R/TWT2.R

In alessiapini/fdatest: Interval Wise Testing for Functional Data

#' @title Two population Threshold Wise Testing procedure
#'
#' @description The function implements the Threshold Wise Testing procedure for testing mean differences between two
#' functional populations. Functional data are tested locally and unadjusted and adjusted p-value
#' functions are provided. The unadjusted p-value function controls the point-wise error rate. The adjusted p-value function controls the
#' family-wise error rate asymptotically.
#'
#' @param data1 First population's data. Either pointwise evaluations of the functional data set on a uniform grid, or a \code{fd} object from the package \code{fda}.
#' If pointwise evaluations are provided, \code{data2} is a matrix of dimensions \code{c(n1,J)}, with \code{J} evaluations on columns and \code{n1} units on rows.
#'
#' @param data2 Second population's data. Either pointwise evaluations of the functional data set on a uniform grid, or a \code{fd} object from the package \code{fda}.
#' If pointwise evaluations are provided, \code{data2} is a matrix of dimensions \code{c(n1,J)}, with \code{J} evaluations on columns and \code{n2} units on rows.
#'
#' @param mu Functional mean difference under the null hypothesis. Three possibilities are available for \code{mu}:
#' a constant (in this case, a constant function is used);
#' a \code{J}-dimensional vector containing the evaluations on the same grid which \code{data} are evaluated;
#' a \code{fd} object from the package \code{fda} containing one function.
#' The default is \code{mu=0}.
#'
#' @param B The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. The defualt is \code{B=1000}.
#'
#' @param paired Flag indicating whether a paired test has to be performed. Default is \code{FALSE}.
#'
#' @param dx Used only if a \code{fd} object is provided. In this case, \code{dx} is the size of the discretization step of the grid  used to evaluate functional data.
#' If set to \code{NULL}, a grid of size 100 is used. Default is \code{NULL}.
#'
#' @param alternative A character string specifying the alternative hypothesis, must be one of "\code{two.sided}" (default), "\code{greater}" or "\code{less}".
#'
#' @return \code{TWT2} returns an object of \code{\link{class}} "\code{fdatest2}" containing the following components:
#' \item{test}{String vector indicating the type of test performed. In this case equal to \code{"2pop"}.}
#' \item{mu}{Evaluation on a grid of the functional mean difference under the null hypothesis (as entered by the user).}
#' \item{unadjusted_pval}{Evaluation on a grid of the unadjusted p-value function.}
#' \item{adjusted_pval}{Evaluation on a grid of the adjusted p-value function.}
#' \item{data.eval}{Evaluation on a grid of the functional data.}
#' \item{ord_labels}{Vector of labels indicating the group membership of data.eval}
#'
#' @seealso See also \code{\link{plot.fdatest2}} for plotting the results.
#'
#' @examples
#' # Importing the NASA temperatures data set
#' data(NASAtemp)
#'
#' # Performing the TWT for two populations
#' TWT.result <- TWT2(NASAtemp$paris,NASAtemp$milan)
#'
#' # Plotting the results of the TWT
#' plot(TWT.result,xrange=c(0,12),main='TWT results for testing mean differences')
#'
#'
#' # Selecting the significant components at 5% level
#' which(TWT.result$adjusted_pval < 0.05)
#'
#' @references
#' Abramowicz, K., Pini, A., Schelin, L., Stamm, A., & Vantini, S. (2022).
#' “Domain selection and familywise error rate for functional data: A unified framework. 
#' \emph{Biometrics} 79(2), 1119-1132.
#'
#' Pini, A., & Vantini, S. (2017). Interval-wise testing for functional data. \emph{Journal of Nonparametric Statistics}, 29(2), 407-424
#'
#' @export

TWT2 <- function(data1,data2,mu=0,B=1000,paired=FALSE,dx=NULL,alternative="two.sided"){
  possible_alternatives <- c("two.sided", "less", "greater")
  if(!(alternative %in% possible_alternatives)){
    stop(paste0('Possible alternatives are ',paste0(possible_alternatives,collapse=', ')))
  }
  
  if(is.fd(data1)){ # data1 is a functional data object
    rangeval1 <- data1$basis$rangeval
    rangeval2 <- data2$basis$rangeval
    if(is.null(dx)){
      dx <- (rangeval1[2]-rangeval1[1])*0.01
    }
    if(sum(rangeval1 == rangeval2)!=2){
      stop("rangeval of data1 and data2 must coincide.")
    }
    abscissa <- seq(rangeval1[1],rangeval1[2],by=dx)
    coeff1 <- t(eval.fd(fdobj=data1,evalarg=abscissa))
    coeff2 <- t(eval.fd(fdobj=data2,evalarg=abscissa))
    
  }else if(is.matrix(data1)){
    coeff1 <- data1
    coeff2 <- data2
  }else{
    stop("First argument must be either a functional data object or a matrix.")
  }
  
  if (is.fd(mu)){ # mu is a functional data
    rangeval.mu <- mu$basis$rangeval
    if(sum(rangeval.mu == rangeval1)!=2){
      stop("rangeval of mu must be the same as rangeval of data.")
    }
    if(is.null(dx)){
      dx <- (rangeval.mu[2]-rangeval.mu[1])*0.01
    }
    abscissa <- seq(rangeval.mu[1],rangeval.mu[2],by=dx)
    mu.eval <- t(eval.fd(fdobj=mu,evalarg=abscissa))
  }else if(is.vector(mu)){
    mu.eval <- mu
  }else{
    stop("Second argument must be either a functional data object or a numeric vector.")
  }
  # at the end you have two matrices coeff1 and coeff2 with the point wise evaluation of the functional data pver a grid
  
  n1 <- dim(coeff1)[1]
  n2 <- dim(coeff2)[1]
  n <- n1+n2
  data.eval <- rbind(coeff1,coeff2)
  coeff1 <- coeff1 - matrix(data=mu.eval,nrow=n1,ncol=p)
  
  coeff <- rbind(coeff1,coeff2)
  p <- dim(coeff)[2]
  etichetta_ord <- c(rep(1,n1),rep(2,n2))
  #print('Point-wise tests')
  # First part:
  # univariate permutation test for each point
  # this is the computation that needs to be done for each voxel
  meandiff <- colMeans(coeff[1:n1,,drop=FALSE],na.rm=TRUE) - colMeans(coeff[(n1+1):n,,drop=FALSE],na.rm=TRUE)
  sign.diff <- sign(meandiff)
  sign.diff[which(sign.diff==-1)] <- 0
  T0 <- switch(alternative,
               two.sided =  (meandiff)^2,
               greater   =  (meandiff*sign.diff)^2,
               less      =  (meandiff*(sign.diff-1))^2)
  
  T_coeff <- matrix(ncol=p,nrow=B)
  for (perm in 1:B){ # loop on random permutations
    if(paired==TRUE){ # paired test (for brain data we will not need it)
      if.perm <- rbinom(n1,1,0.5) 
      coeff_perm <- coeff
      for(couple in 1:n1){
        if(if.perm[couple]==1){
          coeff_perm[c(couple,n1+couple),] <- coeff[c(n1+couple,couple),]
        }
      }
    }else if(paired==FALSE){ # unpaired test
      permutazioni <- sample(n)
      coeff_perm <- coeff[permutazioni,]
    }
    
    meandiff <- colMeans(coeff_perm[1:n1,,drop=FALSE],na.rm=TRUE) - colMeans(coeff_perm[(n1+1):n,,drop=FALSE],na.rm=TRUE)
    sign.diff <- sign(meandiff)
    sign.diff[which(sign.diff==-1)] <- 0
    T_coeff[perm,] <- switch(alternative,
                             two.sided =  (meandiff)^2,
                             greater   =  (meandiff*sign.diff)^2,
                             less      =  (meandiff*(sign.diff-1))^2)
    
  }
  
  # p-value computation
  pval <- numeric(p)
  for(i in 1:p){ 
    pval[i] <- sum(T_coeff[,i]>=T0[i])/B
  }
  
  # Second part:
  # combination into subsets
  print('Threshold-wise tests')
  
  thresholds = c(0,sort(unique(pval)),1)
  adjusted.pval <- pval # we initialize the adjusted p-value as unadjusted one
  pval.tmp <- rep(0,p) # inizialize p-value vector resulting from combined test
  for(test in 1:length(thresholds)){
    #print(paste(test,length(thresholds)))
    # test below threshold
    points.1 = which(pval <= thresholds[test])
    T0_comb = sum(T0[points.1],na.rm=TRUE) # combined test statistic
    T_comb <- (rowSums(T_coeff[,points.1,drop=FALSE],na.rm=TRUE))
    pval.test <- mean(T_comb>=T0_comb)
    pval.tmp[points.1] <- pval.test
    # compute maximum
    adjusted.pval = apply(rbind(adjusted.pval,pval.tmp),2,max) 
    
    # test above threshold
    points.2 = which(pval > thresholds[test])
    T0_comb = sum(T0[points.2]) # combined test statistic
    T_comb <- (rowSums(T_coeff[,points.2,drop=FALSE],na.rm=TRUE))
    pval.test <- mean(T_comb>=T0_comb)
    pval.tmp[points.2] <- pval.test
    # compute maximum
    adjusted.pval = apply(rbind(adjusted.pval,pval.tmp),2,max) 
    
  }
  
  result = list(
    test = '2pop', 
    mu = mu.eval,
    adjusted_pval = adjusted.pval,
    unadjusted_pval = pval,
    data.eval=data.eval,
    ord_labels = etichetta_ord
  )
  class(result) = 'fdatest2'
  return(result)
}