R/homophily.re.R

In CDatanet: Econometrics of Network Data

#' @title Estimating network formation models with degree heterogeneity: the Bayesian random effect approach
#' @param network matrix or list of sub-matrix of social interactions containing 0 and 1, where links are represented by 1.
#' @param formula an object of class \link[stats]{formula}: a symbolic description of the model. The `formula` should be as for example \code{~ x1 + x2}
#' where `x1`, `x2` are explanatory variable of links formation.
#' @param data an optional data frame, list or environment (or object coercible by \link[base]{as.data.frame} to a data frame) containing the variables
#' in the model. If not found in data, the variables are taken from \code{environment(formula)}, typically the environment from which `homophily` is called.
#' @param symmetry indicates whether the network model is symmetric (see details).
#' @param group.fe indicates whether the model includes group fixed effects.
#' @param re.way indicates whether it is a one-way or two-way fixed effect model. The expected value is 1 or 2 (see details).
#' @param init (optional) list of starting values containing `beta`, an K-dimensional vector of the explanatory variables parameter, 
#' `mu` an n-dimensional vector, and `nu` an n-dimensional vector, `smu2` the variance of `mu`, 
#' and `snu2` the variance of `nu`, 
#' where K is the number of explanatory variables and n is the number of individuals.  
#' @param iteration the number of iterations to be performed. 
#' @param print boolean indicating if the estimation progression should be printed.
#' @return A list consisting of:
#'     \item{model.info}{list of model information, such as the type of random effects, whether the model is symmetric,
#'      number of observations, etc.}
#'     \item{posterior}{list of simulations from the posterior distribution.}
#'     \item{init}{returned list of starting values.}
#' @description 
#' `homophily.re` implements a Bayesian Probit estimator for network formation model with homophily. The model includes degree heterogeneity using random effects (see details).
#' @details
#' Let \eqn{p_{ij}}{Pij} be a probability for a link to go from the individual \eqn{i} to the individual \eqn{j}.
#' This probability is specified for two-way effect models (`fe.way = 2`) as
#' \deqn{p_{ij} = F(\mathbf{x}_{ij}'\beta + \mu_j + \nu_j)}{Pij = F(Xij'*\beta + \mu_i + \nu_j),}
#' where \eqn{F} is the cumulative of the standard normal distribution. Unobserved degree heterogeneity is captured by
#' \eqn{\mu_i} and \eqn{\nu_j}. The latter are treated as random effects (see \code{\link{homophily.fe}} for fixed effect models).\cr
#' For one-way random effect models (`fe.way = 1`), \eqn{\nu_j = \mu_j}. For symmetric models, the network is not directed and the 
#' random effects need to be one way.
#' @seealso \code{\link{homophily.fe}}.
#' @importFrom ddpcr quiet
#' @importFrom stats lm
#' @importFrom stats var
#' @importFrom stats cov
#' @examples 
#' \donttest{
#' set.seed(1234)
#' library(MASS)
#' M            <- 4 # Number of sub-groups
#' nvec         <- round(runif(M, 100, 500))
#' beta         <- c(.1, -.1)
#' Glist        <- list()
#' dX           <- matrix(0, 0, 2)
#' mu           <- list()
#' nu           <- list()
#' cst          <- runif(M, -1.5, 0)
#' smu2         <- 0.2
#' snu2         <- 0.2
#' rho          <- 0.8
#' Smunu        <- matrix(c(smu2, rho*sqrt(smu2*snu2), rho*sqrt(smu2*snu2), snu2), 2)
#' for (m in 1:M) {
#'   n          <- nvec[m]
#'   tmp        <- mvrnorm(n, c(0, 0), Smunu)
#'   mum        <- tmp[,1] - mean(tmp[,1])
#'   num        <- tmp[,2] - mean(tmp[,2])
#'   X1         <- rnorm(n, 0, 1)
#'   X2         <- rbinom(n, 1, 0.2)
#'   Z1         <- matrix(0, n, n)  
#'   Z2         <- matrix(0, n, n)
#'   
#'   for (i in 1:n) {
#'     for (j in 1:n) {
#'       Z1[i, j] <- abs(X1[i] - X1[j])
#'       Z2[i, j] <- 1*(X2[i] == X2[j])
#'     }
#'   }
#'   
#'   Gm           <- 1*((cst[m] + Z1*beta[1] + Z2*beta[2] +
#'                        kronecker(mum, t(num), "+") + rnorm(n^2)) > 0)
#'   diag(Gm)     <- 0
#'   diag(Z1)     <- NA
#'   diag(Z2)     <- NA
#'   Z1           <- Z1[!is.na(Z1)]
#'   Z2           <- Z2[!is.na(Z2)]
#'   
#'   dX           <- rbind(dX, cbind(Z1, Z2))
#'   Glist[[m]]   <- Gm
#'   mu[[m]]      <- mum
#'   nu[[m]]      <- num
#' }
#' 
#' mu  <- unlist(mu)
#' nu  <- unlist(nu)
#' 
#' out   <- homophily.re(network =  Glist, formula = ~ dX, group.fe = TRUE, 
#'                       re.way = 2, iteration = 1e3)
#' 
#' # plot simulations
#' plot(out$posterior$beta[,1], type = "l")
#' abline(h = cst[1], col = "red")
#' plot(out$posterior$beta[,2], type = "l")
#' abline(h = cst[2], col = "red")
#' plot(out$posterior$beta[,3], type = "l")
#' abline(h = cst[3], col = "red")
#' plot(out$posterior$beta[,4], type = "l")
#' abline(h = cst[4], col = "red")
#' 
#' plot(out$posterior$beta[,5], type = "l")
#' abline(h = beta[1], col = "red")
#' plot(out$posterior$beta[,6], type = "l")
#' abline(h = beta[2], col = "red")
#' 
#' plot(out$posterior$sigma2_mu, type = "l")
#' abline(h = smu2, col = "red")
#' plot(out$posterior$sigma2_nu, type = "l")
#' abline(h = snu2, col = "red")
#' plot(out$posterior$rho, type = "l")
#' abline(h = rho, col = "red")
#' 
#' i <- 10
#' plot(out$posterior$mu[,i], type = "l")
#' abline(h = mu[i], col = "red")
#' plot(out$posterior$nu[,i], type = "l")
#' abline(h = nu[i], col = "red")
#' }
#' @export
homophily.re <- function(network,
                         formula,
                         data,
                         symmetry  = FALSE,
                         group.fe  = FALSE,
                         re.way    = 1,
                         init      = list(),
                         iteration = 1e3,
                         print     = TRUE) {
  t1              <- Sys.time()
  re.way  <- as.numeric(re.way[1])
  if(symmetry & re.way == 2) stop("Two side random effects are not allowed for symmetric network models.")
  stopifnot(re.way %in% (1:2))
  # Data and dimensions
  if (!is.list(network)) {
    network       <- list(network)
  }
  
  M               <- length(network)
  nvec            <- unlist(lapply(network, nrow))
  n               <- sum(nvec)
  Nvec            <- NULL
  if(symmetry){
    Nvec          <- nvec*(nvec- 1)/2
    stopifnot(sapply(network, is.symmetric.matrix))
    network       <- frMtoVbyCOLsym(network, nvec, M)
  } else {
    Nvec          <- nvec*(nvec- 1)
    # network         <- unlist(lapply(network, function(x){diag(x) = NA; x}))
    # network         <- network[!is.na(network)]
    network       <- frMtoVbyCOL(network, nvec, M)
  }
  N               <- sum(Nvec)

  quiet(gc())
  if (sum(!((network == 0) | (network == 1))) != 0) {
    stop("Network should contain only 0 and 1.")
  } 
  
  tmp1    <- NULL
  if(symmetry){
    tmp1  <- cumsum(unlist(lapply(nvec, function(x) (x - 1):0))) - 1
  } else {
    tmp1  <- cumsum(unlist(lapply(nvec, function(x) rep(x - 1, x)))) - 1
  }
  tmp2    <- c(0, tmp1[-n] + 1)
  index   <- cbind(tmp2, tmp1) 
  rm(list = c("tmp1", "tmp2"))
  quiet(gc())
  
  indexgr         <- matrix(c(cumsum(c(0, nvec[-M])), cumsum(nvec) - 1), ncol = 2)
  INDEXgr         <- matrix(c(cumsum(c(0, Nvec[-M])), cumsum(Nvec) - 1), ncol = 2)
  # Formula to data
  f.t.data        <- formula.to.data(formula, FALSE, NULL, NULL, NULL, data,
                                     type = "network", theta0 =  NA)
  if(!missing(data)) {
    rm("data")
    quiet(gc())
  }
  
  formula         <- f.t.data$formula
  dX              <- f.t.data$X
  if(nrow(dX) != N) stop("The number of observations in X does not match the network.")
  rm("f.t.data")
  quiet(gc())
  coln            <- colnames(dX)
  nfix            <- ifelse("(Intercept)" %in% coln, 1, 0)
  K               <- ncol(dX)
  if (group.fe) {
    if(M < 2){
      stop("Group fixed effects can be added for only one subnetwork.")
    }
    K             <- K + M - nfix
    nfix          <- M
    dX            <- dX[,coln != "(Intercept)"]
    coln          <- c(paste0("(Intercept-", 1:M, ")"), coln[coln != "(Intercept)"])
  }
  
  Kx              <- ncol(dX)
  dXdX            <- crossprod(dX)
  sumnetwork      <- NULL
  invdXdX         <- NULL
  if (nfix >= 2){
    sumdX         <- do.call(cbind, lapply(1:M, function(m){colSums(dX[(INDEXgr[m,1] + 1):(INDEXgr[m,2] + 1),])}))
    sumnetwork    <- sapply(1:M, function(m){sum(network[(INDEXgr[m,1] + 1):(INDEXgr[m,2] + 1)])})
    dXdX          <- rbind(cbind(diag(Nvec), t(sumdX)), cbind(sumdX, dXdX))
    invdXdX       <- solve(as.matrix(dXdX))
    rm("sumdX")
    quiet(gc())
  } else{
    invdXdX       <- solve(as.matrix(dXdX))
  }
  rm("dXdX")
  quiet(gc())
  
  #starting value
  beta            <- init$beta
  mu              <- init$mu
  nu              <- init$nu
  smu2            <- init$smu2
  snu2            <- init$snu2
  rho             <- init$rho

  quiet(gc())
  if (is.null(beta)) {
    # print(dim(invdXdX))
    # print(length(sumnetwork))
    # print(dim(dX))
    # print(length(network))
    beta          <- c(invdXdX %*% c(sumnetwork, crossprod(dX, network)))
  } else{
    stopifnot(length(beta) == K)
  }
  
  if (is.null(mu)) {
    mu            <- rep(0, n)
  } else{
    stopifnot(length(mu) == n)
  }
  
  if(re.way == 2){
    if (is.null(nu)) {
      nu          <- rep(0, n)
    } else{
      stopifnot(length(nu) == n)
    }
  }

  if (is.null(smu2)) {
    tmp           <- var(mu)
    smu2          <- ifelse(tmp > 0, tmp, 1)
  } 
  
  if (is.null(snu2) & re.way == 2) {
    tmp           <- var(nu)
    snu2          <- ifelse(tmp > 0, tmp, 1)
  }
  
  if (is.null(rho) & re.way == 2) {
    rho           <- cov(mu, nu)/sqrt(smu2*snu2)
    rho           <- (rho >= 1) - (rho <= -1) + rho*((rho >= -1) & (rho <= 1))
  } 
  
  if(re.way == 1){
    nu            <- NULL
    snu2          <- NULL
    rho           <- NULL
  }
  
  init            <- list(beta    = beta,
                          mu      = mu,
                          nu      = nu,
                          smu2    = smu2,
                          snu2    = snu2,
                          rho     = rho)
  estima          <- NULL
  if(re.way == 1){
    estim         <- bayesmu(network, dX, invdXdX, beta, mu, smu2, index, indexgr,
                               INDEXgr, nfix, N, M, K, Kx, nvec, n, iteration, symmetry, print)
  } else{
    estim         <- bayesmunu(network, dX, invdXdX, beta, mu, nu, smu2, snu2, rho, index, indexgr,
                               INDEXgr, nfix, N, M, K, Kx, nvec, n, iteration, print)
  }
  
  colnames(estim$beta)     <- coln
  
  t2          <- Sys.time()
  timer       <- as.numeric(difftime(t2, t1, units = "secs"))
  
  nlinks      <- sum(network)
  out         <- list("model.info"     = list("model"       = "probit", 
                                              "sym.network" = symmetry,
                                              "re.way"      = re.way, 
                                              "n"           = nvec,
                                              "n.obs"       = N,
                                              "n.links"     = nlinks,
                                              "K"           = K,
                                              "iteration"   = iteration),
                      "posterior"       = estim,
                      "init"            = init)
  
  class(out)  <- "homophily.re"
  if(print) {
    cat("\n\n")
    cat("The program successfully executed \n")
    cat("\n")
    cat("********SUMMARY******** \n")
    cat("n.obs          : ", N, "\n")
    cat("n.links        : ", nlinks, "\n")
    cat("K              : ", K, "\n")
    cat("Group FE       : ", ifelse(group.fe, "Yes", "No"), "\n")
    cat("Iteration      : ", iteration, "\n\n")
    
    
    # Print the processing time
    nhours     <- floor(timer/3600)
    nminutes   <- floor((timer-3600*nhours)/60)%%60
    nseconds   <- timer-3600*nhours-60*nminutes
    cat("Elapsed time   : ", nhours, " HH ", nminutes, " mm ", round(nseconds), " ss \n \n")
  }
  
  out
}