R/REML.r

In nshi-stat/netiim3: Improved Inference Methods for Network Meta-Analysis

#' Restricted maximum likelihood estimation for a random effects multivariate meta-analysis
#'
#' @inheritParams ML
#' @family ML
#' @export
REML <- function(y, S, maxitr = 200){

  # ML algorithm for the consistent model (Sigma is exchangeble)

  N <- dim(y)[1]
  p <- dim(y)[2]

  mu <- rnorm(p)	# initial values
  g1 <- 0.2
  g2 <- 0.1

  Qc0 <- c(mu, g1, g2)

  LL1 <- function(g){

    #G <- gmat(g, g2, p)
    G <- gmat(g, (g/2), p)

    ll1 <- 0; XWX <- gmat(0, 0, p)

    for(i in 1:N){

      yi <- as.vector(y[i,])
      wi <- which(is.na(yi) == FALSE)
      pl <- length(wi)

      Si <- vmat(S[i,], p)

      yi <- yi[wi]
      Si <- pmat(Si, wi)
      mui <- mu[wi]
      Gi <- pmat(G, wi)

      B1 <- (yi - mui)
      B2 <- ginv(Gi + Si)

      A1 <- log(det(Gi + Si))
      A2 <- t(B1) %*% B2 %*% B1
      A3 <- pl * log(2*pi)

      XWX <- XWX + imat(B2, wi, p)

      ll1 <- ll1 + A1 + A2 + A3

    }

    ll2 <- ll1 + log(det(XWX)) - p*log(2*pi)

    return(ll2)

  }

  LL2 <- function(g){

    G <- gmat(g1, g, p)

    ll1 <- 0; XWX <- gmat(0, 0, p)

    for(i in 1:N){

      yi <- as.vector(y[i,])
      wi <- which(is.na(yi) == FALSE)
      pl <- length(wi)

      Si <- vmat(S[i,], p)

      yi <- yi[wi]
      Si <- pmat(Si, wi)
      mui <- mu[wi]
      Gi <- pmat(G, wi)

      B1 <- (yi - mui)
      B2 <- ginv(Gi + Si)

      A1 <- log(det(Gi + Si))
      A2 <- t(B1) %*% B2 %*% B1
      A3 <- pl * log(2*pi)

      XWX <- XWX + imat(B2, wi, p)

      ll1 <- ll1 + A1 + A2 + A3

    }

    ll2 <- ll1 + log(det(XWX)) - p*log(2*pi)

    return(ll2)

  }

  for(itr in 1:maxitr){

    A1 <- numeric(p)
    A2 <- matrix(numeric(p*p), p)

    G <- gmat(g1, g2, p)

    for(i in 1:N){

      yi <- as.vector(y[i,])
      wi <- which(is.na(yi) == FALSE)
      pl <- length(wi)

      Si <- vmat(S[i,], p)

      yi <- yi[wi]
      Si <- pmat(Si, wi)
      Gi <- pmat(G, wi)

      Wi <- ginv(Gi + Si)

      A1 <- A1 + ivec(yi %*% Wi, wi, p)
      A2 <- A2 + imat(Wi, wi, p)

    }

    mu <- A1 %*% ginv(A2)
    g1 <- optimize(LL1, lower = 0, upper = 5)$minimum
    g2 <- 0.5*g1

    V.mu <- ginv(A2)

    Qc <- c(mu, g1, g2)

    rb <- abs(Qc - Qc0)/abs(Qc0); rb[is.nan(rb)] <- 0
    if(max(rb) < 10^-4) break

    Qc0 <- Qc

  }


  SE <- sqrt(diag(V.mu))

  R1 <- as.vector(mu)
  R2 <- as.vector(SE)
  R3 <- as.vector(mu - qnorm(.975)*SE)
  R4 <- as.vector(mu + qnorm(.975)*SE)

  R5 <- cbind(R1, R2, R3, R4); colnames(R5) <- c("Coef.", "SE", "95%CL", "95%CU")

  R6 <- sqrt(g1)
  R7 <- g2/g1

  R8 <- list("Coefficients" = R5, "Between-studies_SD" = R6,
             "Between-studies_COR" = R7)

  return(R8)

}