R/errorModels.R

In dada2: Accurate, high-resolution sample inference from amplicon sequencing data

#' Use a loess fit to estimate error rates from transition counts.
#' 
#' This function accepts a matrix of observed transitions, with each transition
#' corresponding to a row (eg. row 2 = A->C) and each column to a quality score
#' (eg. col 31 = Q30). It returns a matrix of estimated error
#' rates of the same shape. Error rates are estimates by a \code{\link{loess}} fit
#' of the observed rates of each transition as a function of the quality score.
#' Self-transitions (i.e. A->A) are taken to be the left-over probability.
#' 
#' @param trans (Required). A matrix of the observed transition counts. Must be 16 rows,
#' with the rows named "A2A", "A2C", ...
#' 
#' @return A numeric matrix with 16 rows and the same number of columns as trans.
#'  The estimated error rates for each transition (row, eg. "A2C") and quality score
#'  (column, eg. 31), as determined by \code{\link{loess}} smoothing over the quality
#'  scores within each transition category.
#' 
#' @importFrom stats loess
#' @importFrom stats predict
#' 
#' @export
#' 
#' @examples
#' derep1 <- derepFastq(system.file("extdata", "sam1F.fastq.gz", package="dada2"))
#' dada1 <- dada(derep1, err=tperr1)
#' err.new <- loessErrfun(dada1$trans)
#' 
loessErrfun <- function(trans) {
  qq <- as.numeric(colnames(trans))
  est <- matrix(0, nrow=0, ncol=length(qq))
  for(nti in c("A","C","G","T")) {
    for(ntj in c("A","C","G","T")) {
      if(nti != ntj) {
        errs <- trans[paste0(nti,"2",ntj),]
        tot <- colSums(trans[paste0(nti,"2",c("A","C","G","T")),])
        rlogp <- log10((errs+1)/tot)  # 1 psuedocount for each err, but if tot=0 will give NA
        rlogp[is.infinite(rlogp)] <- NA
        df <- data.frame(q=qq, errs=errs, tot=tot, rlogp=rlogp)
        ###! mod.lo <- loess(rlogp ~ q, df, weights=errs) ###!
        mod.lo <- loess(rlogp ~ q, df, weights=tot) ###!
        #        mod.lo <- loess(rlogp ~ q, df)
        pred <- predict(mod.lo, qq)
        maxrli <- max(which(!is.na(pred)))
        minrli <- min(which(!is.na(pred)))
        pred[seq_along(pred)>maxrli] <- pred[[maxrli]]
        pred[seq_along(pred)<minrli] <- pred[[minrli]]
        est <- rbind(est, 10^pred)
      } # if(nti != ntj)
    } # for(ntj in c("A","C","G","T"))
  } # for(nti in c("A","C","G","T"))
  
  # HACKY
  MAX_ERROR_RATE <- 0.25
  MIN_ERROR_RATE <- 1e-7
  est[est>MAX_ERROR_RATE] <- MAX_ERROR_RATE
  est[est<MIN_ERROR_RATE] <- MIN_ERROR_RATE
  
  # Expand the err matrix with the self-transition probs
  err <- rbind(1-colSums(est[1:3,]), est[1:3,],
               est[4,], 1-colSums(est[4:6,]), est[5:6,],
               est[7:8,], 1-colSums(est[7:9,]), est[9,],
               est[10:12,], 1-colSums(est[10:12,]))
  rownames(err) <- paste0(rep(c("A","C","G","T"), each=4), "2", c("A","C","G","T"))
  colnames(err) <- colnames(trans)
  # Return
  return(err)
}

#' Estimate error rates from transition counts in PacBio CCS data.
#' 
#' This function accepts a matrix of observed transitions from PacBio CCS amplicon
#' sequencing data, with each transition
#' corresponding to a row (eg. row 2 = A->C) and each column to a quality score
#' (eg. col 31 = Q30). It returns a matrix of estimated error
#' rates of the same shape. Error rates are estimates by \code{\link{loessErrfun}}
#' for quality scores 0-92, and individually by the maximum likelihood estimate
#' for the maximum quality score of 93.
#' 
#' @param trans (Required). A matrix of the observed transition counts. Must be 16 rows,
#' with the rows named "A2A", "A2C", ...
#' 
#' @return A numeric matrix with 16 rows and the same number of columns as trans.
#'  The estimated error rates for each transition (row, eg. "A2C") and quality score
#'  (column, eg. 31), as determined by \code{\link{loess}} smoothing over the quality
#'  scores within each transition category.
#' 
#' @export
#' 
#' @examples
#' derep.PB <- derepFastq(system.file("extdata", "samPB.fastq.gz", package="dada2"))
#' dada.PB <- dada(derep.PB, errorEstimationFunction=PacBioErrfun, BAND_SIZE=32, selfConsist=TRUE)
#' err.PB <- PacBioErrfun(dada.PB$trans)
#' 
PacBioErrfun <- function(trans) {
  if("93" %in% colnames(trans)) {
    i.93 <- which(colnames(trans) %in% "93")
    if(i.93 != ncol(trans)) stop("Max qual score of 93 not the last column as expected.")
    err <- loessErrfun(trans[,1:(i.93-1)])
    tot93 <- rep(c(sum(trans[1:4,"93"]), sum(trans[5:8,"93"]), sum(trans[9:12,"93"]), sum(trans[13:16,"93"])), each=4)
    err93 <- (trans[,"93"] + 1)/(tot93 + 1)
    err <- cbind(err, "93"=err93)
  } else {
    message("The max qual score of 93 was not detected. Using standard error fitting.")
    err <- loessErrfun(trans)
  }
  return(err)
}

#' Estimate error rates for each type of transition while ignoring quality scores.
#' 
#' This function accepts a matrix of observed transitions, groups together all observed
#' transitions regardless of quality scores, and estimates the error rate for that transition
#' as the observed fraction of those transitions. This can be used in place of the default
#' \code{\link{loessErrfun}} when calling \code{\link{learnErrors}} or \code{link{dada}}
#' with the effect that quality scores will be effectively ignored.
#' 
#' @param trans (Required). A matrix of the observed transition counts. Must be 16 rows,
#' with the rows named "A2A", "A2C", ...
#' 
#' @param pseudocount (Optional). Default 1. 
#'  Added to each type of transition.
#' 
#' @return A numeric matrix with 16 rows and the same number of columns as trans.
#'  The estimated error rates for each transition (row, eg. "A2C") are identical across
#'  all columns (which correspond to quality scores).
#' 
#' @export
#' 
#' @examples
#' fl1 <- system.file("extdata", "sam1F.fastq.gz", package="dada2")
#' err.noqual <- learnErrors(fl1, errorEstimationFunction=noqualErrfun)
#' 
noqualErrfun <- function(trans, pseudocount=1) {
  # Init matrix to record the estimated transition probabilities
  est <- matrix(0, nrow=0, ncol=ncol(trans))
  obs <- rowSums(trans) + pseudocount
  for(nti in c("A","C","G","T")) {
    for(ntj in c("A","C","G","T")) {
      if(nti != ntj) {
        row.name <- paste0(nti,"2",ntj)
        # Estimate transition rate by aggregating across all quality scores
        #        tot.trans <- sum(trans[row.name,])
        #        tot.init.nt <- sum(trans[paste0(nti,"2",c("A","C","G","T")),])
        tot.trans <- obs[row.name]
        tot.init.nt <- sum(obs[paste0(nti,"2",c("A","C","G","T"))])
        est <- rbind(est, rep(tot.trans/tot.init.nt, ncol(trans)))
      } # if(nti != ntj)
    } # for(ntj in c("A","C","G","T"))
  } # for(nti in c("A","C","G","T"))
  
  # Expand the err matrix with the self-transition probs
  err <- rbind(1-colSums(est[1:3,]), est[1:3,],
               est[4,], 1-colSums(est[4:6,]), est[5:6,],
               est[7:8,], 1-colSums(est[7:9,]), est[9,],
               est[10:12,], 1-colSums(est[10:12,]))
  rownames(err) <- paste0(rep(c("A","C","G","T"), each=4), "2", c("A","C","G","T"))
  colnames(err) <- colnames(trans)
  # Return
  return(err)
}

#' Learns the error rates from an input list, or vector, of file names or a list of \code{\link{derep-class}} objects.
#' 
#' Error rates are learned by alternating between sample inference and error rate estimation 
#'  until convergence. Sample inferences is performed by the \code{\link{dada}} function.
#'  Error rate estimation is performed by \code{errorEstimationFunction}.
#'  The output of this function serves as input to the dada function call as the \code{err} parameter.
#'   
#' @param fls (Required). \code{character}.
#'  The file path(s) to the fastq file(s), or a directory containing fastq file(s).
#'  Compressed file formats such as .fastq.gz and .fastq.bz2 are supported.
#'  A list of \code{\link{derep-class}} ojects can also be provided.
#'  
#' @param nbases (Optional). Default 1e8.
#'  The minimum number of total bases to use for error rate learning. Samples are read into memory
#'  until at least this number of total bases has been reached, or all provided samples have been
#'  read in.
#'    
#' @param nreads (Optional). Default NULL. DEPRECATED.
#'  Please update your code to use the nbases parameter.
#'  
#' @param errorEstimationFunction (Optional). Function. Default \code{\link{loessErrfun}}.
#' 
#'  \code{errorEstimationFunction} is computed on the matrix of observed transitions
#'  after each sample inference step in order to generate the new matrix of estimated error rates.
#'    
#' @param multithread (Optional). Default is FALSE.
#'  If TRUE, multithreading is enabled and the number of available threads is automatically determined.   
#'  If an integer is provided, the number of threads to use is set by passing the argument on to
#'  \code{\link{setThreadOptions}}.
#'   
#' @param randomize (Optional). Default FALSE.
#'  If FALSE, samples are read in the provided order until enough reads are obtained.
#'  If TRUE, samples are picked at random from those provided.
#'  
#' @param MAX_CONSIST (Optional). Default 10.
#'  The maximum number of times to step through the self-consistency loop. If convergence was not
#'  reached in MAX_CONSIST steps, the estimated error rates in the last step are returned.
#'  
#' @param OMEGA_C (Optional). Default 0.
#' The threshold at which unique sequences inferred to contain errors are corrected in the final output,
#'  and used to estimate the error rates (see more at \code{\link{setDadaOpt}}). For reasons of convergence,
#'  and because it is more conservative, it is recommended to set this value to 0, which means that all
#'  reads are counted and contribute to estimating the error rates. 
#'  
#' @param qualityType (Optional). \code{character(1)}.
#'  The quality encoding of the fastq file(s). "Auto" (the default) means to
#'  attempt to auto-detect the encoding. This may fail for PacBio files with
#'  uniformly high quality scores, in which case use "FastqQuality". This
#'  parameter is passed on to \code{\link[ShortRead]{readFastq}}; see
#'  information there for details.
#'  
#' @param verbose (Optional). Default TRUE 
#'  Print verbose text output. More fine-grained control is available by providing an integer argument.
#' \itemize{ 
#'  \item{0: Silence. No text output (same as FALSE).}
#'  \item{1: Basic text output (same as TRUE). }
#'  \item{2: Detailed text output, mostly intended for debugging. }
#' }
#'  
#' @param ... (Optional). Additional arguments will be passed on to the \code{\link{dada}} function.
#'  
#' @return A named list with three entries:
#'  $err_out: A numeric matrix with the learned error rates.
#'  $err_in: The initialization error rates (unimportant).
#'  $trans: A feature table of observed transitions for each type (eg. A->C) and quality score.
#'  
#' @importFrom methods is
#' 
#' @export
#' 
#' @seealso 
#'  \code{\link{derepFastq}}, \code{\link{plotErrors}}, \code{\link{loessErrfun}}, \code{\link{dada}}
#'
#' @examples
#'  fl1 <- system.file("extdata", "sam1F.fastq.gz", package="dada2")
#'  fl2 <- system.file("extdata", "sam2F.fastq.gz", package="dada2")
#'  err <- learnErrors(c(fl1, fl2))
#'  err <- learnErrors(c(fl1, fl2), nbases=5000000, randomize=TRUE)
#'  # Using a list of derep-class objects
#'  dereps <- derepFastq(c(fl1, fl2))
#'  err <- learnErrors(dereps, multithread=TRUE, randomize=TRUE, MAX_CONSIST=20)
#' 
learnErrors <- function(fls, nbases=1e8, nreads=NULL, errorEstimationFunction = loessErrfun, multithread=FALSE, 
                        randomize=FALSE, MAX_CONSIST=10, OMEGA_C=0, qualityType = "Auto", verbose=FALSE, ...) {
  if(!is.null(nreads)) {
    warning("The nreads parameter is DEPRECATED. Please update your code with the nbases parameter.")
  }
  NBASES <- 0
  NREADS <- 0
  if(is(fls, "derep")) { fls <- list(fls) } # A single derep-class object
  if(is.character(fls) && length(fls) == 1 && dir.exists(fls)) { fls <- parseFastqDirectory(fls) }
  drps <- vector("list", length(fls))
  if(randomize) { fls <- sample(fls) }
  for(i in seq_along(fls)) {
    if (is.list.of(fls, "derep")){
        drps[[i]] <- fls[[i]]
    } else {
        drps[[i]] <- derepFastq(fls[[i]], qualityType = qualityType)
    }
    NREADS <- NREADS + sum(drps[[i]]$uniques)
    NBASES <- NBASES + sum(drps[[i]]$uniques * nchar(names(drps[[i]]$uniques)))
    if(is.null(nreads) && NBASES > nbases) { break }
    if(!is.null(nreads) && NREADS > nreads) { break }
  }
  drps <- drps[1:i]
  if(is.logical(verbose) || verbose > 0) {
    cat(NBASES, "total bases in", NREADS, "reads from", i, "samples will be used for learning the error rates.\n")
  }
  # Run dada in self-consist mode on those samples
  dds <- dada(drps, err=NULL, errorEstimationFunction=errorEstimationFunction, selfConsist=TRUE, 
              multithread=multithread, verbose=verbose, MAX_CONSIST=MAX_CONSIST, OMEGA_C=OMEGA_C, ...)
  return(getErrors(dds, detailed=TRUE))
}

#' Extract already computed error rates.
#' 
#' @param obj (Required). An R object with error rates.
#'  Supported objects: dada-class; list of dada-class; numeric matrix; named list with $err_out, $err_in, $trans.
#' 
#' @param detailed (Optional). Default FALSE.
#'  If FALSE, an error rate matrix corresponding to $err_out is returned.
#'  If TRUE, a named list with $err_out, $err_in and $trans. $err_in and $trans can be NULL.
#'  
#' @param enforce (Optional). Default TRUE.
#'  If TRUE, will check validity of $err_out and error if invalid or NULL.
#'  
#' @return A numeric matrix of error rates.
#'  Or, if detailed=TRUE, a named list with $err_out, $err_in and $trans.
#'  
#' @importFrom methods is
#' 
#' @export
#' 
#' @examples
#'  fl1 <- system.file("extdata", "sam1F.fastq.gz", package="dada2")
#'  drp <- derepFastq(fl1)
#'  dd <- dada(drp, err=NULL, selfConsist=TRUE)
#'  err <- getErrors(dd)
#' 
getErrors <- function(obj, detailed=FALSE, enforce=TRUE) {
  rval <- list(err_out=NULL, err_in=NULL, trans=NULL)
  if(is(obj, "matrix") && is.numeric(obj)) {
    rval$err_out <- obj
  } else if(is(obj, "dada")) {
    if(!is.null(obj$err_out)) rval$err_out <- obj$err_out
    rval$err_in <- obj$err_in
    rval$trans <- obj$trans
  } else if(is.list.of(obj, "dada")) {
    if(!all(sapply(obj, function(x) identical(x$err_out, obj[[1]]$err_out)))) {
      stop("If list of dada-class objects provided, all must have the same output error rates.")
    }
    if(!is.null(obj[[1]]$err_out)) rval$err_out <- obj[[1]]$err_out
    rval$err_in <- obj[[1]]$err_in
    rval$trans <- accumulateTrans(lapply(obj, function(x) x$trans))
  } else if(is.list(obj) && "err_out" %in% names(obj) && "err_in" %in% names(obj) && "trans" %in% names(obj)) {
    rval <- obj
  }
  
  if(enforce) {
    if(is.null(rval$err_out)) stop("Error matrix is NULL.")
    if(!is.numeric(rval$err_out)) stop("Error matrix must be numeric.")
    if(!(nrow(rval$err_out)==16)) stop("Error matrix must have 16 rows (A2A, A2C, ...).")
    if(!all(rval$err_out>=0)) stop("All error matrix entries must be >= 0.")
    if(!all(rval$err_out<=1)) stop("All error matrix entries must be <=1.")
    if(any(rval$err_out==0)) warning("Zero in error matrix.")
  }
  
  if(detailed) {
    return(rval)
  } else {
    return(rval$err_out)
  }
}

#' Inflates an error rate matrix by a specified factor, while accounting for saturation.
#' 
#' Error rates are "inflated" by the specified factor, while appropriately saturating so that rates
#' cannot exceed 1. The formula is:
#'   new_err_rate <- err_rate * inflate / (1 + (inflate-1) * err_rate)
#'   
#' @param err (Required). A numeric matrix of transition rates (16 rows, named "A2A", "A2C", ...).
#' 
#' @param inflation (Required). The fold-factor by which to inflate the transition rates.
#' 
#' @param inflateSelfTransitions (Optional). Default FALSE.
#'  If True, self-transitions (eg. A->A) are also inflated.
#'  
#' @return An error rate matrix of the same dimensions as the input error rate matrix.
#'  
#' @export
#' 
#' @examples
#'  tperr2 <- inflateErr(tperr1, 2)
#'  tperr3.all <- inflateErr(tperr1, 3, inflateSelfTransitions=TRUE)
#' 
inflateErr <- function(err, inflation, inflateSelfTransitions = FALSE) {
  err <- getErrors(err)
  t_errs <- c("A2C", "A2G", "A2T", "C2A", "C2G", "C2T", "G2A", "G2C", "G2T", "T2A", "T2C", "T2G")
  err[t_errs,] <- (err[t_errs,] * inflation)/(1 + (inflation-1) * err[t_errs,])
  if(inflateSelfTransitions) { # Also inflate the non-substitution probabilities
    t_nonsubs <- c("A2A", "C2C", "G2G", "T2T")
    err[t_nonsubs,] <- (err[t_nonsubs,] * inflation)/(1 + (inflation-1) * err[t_nonsubs,])
  }
  return(err)
}

## Sum matrices of transition counts together, accounting for the possibility
## of variation in the number of columns present in each.
## 
## @param trans (Required). A list of matrices recording the counts of transitions in each sample.
## 
accumulateTrans <- function(trans) {
  maxcol <- max(sapply(trans, ncol))
  rval <- matrix(0L, nrow=16, ncol=maxcol)
  rownames(rval) <- c("A2A", "A2C", "A2G", "A2T", "C2A", "C2C", "C2G", "C2T", "G2A", "G2C", "G2G", "G2T", "T2A", "T2C", "T2G", "T2T")
  colnames(rval) <- seq(0, maxcol-1)  # One col for each integer starting at 0
  for(tt in trans) {
    rval[,1:ncol(tt)] <- rval[,1:ncol(tt)] + tt
  }
  rval
}
  
################################################################################
#  --------------------- REQUIRES FURTHER TESTING --------------------------
# Identify False Positive inferred sequences due to bad bases.
# 
# Illumina sequencing sometimes produces "bad bases", positions at which 
# error rates are significantly higher than expected by the assigned quality
# score. This function identifies the inferred sequences that are likely to
# have been driven by those bad bases.
# 
# @param clust (Required). The $clustering data frame from the dada() output.
#   May be subsetted from the original prior to using this function.
# 
# @param birth_subs (Required). The $birth_subs data frame from the dada() output.
# 
# @param minFraction (Optional). A \code{numeric(1)}. Default is 0.51.
#  The minimum fraction of bad bases among the base positions used to infer the
#  sequence required to call the inferred sequence a false positive.
#   
# @param omegaB (Optional). A \code{numeric(1)}. Default is 1e-10.
#  The p-value threshold below which a base is assigned as "bad".
#  The p-value is calculated by the number of repeated occurrences of a particular
#    base position individually driving the formation of a new cluster. Bad bases 
#    drive many new "1-away" clusters.
#  The null hypothesis being tested is that real differences are distributed
#    uniformly along the sequence. This is not true, biological differences are
#    non-uniform, so this pvalue threshold should be set conservatively.
# 
# @param minOccurence (Optional). A \code{numeric(1)}. Default is 4.
#  The minimum times a single base position must drive the formation of a new cluster
#    before it can be considered a "bad base".
#
# @param verbose (Optional). \code{logical(1)} indicating verbose text output. Default FALSE.
#
# @return Logical vector of length the number of inferred sequences. 
#  TRUE if inferred sequence a false positive.
#  FALSE otherwise.
#
# @seealso \code{\link{getBadBases}}
#
isBadBaseFP <- function(clust, birth_subs, minFraction = 0.51, omegaB = 1e-10, minOccurence = 4, verbose=FALSE) {
  bb <- getBadBases(clust, birth_subs, omegaB, minOccurence, verbose=verbose)
  fps <- tapply(birth_subs$pos, birth_subs$clust, function(x) mean(x %in% bb) >= minFraction)
  fps <- names(fps)[fps]
  rval <- rownames(clust) %in% fps
  if(verbose) {
    cat(sum(rval), "false positives caused by bad bases identified from", nrow(clust), "input sequences.\n")
  }
  rval
}

################################################################################
#  --------------------- REQUIRES FURTHER TESTING --------------------------
# Identify bad base positions.
# 
# Illumina sequencing sometimes produces "bad bases", positions at which 
# error rates are significantly higher than expected by the assigned quality
# score. This function identifies those bad bases.
# 
# @param clust (Required). The $clustering data frame from the dada() output.
#   May be subsetted from the original prior to using this function.
#   
# @param birth_subs (Required). The $birth_subs data frame from the dada() output.
# 
# @param omegaB (Optional). A \code{numeric(1)}. Default is 1e-10.
#  The p-value threshold below which a base is assigned as "bad".
#  The p-value is calculated by the number of repeated occurrences of a particular
#    base position individually driving the formation of a new cluster. Bad bases 
#    drive many new "1-away" clusters.
#  The null hypothesis being tested is that real differences are distributed
#    uniformly along the sequence. This is not true, biological differences are
#    non-uniform, so this pvalue threshold should be set conservatively.
# 
# @param minOccurence (Optional). A \code{numeric(1)}. Default is 4.
#  The minimum times a single base position must drive the formation of a new cluster
#    before it can be considered a "bad base".
#
# @param verbose (Optional). \code{logical(1)} indicating verbose text output. Defaults FALSE.
#
# @return Integer vector of the bad base positions. 
#
# @seealso \code{\link{isBadBaseFP}}
#
#' @importFrom stats ppois
#' @keywords internal
getBadBases <- function(clust, birth_subs, omegaB = 1e-20, minOccurence = 4, verbose=FALSE) {
  oos <- which(clust$birth_ham == 1)
  oopos <- birth_subs[birth_subs$clust %in% oos,]
  tab <- table(oopos$pos)
  if(length(unique(nchar(clust$sequence)))>1) stop("Requires same length sequences.")
  seqlen <- nchar(clust$sequence[[1]])
  posp <- ppois(tab, length(oos)/seqlen, lower.tail=FALSE) * seqlen
  bad_bases <- as.integer(names(posp)[posp<omegaB & tab>=minOccurence])
  if(verbose) {
    cat(length(bad_bases), "bad bases identified.\n")
  }
  return(bad_bases)
}

#' An empirical error matrix.
#'
#' A dataset containing the error matrix estimated by fitting a piecewise linear model to
#' the errors observed in the mock community featured in Schirmer 2015 (metaID 35).
#'
#' @format A numerical matrix with 16 rows and 41 columns.
#'  Rows correspond to the 16 transition (eg. A2A, A2C, ...)
#'  Columns correspond to consensus quality scores 0 to 40.
#'  
#' @name tperr1
NULL

#' An empirical error matrix.
#'
#' A dataset containing the error matrix estimated by DADA2 from the forward reads of the
#' Illumina Miseq 2x250 sequenced Balanced mock community (see manuscript).
#'
#' @format A numerical matrix with 16 rows and 41 columns.
#'  Rows correspond to the 16 transition (eg. A2A, A2C, ...)
#'  Columns correspond to consensus quality scores 0 to 40.
#'  
#' @name errBalancedF
NULL

#' An empirical error matrix.
#'
#' A dataset containing the error matrix estimated by DADA2 from the reverse reads of the
#' Illumina Miseq 2x250 sequenced Balanced mock community (see manuscript).
#'
#' @format A numerical matrix with 16 rows and 41 columns.
#'  Rows correspond to the 16 transition (eg. A2A, A2C, ...)
#'  Columns correspond to consensus quality scores 0 to 40.
#'  
#' @name errBalancedR
NULL