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#' Average Relative Increase in Variance
#'
#' @author Ivan Jacob Agaloos Pesigan
#'
#' @details The average relative increase in variance
#' is given by
#' \deqn{
#' \mathrm{ARIV}
#' =
#' \left( 1 + M^{-1} \right)
#' \mathrm{tr}
#' \left(
#' \mathbf{V}_{\mathrm{between}}
#' \mathbf{V}_{\mathrm{within}}^{-1}
#' \right)
#' }
#'
#' @param between Numeric matrix.
#' Covariance between imputations
#' \eqn{\mathbf{V}_{\mathrm{between}}}.
#' @param within Numeric matrix.
#' Covariance within imputations
#' \eqn{\mathbf{V}_{\mathrm{within}}}.
#' @param M Positive integer.
#' Number of imputations.
#' @param k Positive integer.
#' Number of parameters.
#'
#' @return Returns a numeric vector of length one.
#'
#' @references
#' Li, K. H., Raghunathan, T. E., & Rubin, D. B. (1991).
#' Large-sample significance levels from multiply imputed data
#' using moment-based statistics and an F reference distribution.
#' *Journal of the American Statistical Association*, 86 (416), 1065–1073.
#' \doi{10.1080/01621459.1991.10475152}
#'
#' Rubin, D. B. (1987).
#' *Multiple imputation for nonresponse in surveys*.
#' John Wiley & Sons, Inc.
#' \doi{10.1002/9780470316696}
#'
#' @family Multiple Imputation Helper Functions
#' @keywords miHelper combine
#' @noRd
.ARIV <- function(between,
within,
M,
k) {
return(
(
(
1 + (
1 / M
)
) * sum(
diag(
between %*% chol2inv(
chol(
within
)
)
)
)
) / k
)
}
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