#' Maximum likelihood estimation for a random effects multivariate meta-analysis
#'
#' Maximum likelihood estimation for a random effects model for multivariate meta-analysis.
#'
#' The correlation matrix of the between-studies covariance matrix is set to compound symmetry
#'with the correlation coefficient 0.50. It can be changed by modifying the source code.
#' Please see Noma et al. (2017) for details.
#'
#' @param y N x p matrix of outcome variables.
#' @param S Series of within-study covariance matrices of the outcome variables.
#' A matrix or data frame with N rows and p(p+1)/2 columns.
#' @param maxitr The maximum iteration number of the Newton-Raphson algorithm.
#' @return
#' \item{Coefficients}{The maximum likeliood estimate of the grand mean vector and its standard error (SE) with the 95\% confidence interval.}
#' \item{Between-studies_SD}{The maximum likeliood estimate of the between-studies standard deviance (SD).}
#' \item{Between-studies_COR}{The correlation coefficient of the between-studies correlation matrix.}
#' \item{Loglikelihood}{The loglikelihood at the converged point.}
#' @references
#' Jackson, D., Riley, R., White, I. R. (2011).
#' Multivariate meta-analysis: Potential and promise.
#' \emph{Statistics in Medicine}. \strong{30}: 2481-2498.
#'
#' Noma, H., Nagashima, K., Maruo, K., Gosho, M., Furukawa, T. A. (2017).
#' Bartlett-type corrections and bootstrap adjustments of likelihood-based inference methods for network meta-analysis.
#' \emph{ISM Research Memorandum} 1205.
#' @examples
#' # dae <- data.aug.edit(smoking)
#' # y <- dae$y
#' # S <- dae$S
#'
#' # ML(y, S)
#' @family ML
#' @export
ML <- 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
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)
ll1 <- ll1 + A1 + A2 + A3
}
return(ll1)
}
LL2 <- function(g){
G <- gmat(g1, g, p)
ll1 <- 0
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)
ll1 <- ll1 + A1 + A2 + A3
}
return(ll1) # return "minus loglikelihood"
}
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 <- -LL1(g1)
R9 <- list("Coefficients" = R5, "Between-studies_SD" = R6,
"Between-studies_COR" = R7, "Loglikelihood" = R8)
return(R9)
}
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