R/generateBounds.R

In gMCP: Graph Based Multiple Comparison Procedures

#' generateBounds
#' 
#' compute rejection bounds for z-scores of each elementary hypotheses within
#' each intersection hypotheses
#' 
#' 
#' It is assumed that under the global null hypothesis
#' \eqn{(\Phi^{-1}(1-p_1),...,\Phi^{-1}(1-p_m))} follow a multivariate normal
#' distribution with correlation matrix \code{cr} where \eqn{\Phi^{-1}} denotes
#' the inverse of the standard normal distribution function.
#' 
#' For example, this is the case if \eqn{p_1,..., p_m} are the raw p-values
#' from one-sided z-tests for each of the elementary hypotheses where the
#' correlation between z-test statistics is generated by an overlap in the
#' observations (e.g. comparison with a common control, group-sequential
#' analyses etc.). An application of the transformation \eqn{\Phi^{-1}(1-p_i)}
#' to raw p-values from a two-sided test will not in general lead to a
#' multivariate normal distribution. Partial knowledge of the correlation
#' matrix is supported. The correlation matrix has to be passed as a numeric
#' matrix with elements of the form: \eqn{correlation[i,i] = 1} for diagonal
#' elements, \eqn{correlation[i,j] = \rho_{ij}}, where \eqn{\rho_{ij}} is the
#' known value of the correlation between \eqn{\Phi^{-1}(1-p_i)} and
#' \eqn{\Phi^{-1}(1-p_j)} or \code{NA} if the corresponding correlation is
#' unknown. For example correlation[1,2]=0 indicates that the first and second
#' test statistic are uncorrelated, whereas correlation[2,3] = NA means that
#' the true correlation between statistics two and three is unknown and may
#' take values between -1 and 1. The correlation has to be specified for
#' complete blocks (ie.: if cor(i,j), and cor(i,k) for i!=j!=k are specified
#' then cor(j,k) has to be specified as well) otherwise the corresponding
#' intersection null hypotheses tests are not uniquely defined and an error is
#' returned.
#' 
#' The parametric tests in (Bretz et al. (2011)) are defined such that the
#' tests of intersection null hypotheses always exhaust the full alpha level
#' even if the sum of weights is strictly smaller than one. This has the
#' consequence that certain test procedures that do not test each intersection
#' null hypothesis at the full level alpha may not be implemented (e.g., a
#' single step Dunnett test). If \code{upscale} is set to \code{FALSE}
#' (default) the parametric tests are performed at a reduced level alpha of
#' sum(w) * alpha and p-values adjusted accordingly such that test procedures
#' with non-exhaustive weighting strategies may be implemented. If set to
#' \code{TRUE} the tests are performed as defined in Equation (3) of (Bretz et
#' al. (2011)).
#' 
#' @param g graph defined as a matrix, each element defines how much of the
#' local alpha reserved for the hypothesis corresponding to its row index is
#' passed on to the hypothesis corresponding to its column index
#' @param w vector of weights, defines how much of the overall alpha is
#' initially reserved for each elementary hypothesis
#' @param cr correlation matrix if p-values arise from one-sided tests with
#' multivariate normal distributed test statistics for which the correlation is
#' partially known. Unknown values can be set to NA. (See details for more
#' information)
#' @param al overall alpha level at which the family error is controlled
#' @param hint if intersection hypotheses weights have already been computed
#' (output of \code{\link{generateWeights}}) can be passed here otherwise will
#' be computed during execution
#' @param upscale if \code{FALSE} (default) the parametric test is performed at
#' the reduced level alpha of sum(w)*alpha. (See details)
#' @return Returns a matrix of rejection bounds. Each row corresponds to an
#' intersection hypothesis. The intersection corresponding to each line is
#' given by conversion of the line number into binary (eg. 13 is binary 1101
#' and corresponds to (H1,H2,H4))
#' @author Florian Klinglmueller
#' @references Bretz F, Maurer W, Brannath W, Posch M; (2008) - A graphical
#' approach to sequentially rejective multiple testing procedures. - Stat Med -
#' 28/4, 586-604
#' 
#' Frank Bretz, Martin Posch, Ekkehard Glimm, Florian Klinglmueller, Willi
#' Maurer, Kornelius Rohmeyer (2011): Graphical approaches for multiple
#' comparison procedures using weighted Bonferroni, Simes or parametric tests.
#' Biometrical Journal 53 (6), pages 894-913, Wiley.
#' \url{http://onlinelibrary.wiley.com/doi/10.1002/bimj.201000239/full}
#' @keywords htest
#' @examples
#' 
#'  ## Define some graph as matrix
#'  g <- matrix(c(0,0,1,0,
#'                0,0,0,1,
#'                0,1,0,0,
#'                1,0,0,0), nrow = 4,byrow=TRUE)
#'  ## Choose weights
#'  w <- c(.5,.5,0,0)
#'  ## Some correlation (upper and lower first diagonal 1/2)
#'  c <- diag(4)
#'  c[1:2,3:4] <- NA
#'  c[3:4,1:2] <- NA
#'  c[1,2] <- 1/2
#'  c[2,1] <- 1/2
#'  c[3,4] <- 1/2
#'  c[4,3] <- 1/2
#' 
#'  ## Boundaries for correlated test statistics at alpha level .05:
#'  generateBounds(g,w,c,.05)
#' 
#' @export generateBounds
#' 
generateBounds <-
function(g,w,cr,al=.05,hint=generateWeights(g,w),upscale=FALSE){ #, alternatives="less"){
  res <- t(apply(hint,1,b.dunnett,a=al,cr=cr,upscale=upscale)) #, alternatives=alternatives))
  res
}