# Spatial Modeling by a Bayesian Hierarchical Linear Mixed-effects Model A linear mixed-effects model that combines unstructured variance/covariance matrix for inter-regional (long-range) correlations and an exchangeable correlation structure for intra-regional (short-range) correlations. Estimation is performed using the Gibbs sampling. ```r spatial.lme.mcmc(spatialMat, nlr, nsweep, verbose=TRUE) ``` ## Arguments - `spatialMat`: a matrix of observations with one subject per column and one location per row. For each subject, the observations are arranged one location after another. - `nlr`: a vector of number of locations within each region. One component per region. - `nsweep`: the number of iterations. - `verbose`: logical. If `TRUE`, then indicate the level of output after every 1000 iterations. The default is `TRUE`. ## Returns - **mu.save**: a matrix of means at every location. One column per iteration. - **sigma2.save**: a matrix of intra-regional variances. One column per iteration. - **lambda2.save**: a matrix of variances of locations' means within regions. One column per iteration. - **Gamma.save**: a matrix of inter-regional variance/covariance matrix. One column per iteration and within each column the elements of the variance/covariance matrix are arranged column-wise. ## Details The function was proposed to study the fMRI data. The original MATLAB code written by DuBois Bowman and Brian Caffo can be found at: [http://www.biostat.jhsph.edu/~bcaffo/downloads/clusterBayes.m](http://www.biostat.jhsph.edu/~bcaffo/downloads/clusterBayes.m). Instead of stacking the data from two conditions, the R version fits the model for one condition and the user needs to use the function multiple times for separate conditions. The initial values are obtained based on sample moments. The hyper-parameters for the prior distributions of the intra-regional variances and variances of locations' means are set up in the way that the mean is equal to the sample mean and the variance is large. ## Note There seemed to be no easy way to use `lmer` or `lme` to fit the variance/covariance structure in this model and **SAS proc mixed** failed for certain cases. ## References Brian Caffo, DuBois Bowman, Lynn Eberly and Susan Spear Bassett (2009) A Markov Chain Monte Carlo Based Analysis of a Multilevel Model for Functional MRI Data **Handbook of Markov Chain Monte Carlo** F. DuBois Bowman, Brian Caffo, Susan Spear Bassett, and Clinton Kilts (2008) A Bayesian Hierarchical Framework for Spatial Modeling of fMRI Data **Neuroimage** vol. 39, no. 1 146-156 ## Examples ```r ## Not run: #simulate the data ns=100; nr=2; nlr <- c(20, 20) mu0 <- c(0, 0) sigma2 <- c(1., 1.) Gamma <- matrix(c(3, 0, 0, 3), nrow=2) sample <- matrix(0, nrow=sum(nlr), ncol=ns) for(i in 1:ns){ alpha <- mvrnorm(1, rep(0, nr), Gamma) sampleR <- NULL for(g in 1:nr){ beta <- rnorm(nlr[g], mean=alpha[g] + mu0[g], sd=sqrt(sigma2[g])) sampleR <- c(sampleR, beta) } sample[,i] <- sampleR } #run mcmc mcmc.result <- spatial.lme.mcmc(sample, nlr, 10000) #check the results Gamma <- mcmc.result$Gamma.save sigma2 <- mcmc.result$sigma2.save mu <- mcmc.result$mu.save matrix(rowMeans(Gamma), nr, nr) apply(sigma2, 1, function(x) quantile(x, prob=c(0.025, 0.5, 0.975))) summary(rowMeans(mu[1:nlr[1],])) summary(rowMeans(mu[(nlr[1]+1):sum(nlr),])) ## End(Not run) ```

spatial.lme.mcmc function