Theta` with `Theta[1,1] = A`, `Theta[1,2] = C` and `Theta[2,2] = B`. When `K = 1` (two-level data), the single-level model coefficients $A_{i}$, $B_{i}$ and $C_{i}$ are assumed to follow a trivariate linear regression model, where the population level coefficients are specified in `Theta` and the model errors are from a trivariate normal distribution with mean zero and covariance `Lambda`. See Section 5.2 of the reference for details. ## Returns - **data**: a list of data. When `K = 1`, each list is a contains a dataframe; when `K > 1`, each list is a list of length `K`, and within each list is a dataframe. - **A**: the value of $A$s. When `K = 1`, it is a vector of length `N`; when `K > 1`, it is a `N` by `K` matrix. - **B**: the value of $B$s. When `K = 1`, it is a vector of length `N`; when `K > 1`, it is a `N` by `K` matrix. - **C**: the value of $C$s. When `K = 1`, it is a vector of length `N`; when `K > 1`, it is a `N` by `K` matrix. - **type**: a character indicates the type of the dataset. When `K = 1`, `type = twolevel`; when `K > 1`, `type = multilevel` ## References Zhao, Y., & Luo, X. (2014). **Estimating Mediation Effects under Correlated Errors with an Application to fMRI.** arXiv preprint arXiv:1410.7217. ## Author(s) Yi Zhao, Brown University, yi_zhao@brown.edu ; Xi Luo, Brown University, xi.rossi.luo@gmail.com ## Examples ```r ################################################### # Generate a two-level dataset # covariance matrix of errors delta<-0.5 Sigma<-matrix(c(1,2*delta,2*delta,4),2,2) # model coefficients A0<-0.5 B0<--1 C0<-0.5 Theta<-matrix(c(A0,0,C0,B0),2,2) # number of subjects, and trials of each set.seed(2000) N<-50 K<-1 n<-matrix(NA,N,K) for(i in 1:N) { n0<-rpois(1,100) n[i,]<-rpois(K,n0) } # treatment assignment list set.seed(100000) Z.list<-list() for(i in 1:N) { Z.list[[i]]<-rbinom(n[i,1],size=1,prob=0.5) } # Lambda rho.AB=rho.AC=rho.BC<-0 Lambda<-matrix(0,3,3) lambda2.alpha=lambda2.beta=lambda2.gamma<-0.5 Lambda[1,2]=Lambda[2,1]<-rho.AB*sqrt(lambda2.alpha*lambda2.beta) Lambda[1,3]=Lambda[3,1]<-rho.AC*sqrt(lambda2.alpha*lambda2.gamma) Lambda[2,3]=Lambda[3,2]<-rho.BC*sqrt(lambda2.beta*lambda2.gamma) diag(Lambda)<-c(lambda2.alpha,lambda2.beta,lambda2.gamma) # Data set.seed(5000) re.dat<-sim.data.multi(Z.list=Z.list,N=N,K=K,Theta=Theta,Sigma=Sigma,Lambda=Lambda) data2<-re.dat$data ################################################### ################################################### # Generate a three-level dataset # covariance matrix of errors delta<-0.5 Sigma<-matrix(c(1,2*delta,2*delta,4),2,2) # model coefficients A0<-0.5 B0<--1 C0<-0.5 Theta<-matrix(c(A0,0,C0,B0),2,2) # number of subjects, and trials of each set.seed(2000) N<-50 K<-4 n<-matrix(NA,N,K) for(i in 1:N) { n0<-rpois(1,100) n[i,]<-rpois(K,n0) } # treatment assignment list set.seed(100000) Z.list<-list() for(i in 1:N) { Z.list[[i]]<-list() for(j in 1:K) { Z.list[[i]][[j]]<-rbinom(n[i,j],size=1,prob=0.5) } } # Psi and Lambda sigma2.alpha=sigma2.beta=sigma2.gamma<-0.5 theta2.alpha=theta2.beta=theta2.gamma<-0.5 Psi<-diag(c(sigma2.alpha,sigma2.beta,sigma2.gamma)) Lambda<-diag(c(theta2.alpha,theta2.beta,theta2.gamma)) # Data set.seed(5000) re.dat<-sim.data.multi(Z.list,N,K,Theta,Sigma,Psi,Lambda) data3<-re.dat$data ################################################### ```

sim.data.multi function