R/manova.update.r

In RRPP: Linear Model Evaluation with Randomized Residuals in a Permutation Procedure

#' MANOVA update for lm.rrpp model fits
#' 
#' @description 
#' Function updates a lm.rrpp fit to add $MANOVA, which like 
#' $ANOVA, provides statistics
#' or matrices typically associated with multivariate analysis 
#' of variance (MANOVA).
#' 
#' MANOVA statistics or sums of squares and cross-products (SSCP) matrices
#' are calculated over the random permutations of a \code{\link{lm.rrpp}} 
#' fit.  SSCP matrices are 
#' computed, as are the inverse of R times H (invR.H), where R is a SSCP 
#' for the residuals or random effects and H is
#' the difference between SSCP matrices of full and reduced models 
#' (see below).   From invR.H,
#' MANOVA statistics are calculated, including Roy's maximum root 
#' (eigenvalue), Pillai trace, Hotelling-Lawley trace,
#' and Wilks lambda (via \code{\link{summary.manova.lm.rrpp}}).
#' 
#' The manova.update to add $MANOVA to \code{\link{lm.rrpp}} fits 
#' requires more computation time than the $ANOVA
#' statistics that are computed automatically in \code{\link{lm.rrpp}}.  
#' Generally, the same inferential conclusions will
#' be found with either approach, when observations outnumber response 
#' variables.  For high-dimensional data (more variables
#' than observations) data are projected into a Euclidean space of 
#' appropriate dimensions (rank of residual covariance matrix).  
#' One can vary the tolerance for eigenvalue decay or specify the number 
#' of PCs, if a smaller set of PCs than the maximum is desired.  
#' This is advised if there is strong correlation among variables 
#' (the data space could be simplified to fewer dimensions), as spurious
#' results are possible.  Because distributions of MANOVA stats 
#' can be generated from the random permutations,
#' there is no need to approximate F-values, like with parametric MANOVA. 
#' By restricting analysis to the real, positive eigenvalues calculated,
#' all statistics can be calculated (but Wilks lambda, as a product but 
#' not a trace, might be less reliable as variable number approaches
#' the number of observations).
#' 
#'  \subsection{ANOVA vs. MANOVA}{ 
#'  Two SSCP matrices are calculated for each linear model effect, for 
#'  every random permutation: R (Residuals or Random effects) and
#'  H, the difference between SSCPs for "full" and "reduced" models. 
#'  (Full models contain and reduced models lack
#'  the effect tested; SSCPs are hypothesized to be the same under 
#'  a null hypothesis, if there is no effect.  The 
#'  difference, H, would have a trace of 0 if the null hypothesis 
#'  were true.)  In RRPP, ANOVA and MANOVA correspond to
#'  two different ways to calculate statistics from R and H matrices.
#'  
#'  ANOVA statistics are those that find the trace of R and H SSCP 
#'  matrices before calculating subsequent statistics,
#'  including sums of squares (SS), mean squares (MS), and F-values.  
#'  These statistics can be calculated with univariate data
#'  and provide univariate-like statistics for multivariate data.  
#'  These statistics are dispersion measures only (covariances
#'  among variables do not contribute) and are the same as "distance-based" 
#'  stats proposed by Goodall (1991) and Anderson (2001).
#'  MANOVA stats require multivariate data and are implicitly 
#'  affected by variable covariances.  For MANOVA, the inverse of R times
#'  H (invR.H) is first calculated for each effect, then eigen-analysis 
#'  is performed on these matrix products.  Multivariate
#'  statistics are calculated from the positive, real eigenvalues.  
#'  In general, inferential
#'  conclusions will be similar with either approach, but effect 
#'  sizes might differ.
#'  
#'  Two important differences between manova.update and 
#'  \code{\link{summary.manova}} (for \code{\link{lm}} objects) 
#'  are that manova.update
#'  does not attempt to normalize residual SSCP matrices (unneeded 
#'  for non-parametric statistical solutions) and (2) uses a generalized
#'  inverse of the residual SSCP, if needed, when the number of 
#'  variables could render eigen-analysis problematic.  This approach 
#'  is consistent
#'  with covariance regularization methods that attempt to make covariance 
#'  matrices positive-definite for calculating model likelihoods or
#'  multivariate statistics.  If the number of observations far exceeds 
#'  the number of response variables, observed statistics from 
#'  manova.update and 
#'  \code{\link{summary.manova}} will be quite similar.  If the 
#'  number of response variables approaches or exceeds the number 
#'  of observations, manova.update
#'  statistics will be much more reliable.
#'  
#'  ANOVA tables are generated by \code{\link{anova.lm.rrpp}} on 
#'  lm.rrpp fits and MANOVA tables are generated
#'  by \code{\link{summary.manova.lm.rrpp}}, after running 
#'  manova.update on lm.rrpp fits.
#'  
#'  Currently, mixed model effects are only possible with 
#'  $ANOVA statistics, not $MANOVA.
#'  
#'  More detail is found in the vignette, ANOVA versus MANOVA.  
#' }
#' 
#' @references Goodall, C.R. 1991. Procrustes methods in the 
#' statistical analysis of shape. Journal of the Royal Statistical 
#' Society B 53:285-339.
#' @references Anderson MJ. 2001. A new method for non-parametric 
#' multivariate analysis of variance.
#'    Austral Ecology 26: 32-46.
#' @param fit Linear model fit from \code{\link{lm.rrpp}}
#' @param error An optional character string to define R matrices 
#' used to calculate invR.H.
#' (Currently only Residuals can be used and this argument defaults 
#' to NULL.  Future versions
#' will update this argument.)
#' @param tol A tolerance value for culling data dimensions to 
#' prevent spurious results.  The distribution
#' of eigenvalues for the data will be examined and if the decay 
#' becomes less than the tolerance,
#' the data will be truncated to principal components ahead of this 
#' point.  This will possibly prevent spurious results
#' calculated from eigenvalues near 0.  If tol = 0, all possible 
#' PC axes are used, which is likely
#' not a problem if observations outnumber variables.  If tol = 0 
#' and the number of variables exceeds the number of observations,
#' the value of tol will be made slightly positive to prevent problems 
#' with eigen-analysis.
#' @param PC.no A value that, if not NULL, can override the tolerance 
#' argument, and forces a desired number of 
#' data PCs to use for analysis.  If a value larger than the possible 
#' number of PCs is chosen, the full compliment of PCs
#' (the full data space) will be used.  If a number larger than tol 
#' would permit is chosen, the minimum number of PCs between the tol
#' argument and PC.no argument is returned.
#' @param print.progress A logical value to indicate whether a 
#' progress bar should be printed to the screen.
#' This is helpful for long-running analyses.
#' @param verbose Either a NULL or logical value for whether to retain
#' all MANOVA result (if TRUE).  If NULL, the verbose argument used for
#' the \code{\link{lm.rrpp}} is retained for MANOVA update.  Essentially,
#' verbose indicates whether to retain all SSCP matrices and all invR.H matrices,
#' for every model effect, in every RRPP iteration.
#' @keywords analysis
#' @export
#' @author Michael Collyer
#' @return An object of class \code{lm.rrpp} is updated to include 
#' class \code{manova.lm.rrpp}, and the object,
#' $MANOVA, which includes
#' \item{SSCP}{Terms and Model SSCP matrices.}
#' \item{invR.H}{The inverse of the residuals SSCP times the H SSCP.}
#' \item{eigs}{The eigenvalues of invR.H.}
#' \item{e.rank}{Rank of the error (residuals) covariance matrix.  
#' Currently NULL only.}
#' \item{PCA}{Principal component analysis of data, using either tol 
#' or PC.no.}
#' \item{manova.pc.dims}{Resulting number of PC vectors in the 
#' analysis.}
#' \item{e.rank}{Rank of the residual (error) covariance matrix, 
#' irrespective of the number of dimensions 
#'  used for analysis.}
#' 
#' @examples 
#'    
#' # Body Shape Analysis (Multivariate) ----------------
#' 
#' data(Pupfish)
#' 
#' # Although not recommended as a practice, this example will use only
#' # three principal components of body shape for demonstration.  
#' # A larger number of random permutations should also be used.
#' 
#' Pupfish$shape <- prcomp(Pupfish$coords)$x[, 1:3]
#' 
#' Pupfish$logSize <- log(Pupfish$CS) 
#' 
#' fit <- lm.rrpp(shape ~ logSize + Sex, SS.type = "I", 
#' data = Pupfish, print.progress = FALSE, iter = 499) 
#' summary(fit, formula = FALSE)
#' anova(fit) # ANOVA table
#' 
#' # MANOVA
#' 
#' fit.m <- manova.update(fit, print.progress = FALSE, tol = 0.001)
#' summary(fit.m, test = "Roy")
#' summary(fit.m, test = "Pillai")
#' 
#' fit.m$MANOVA$eigs$obs # observed eigenvalues
#' fit.m$MANOVA$SSCP$obs # observed SSCP
#' fit.m$MANOVA$invR.H$obs # observed invR.H 
#' 
#' # Distributions of test statistics
#' 
#' summ.roy <- summary(fit.m, test = "Roy")
#' dens <- apply(summ.roy$rand.stats, 1, density)
#' par(mfcol = c(1, length(dens)))
#' for(i in 1:length(dens)) {
#'      plot(dens[[i]], xlab = "Roy max root", ylab = "Density",
#'      type = "l", main = names(dens)[[i]])
#'      abline(v = summ.roy$rand.stats[1, i], col = "red")
#' }
#' par(mfcol = c(1,1))
#' 

manova.update <- function(fit, error = NULL, 
                              tol = 1e-7, PC.no = NULL,
                              print.progress = TRUE,
                          verbose = NULL) {
  
  if(inherits(fit, "manova.lm.rrpp")) 
    stop("\nlm.rrpp object has already been updated for MANOVA.\n", 
                                           call. = FALSE)
  if(!inherits(fit, "lm.rrpp")) 
    stop("\nOnly an lm.rrpp object can be updated for MANOVA.\n", 
                                     call. = FALSE)
  if(is.null(verbose)) verbose <- fit$verbose else
    verbose <- as.logical(verbose)
  
  p <- fit$LM$p
  if(p < 2) 
    stop("\n Multiple responses are required for this analysis.\n", 
         call. = FALSE)
  p.prime <- fit$LM$p.prime
  n <- fit$LM$n
  gls <- fit$LM$gls
  w <- if(!is.null(fit$LM$weights)) fit$LM$weights else NULL
  Pcov <- if(!is.null(fit$LM$Pcov)) fit$LM$Pcov else NULL
  Y <- fit$LM$Y
  
  PermInfo <- getPermInfo(fit, attribute = "all")
  
  perm.method <- PermInfo$perm.method
  if(perm.method == "RRPP") RRPP = TRUE else RRPP = FALSE
  ind <- PermInfo$perm.schedule
  perms <- length(ind)
  
  Models <- getModels(fit, attribute = "all")
  
  reduced <- Models$reduced
  full <- Models$full
  Terms <- fit$LM$Terms
  trms <- fit$LM$term.labels
  k <- length(trms)
  
  if(k > length(Models$full)) {
    k <- length(Models$full)
    trms <- names(Models$full)
  }
  
  if(k == 0) stop("\nNo model terms from which to calculate SSCPs.\n",
                  call. = FALSE)
  int <- attr(Terms, "intercept")
  
  E.rank <- getRank(qr(var(as.matrix(resid(fit)))))
  
  PCA <- ordinate(Y, tol = tol, rank. = PC.no)
  d <- PCA$sdev
  if(!is.null(PC.no)) {
    ld <- seq_len(min(PC.no, length(d)))
    d <- PCA$sdev <- PCA$sdev[ld]
    PCA$x <- PCA$x[, ld]
  }

  if(length(d) > min(n - 1, p)) {
    mnp <- seq_len(min(n - 1, p))
    d <- PCA$sdev <- PCA$sdev[mnp]
    PCA$x <- PCA$x[, mnp]
  }
  
  kk <- if (!is.null(PC.no)) {
    stopifnot(length(PC.no) == 1, is.finite(PC.no), as.integer(PC.no) > 
                0)
    min(as.integer(PC.no), n, p)
  } else min(n, p)
  
  
  if (tol > 0) {
    rank <- sum(d > (d[1L] * tol))
    if (rank < kk) {
      j <- seq_len(rank)
      PCA$x <- PCA$x[, j, drop = FALSE]
    }
  }
  
  if(NCOL(Y) > NCOL(PCA$x)) Y <- PCA$x
  
  if(!is.null(error)) {
    if(!inherits(error, "character")) 
      stop("The error description is illogical.  
           It should be a string of character values matching ANOVA terms.",
                                           call. = FALSE)
    kk <- length(error)
    if(kk != k) 
      stop("The error description should match in length the number of ANOVA terms 
           (not including Residuals)",
                     call. = FALSE)
    Ematch <- match(error, c(trms, "Residuals"))
    if(any(is.na(Ematch))) 
      stop("At least one of the error terms is not an ANOVA term",
                                call. = FALSE)
  } else Ematch <- NULL
  
  # Until a better solution is found, this must be forced
  error <- Ematch <- NULL
  
  Qr <- lapply(reduced, function(q) q$qr)
  Qf <- lapply(full, function(q) q$qr)
  
  Ur <- lapply(Qr, qr.Q)
  Uf <- lapply(Qf, qr.Q)
  Ufull <- Uf[[max(1, k)]]
  
  int <- attr(Terms, "intercept")
  Unull <- if(!is.null(Pcov)) 
    qr.Q(qr(Pcov %*% rep(int, n))) else if(!is.null(w)) 
      qr.Q(qr(rep(int, n) * sqrt(w))) else
        qr.Q(qr(rep(int, n)))
  
  if(!is.null(Pcov)) Y <- Pcov %*% Y
  if(!is.null(w)) Y <- Y * sqrt(w)
  
  SS.tot <- sum(diag(crossprod(center(Y))))
  yh0 <- fastFit(Unull, Y, n, p.prime)
  r0 <- Y - yh0
  
  rrpp <- function(FR, ind.i) {
    lapply(FR, function(x) as.matrix(x$fitted) + as.matrix(x$residuals)[ind.i, ])
  }
  
  fitted <- lapply(Ur, function(u) fastFit(u, Y, n, p.prime))
  res <- lapply(fitted, function(f) Y - f)

  if(!RRPP) {
    fitted <- lapply(fitted, function(.) matrix(0, n, p.prime))
    res <- lapply(res, function(.) as.matrix(Y))
  } 
  

  FR <- lapply(1:max(1, k), function(j) list(fitted = fitted[[j]], 
                                             residuals = res[[j]]))
  rrpp.args <- list(FR = FR, ind.i = ind[[1]])
  
  rrpp <- function(FR, ind.i) {
    lapply(FR, function(x) as.matrix(x$fitted) + as.matrix(x$residuals)[ind.i, ])
  }
  
  sscp.args <- list(Uf = Uf, Ur = Ur, 
                    Ufull = Ufull, Unull = Unull,
                    Y = NULL, 
                    n = n, p = p.prime,
                    Yt = Y)
  
  sscp <- function(Uf, Ur, Ufull, Unull, Y, n, p, Yt) {
    ss <- Map(function(uf, ur, y) list(SS = crossprod(
      fastFit(uf, y, n, p.prime) - fastFit(ur, y, n, p.prime)),
              Residuals = crossprod(y - fastFit(Ufull, y, n, p.prime))), 
              Uf, Ur, Y)
    
    names(ss) <- trms
    rss <- crossprod(Yt - fastFit(Ufull, Yt, n, p.prime))
    tss <- crossprod(Yt - fastFit(Unull, Yt, n, p.prime))
    
    result <- c(ss, list(Full.Model = list(SS = tss - rss, 
                                           Residuals = rss)))

    result
  }
  
  invR.H <- function(ss){ # modified from stats::summary.manova version
    tss <- diag(Reduce("+", ss[[length(ss)]]))
    result <- lapply(1:length(ss), function(j){
      H <- ss[[j]]$SS
      R <- ss[[j]]$Residuals
      fast.solve(R) %*% H
    })
    
    names(result) <- c(trms, "Full.Model")
    
    result
  }

  getEigs <- function(x) Re(eigen(x, symmetric = FALSE, 
                                  only.values = TRUE)$values)
  
  getEigsL <- function(L) t(sapply(L, getEigs))
  
  
  if(print.progress){
    cat(paste("\nCalculation of SSCP matrix products:", 
              perms, "permutations.\n")) 
    pb <- txtProgressBar(min = 0, max = perms+1, initial = 0, style=3)
  }
  
  result <- lapply(1: perms, function(j){
    step <- j
    if(print.progress) setTxtProgressBar(pb,step)
    x <- ind[[j]]
    rrpp.args$ind.i <- x
    sscp.args$Y <- do.call(rrpp, rrpp.args)
    sscp.args$Yt <- yh0 + r0[x, ]
    ss <- do.call(sscp, sscp.args)
    invRH <- invR.H(ss)
    eigs <- getEigsL(invRH)
    list(SSCP = if(verbose) ss else NULL, 
         invR.H = if(verbose) invRH else NULL, 
         eigs = eigs)
    
  })
  
  
  if(print.progress) {
    step <- perms + 1
    setTxtProgressBar(pb,step)
    close(pb)
  }
  
  
  SSCP <- if(verbose) lapply(result, function(x) x$SSCP) else NULL
  invR.H <- if(verbose) lapply(result, function(x) x$invR.H) else NULL
  eigs <- lapply(result, function(x) x$eigs)
  
  if(verbose){
    names(SSCP) <- names(invR.H) <- c("obs", paste("iter", 1:(perms - 1), sep = "."))
  }
  names(eigs) <- c("obs", paste("iter", 1:(perms - 1), sep = "."))
  
  
  out <- fit
  out$MANOVA <- list(SSCP = SSCP, invR.H = invR.H, eigs = eigs,
                     error = error, PCA = PCA, manova.pc.dims = length(d), 
                     e.rank = E.rank, SS.tot = SS.tot)
  
  class(out) <- c("manova.lm.rrpp", class(fit))
  out
}