removal.R

#' @title Population estimates for k-, 3-, or 2-pass removal data.
#'
#' @description Computes estimates, with confidence intervals, of the population size and probability of capture from the number of fish removed in k-, 3-, or 2-passes in a closed population.
#'
#' @details The main function computes the estimates and associated standard errors, if possible, for the initial population size, No, and probability of capture, p, for eight methods chosen with \code{method=}. The possible methods are:
#'  \itemize{
#'    \item \code{method="CarleStrub"}: The general weighted k-pass estimator proposed by Carle and Strub (1978). This function iteratively solves for No in equation 7 of Carle and Strub (1978).
#'    \item \code{method="Zippin"}: The general k-pass estimator generally attributed to Zippin. This function iteratively solves for No in bias corrected version of equation 3 (page 622) of Carle and Strub (1978). These results are not yet trustworthy (see Testing section below).
#'    \item \code{method="Seber3"}: The special case for k=3 estimator shown in equation 7.24 of Seber(2002).
#'    \item \code{method="Seber2"}: The special case for k=2 estimator shown on page 312 of Seber(2002).
#'    \item \code{method="RobsonRegier2"}: The special case for k=2 estimator shown by Robson and Regier (1968).
#'    \item \code{method="Moran"}: The likelihood method of Moran (1951) as implemented by Schnute (1983).
#'    \item \code{method="Schnute"}: The likelihood method of Schnute (1983) for the model that has a different probability of capture for the first sample but a constant probability of capture for all ensuing samples.
#'    \item \code{method="Burnham"}: The general k-pass estimator likelihood method created by Ken Burnham and presented by Van Deventer and Platts (1983). This method is used in the Microfish software (Van Deventer 1989).
#'  }
#'
#' Confidence intervals for the first five methods are computed using standard large-sample normal distribution theory. Note that the confidence intervals for the 2- and 3-pass special cases are only approximately correct if the estimated population size is greater than 200. If the estimated population size is between 50 and 200 then a 95\% CI behaves more like a 90\% CI.
#'
#' Confidence intervals for the next two methods use likelihood ratio theory as described in Schnute (1983) and are only produced for the No parameter. Standard errors are not produced with the Moran or Schnute methods.
#'
#' Confidence intervals for the last method are computed as per Ken Burnham's instructions for the Burnham Method (Jack Van Deventer, personal communication). Specifically, they are calculated with the t-statistic and No-1 degrees of freedom. Please note that the MicroFish software rounds the t-statistic before it calculates the confidence intervals about No and p. If you need the confidence interals produced by FSA::removal to duplicate MicroFish, please use CIMicroFish=TRUE.
#'
#' @param catch A numerical vector of catch at each pass.
#' @param method A single string that identifies the removal method to use. See details.
#' @param alpha A single numeric value for the alpha parameter in the CarleStrub method (default is \code{1}).
#' @param beta A single numeric value for the beta parameter in the CarleStrub method (default is \code{1}).
#' @param CS.se A single string that identifies whether the SE in the CarleStrub method should be computed according to Seber or Zippin.
#' @param object An object saved from \code{removal()}.
#' @param parm A specification of which parameters are to be given confidence intervals, either a vector of numbers or a vector of names. If missing, all parameters are considered.
#' @param level Not used, but here for compatibility with generic \code{confint} function.
#' @param conf.level A single number representing the level of confidence to use for constructing confidence intervals. This is sent in the main \code{removal} function rather than \code{confint}.
#' @param just.ests A logical that indicates whether just the estimates (\code{=TRUE}) or the return list (\code{=FALSE}; default; see below) is returned.
#' @param verbose A logical that indicates whether descriptive labels should be printed from \code{summary} and if certain warnings are shown with \code{confint}.
#' @param digits A single numeric that controls the number of decimals in the output from \code{summary} and \code{confint}.
#' @param Tmult A single numeric that will be multiplied by the total catch in all samples to set the upper value for the range of population sizes when minimizing the log-likelihood and creating confidence intervals for the Moran and Schnute methods. Large values are much slower to compute, but values that are too low may result in missing the best estimate. A warning is issued if too low of a value is suspected.
#' @param CIMicroFish A logical that indicates whether the t value used to calculate confidence intervals when \code{method="Burnham"} should be rounded to two or three decimals and whether the confidence intervals for No should be rounded to whole numbers as done in MicroFish 3.0. The default (\code{=FALSE}) is to NOT round the t values or No confidence interval. This option is provided only so that results will exactly match MicroFish results (see testing).
#' @param \dots Additional arguments for methods.
#'
#' @return A vector that contains the estimates and standard errors for No and p if \code{just.ests=TRUE} or (default) a list with at least the following items:
#'  \itemize{
#'    \item catch The original vector of observed catches.
#'    \item method The method used (provided by the user).
#'    \item lbl A descriptive label for the method used.
#'    \item est A matrix that contains the estimates and standard errors for No and p.
#'  }
#'
#' In addition, if the Moran or Schnute methods are used the list will also contain
#'  \itemize{
#'    \item min.nlogLH The minimum value of the negative log-likelihood function.
#'    \item Tmult The Tmult value sent by the user.
#'  }
#'
#' @section testing: The Carle-Strub method matches the examples in Carle and Strub (1978) for No, p, and the variance of No. The Carle-Strub estimates of No and p match the examples in Cowx (1983) but the SE of No does not. The Carle-Strub estimates of No match the results (for estimates that they did not reject) from Jones and Stockwell (1995) to within 1 individual in most instances and within 1\% for all other instances (e.g., off by 3 individuals when the estimate was 930 individuals).
#'
#' The Seber3 results for No match the results in Cowx (1983).
#'
#' The Seber2 results for No, p, and the SE of No match the results in example 7.4 of Seber (2002) and in Cowx (1983).
#'
#' The RobsonRegier2 results for No and the SE of NO match the results in Cowx (1983)
#'
#' The Zippin method results do not match the examples in Seber (2002) or Cowx (1983) because \code{removal} uses the bias-corrected version from Carle and Strub (1978) and does not use the tables in Zippin (1958). The Zippin method is not yet trustworthy.
#'
#' The Moran and Schnute methods match the examples in Schnute (1983) perfectly for all point estimates and within 0.1 units for all confidence intervals.
#'
#' The Burnham method was tested against the free (gratis) Demo Version of MicroFish 3.0. Powell Wheeler used R to simulate 100, three-pass removal samples with capture probabilities between 0 and 1 and population sizes <= 1000. The Burnham method implemented here exactly matched MicroFish in all 100 trials for No and p. In addition, the CIs for No exactly matched all 100 trials when CIMicroFish=TRUE. Powell was not able to check the CIs for p because the MicroFish 'Quick Population Estimate' does not report them. 
#'
#' @author Derek H. Ogle, \email{DerekOgle51@gmail.com}
#'
#' @author A. Powell Wheeler, \email{powell.wheeler@@gmail.com}
#'
#' @section IFAR Chapter: 10-Abundance from Depletion Data.
#'
#' @seealso See \code{\link{depletion}} for related functionality.
#'
#' @references Ogle, D.H. 2016. \href{https://fishr-core-team.github.io/fishR/pages/books.html#introductory-fisheries-analyses-with-r}{Introductory Fisheries Analyses with R}. Chapman & Hall/CRC, Boca Raton, FL.
#'
#' Carle, F.L. and M.R. Strub. 1978. A new method for estimating population size from removal data. Biometrics, 34:621-630.
#'
#' Cowx, I.G. 1983. Review of the methods for estimating fish population size from survey removal data. Fisheries Management, 14:67-82.
#'
#' Moran, P.A.P. 1951. A mathematical theory of animal trapping. Biometrika 38:307-311.
#'
#' Robson, D.S., and H.A. Regier. 1968. Estimation of population number and mortality rates. pp. 124-158 in Ricker, W.E. (editor) Methods for Assessment of Fish Production in Fresh Waters. IBP Handbook NO. 3 Blackwell Scientific Publications, Oxford.
#'
#' Schnute, J. 1983. A new approach to estimating populations by the removal method. Canadian Journal of Fisheries and Aquatic Sciences, 40:2153-2169.
#'
#' Seber, G.A.F. 2002. The Estimation of Animal Abundance. Edward Arnold, second edition (Reprint).
#'
#' van Dishoeck, P. 2009. Effects of catchability variation on performance of depletion estimators: Application to an adaptive management experiment. Masters Thesis, Simon Fraser University. [Was (is?) from http://rem-main.rem.sfu.ca/theses/vanDishoeckPier_2009_MRM483.pdf.]
#'
#' Van Deventer, J.S. 1989. Microcomputer Software System for Generating Population Statistics from Electrofishing Data--User's Guide for MicroFish 3.0. USDA Forest Service, General Technical Report INT-254. 29 p. [Was (is?) from https://relicensing.pcwa.net/documents/Library/PCWA-L%20460.pdf].
#'
#' Van Deventer, J.S., and W.S. Platts. 1983. Sampling and estimating fish populations from streams. Transactions of the 48th North American Wildlife and Natural Resource Conference. pp. 349-354.
#'
#' @keywords manip
#'
#' @aliases removal summary.removal confint.removal
#'
#' @examples
#' ## First example -- 3 passes
#' ct3 <- c(77,50,37)
#'
#' # Carle Strub (default) method
#' p1 <- removal(ct3)
#' summary(p1)
#' summary(p1,verbose=TRUE)
#' summary(p1,parm="No")
#' summary(p1,parm="p")
#' confint(p1)
#' confint(p1,parm="No")
#' confint(p1,parm="p")
#'
#' # Moran method
#' p2 <- removal(ct3,method="Moran")
#' summary(p2,verbose=TRUE)
#' confint(p2,verbose=TRUE)
#'#'
#' # Schnute method
#' p3 <- removal(ct3,method="Schnute")
#' summary(p3,verbose=TRUE)
#' confint(p3,verbose=TRUE)
#'
#' # Burnham method
#' p4 <- removal(ct3,method="Burnham")
#' summary(p4)
#' summary(p4,verbose=TRUE)
#' summary(p4,parm="No")
#' summary(p4,parm="p")
#' confint(p4)
#' confint(p4,parm="No")
#' confint(p4,parm="p")

#' ## Second example -- 2 passes
#' ct2 <- c(77,37)
#'
#' # Seber method
#' p4 <- removal(ct2,method="Seber2")
#' summary(p4,verbose=TRUE)
#' confint(p4)
#'
#'
#' ### Test if catchability differs between first sample and the other samples
#' # chi-square test statistic from  negative log-likelihoods
#' #   from Moran and Schnute fits (from above)
#' chi2.val <- 2*(p2$min.nlogLH-p3$min.nlogLH)
#' # p-value ... no significant difference
#' pchisq(chi2.val,df=1,lower.tail=FALSE)
#'
#' # Another LRT example ... sample 1 from Schnute (1983)
#' ct4 <- c(45,11,18,8)
#' p2a <- removal(ct4,method="Moran")
#' p3a <- removal(ct4,method="Schnute")
#' chi2.val <- 2*(p2a$min.nlogLH-p3a$min.nlogLH)  # 4.74 in Schnute(1983)
#' pchisq(chi2.val,df=1,lower.tail=FALSE)         # significant difference (catchability differs)
#' summary(p3a)
#'
#'
#' ### Using lapply() to use removal() on many different groups
#' ###   with the removals in a single variable ("long format")
#' ## create a dummy data frame
#' lake <- factor(rep(c("Ash Tree","Bark","Clay"),each=5))
#' year <- factor(rep(c("2010","2011","2010","2011","2010","2011"),times=c(2,3,3,2,2,3)))
#' pass <- factor(c(1,2,1,2,3,1,2,3,1,2,1,2,1,2,3))
#' catch <- c(57,34,65,34,12,54,26,9,54,27,67,34,68,35,12)
#' d <- data.frame(lake,year,pass,catch)
#'
#' ## create a variable that indicates each different group
#' d$group <- with(d,interaction(lake,year))
#' d
#' ## split the catch by the different groups (creates a list of catch vectors)
#' ds <- split(d$catch,d$group)
#' ## apply removal() to each catch vector (i.e., different group)
#' res <- lapply(ds,removal,just.ests=TRUE)
#' res <- data.frame(t(data.frame(res,check.names=FALSE)))
#' ## get rownames from above and split into separate columns
#' nms <- t(data.frame(strsplit(rownames(res),"\\.")))
#' attr(nms,"dimnames") <- NULL
#' fnl <- data.frame(nms,res)
#' ## put names together with values
#' rownames(fnl) <- NULL
#' colnames(fnl)[1:2] <- c("Lake","Year")
#' fnl
#'
#'
#' ### Using apply() to use removal() on many different groups
#' ###   with the removals in several variables ("wide format")
#' ## create a dummy data frame (just reshaped from above as
#' ## an example; -5 to ignore the group variable from above)
#' d1 <- reshape(d[,-5],timevar="pass",idvar=c("lake","year"),direction="wide")
#' ## apply restore() to each row of only the catch data
#' res1 <- apply(d1[,3:5],MARGIN=1,FUN=removal,method="CarleStrub",just.ests=TRUE)
#' res1 <- data.frame(t(data.frame(res1,check.names=FALSE)))
#' ## add the grouping information to the results
#' fnl1 <- data.frame(d1[,1:2],res1)
#' ## put names together with values
#' rownames(fnl1) <- NULL
#' fnl1
#'
#' @rdname removal
#' @export
removal <- function(catch,
                    method=c("CarleStrub","Zippin","Seber3","Seber2",
                             "RobsonRegier2","Moran","Schnute","Burnham"),
                    alpha=1,beta=1,CS.se=c("Zippin","alternative"),
                    conf.level=0.95,just.ests=FALSE,Tmult=3,CIMicroFish=FALSE) {
  # some initial checks
  method <- match.arg(method)
  
  ## Check on conf.level
  iCheckConfLevel(conf.level) 
  
  if (Tmult<1) STOP("'Tmult' should be greater than 1.")
  if (!is.vector(catch)) {
    # if a one row or column matrix then convert to a vector
    if (is.matrix(catch) & nrow(catch)==1) {
      catch <- catch[1,]
    } else if (is.matrix(catch) & ncol(catch)==1) {
      catch <- catch[,1]
    } else if (is.data.frame(catch) & ncol(catch)==1) {
      catch <- catch[,1]
    } else {
      # otherwise send an error
      STOP("'catch' must be a vector.")
    }
  }
  #Check if catch vector is not numeric
  if (!is.numeric(catch)){
    stop("'catch' must be a vector of numeric values.")
  }
  #Check is.whole
  if (!all(is.wholenumber(catch),na.rm=TRUE)) WARN("'catch' contains non-whole numbers.")
  
  # Checking for and removing (if necessary) missing values
  if (any(is.na(catch))) {
    WARN("'NA's removed from 'catch' to continue.")
    catch <- catch[!is.na(catch)]
  }

  if (length(catch)<2) STOP("Cannot perform calculations with one catch value.")

  # intermediate calculations
  # Different methods
  switch(method,
    Zippin=        { tmp <- iZippin(catch,conf.level) },
    CarleStrub=    { tmp <- iCarleStrub(catch,conf.level,alpha,beta,CS.se) },
    Seber3=        { tmp <- iSeber3(catch,conf.level) },
    Seber2=        { tmp <- iSeber2(catch,conf.level) },
    RobsonRegier2= { tmp <- iRobsonRegier2(catch,conf.level) },
    Moran=         { tmp <- iMoran(catch,conf.level,Tmult) },
    Schnute=       { tmp <- iSchnute(catch,conf.level,Tmult) },
    Burnham=       { tmp <- iBurnham(catch,conf.level,Tmult,CIMicroFish) }
  )
  if (just.ests) { tmp <- tmp$est }
  else {
    tmp <- c(tmp,method=method,conf.level=conf.level)
    class(tmp) <- "removal"
  }
  # return object
  tmp
}

#=============================================================
# INTERNAL -- Calculate some intermediate values
#=============================================================
iRemovalKTX <- function(catch) {
  # number of samples
  k <- length(catch)
  # total catch
  T <- sum(catch)
  # cumulative catch prior to ith sample
  #   same as sum(cumsum(catch)[-k]) as used in Schnute (1983)
  i <- 1:k
  X <- sum((k-i)*catch)
  # return named vector
  c(k=k,T=T,X=X)
}

#=============================================================
# INTERNAL -- Calculate CIs with normal theory
#=============================================================
iRemovalNCI <- function(est,se,conf.level) {
  est+c(-1,1)*stats::qnorm(0.5+conf.level/2)*se
}

#=============================================================
# INTERNAL -- Calculate CIs with likelihood theory for the
#             Moran and Schnute methods.
#=============================================================
iRemovalLHCI <- function(method,catch,conf.level,k,T,X,min.nlogLH,Tmult){
  ## critical negative log-likelihood value
  nlogLHcrit <- min.nlogLH+stats::qchisq(conf.level,df=1)/2

  ## Determine if the upper limit is going to fail.
  UCImsg <- LCImsg <- NA
  if (method=="Moran") {
    # Modified catches to eliminates zero catches -- required
    #   because formulas below uses log()s.
    mod.catch <- catch[catch>0]
    # This is (Schnute (1983) equation 3.6
    nlogLHfail <- lfactorial(T)-T*log(T)+T*sum(mod.catch*log(k*mod.catch/T))
  } else {
    # Modified catches to eliminates first catch (per Schnute)
    #   and zero catches as described above
    mod.catch <- catch[-1]
    mod.catch <- mod.catch[mod.catch>0]
    # This is Schnute (1983) equation 3.11
    nlogLHfail <- lfactorial(T)-T*log(T)+T+sum(mod.catch*log(((k-1)*mod.catch)/(T-catch[1])))
  }
  # logical for whether the upper limit fails (TRUE) or not
  #   if it does then set UCI to infinity (per Schnute)
  UCIfail <- nlogLHfail<nlogLHcrit
  if (UCIfail) {
    UCImsg <- "An upper confidence value for 'No' cannot be determined."
    UCI <- Inf
  }

  ## Determine if the lower limit should be the total catch
  # find negative log LH at total catch (internal functions are further below)
  if (method=="Moran") { tmp <- iLHMoran(T,catch,k=k,T=T,XX=X) }
  else { tmp <- iLHSchnute(T,catch,k=k,T=T,XX=X) }
  # if nlLH is the less than critical value then fail (logical will be true)
  #   if it is then set LCI to total catch (per Schnute)
  LCIfail <- tmp<nlogLHcrit
  if (LCIfail) {
    LCImsg <- "The lower confidence value for 'No' has been set at the total catch."
    LCI <- T
  }

  ## Find LCI or UCI if need be (i.e., at least one did not fail above)
  if (!LCIfail | !UCIfail) {
    # create a matrix of N values to compute the negative log likelihood at
    Ntrys <- matrix(seq(T,ceiling(Tmult*T),0.02),ncol=1)
    # compute the nlLH at all those values
    nlogLHvals <- numeric(nrow(Ntrys))
    if (method=="Moran") { nlogLHvals <- apply(Ntrys,MARGIN=1,FUN=iLHMoran,catch=catch,k=k,T=T,XX=X) }
      else { nlogLHvals <- apply(Ntrys,MARGIN=1,FUN=iLHSchnute,catch=catch,k=k,T=T,XX=X) }
    # find which values are less than the critical negative log LH
    #   value and then find the first and last position
    tmp <- range(which(nlogLHvals<=nlogLHcrit))
    # If the last position in the last N tried then the Ns tried were
    #   probably not adequate. Tell the user to up the Tmult value.
    if (max(tmp)==length(Ntrys) & !UCIfail)
      WARN("Upper confidence value is ill-formed; try increasing 'Tmult' in 'removal()'.")
    # The LCI is N in the lower position, UCI is N in the higher position.
    if (!LCIfail) LCI <- Ntrys[tmp[1]]
    if (!UCIfail) UCI <- Ntrys[tmp[2]]
  }
  # return value rounded to one decimal place
  list(CI=round(c(No.LCI=LCI,No.UCI=UCI),1),LCImsg=LCImsg,UCImsg=UCImsg)
}

#=============================================================
# INTERNAL -- Calculate Moran estimates
#=============================================================
## Moran negative log-likelihood function
iLHMoran <- function(N,catch,k,T,XX) {
  # Estimated q (Schnute (1983) equation 3.2)
  #   Note sum(Ti[-k]) in Schnute is XX (to play well with apply()) here
  ##   Note q, Tk in Schnute are p, T here
  p_hat <- T/(k*N-XX)
  # Predicted catches and total catches
  i <- seq(1,k)
  ct_hat <- N*p_hat*(1-p_hat)^(i-1)                 # Schnute (1983) equation 1.8
  Ti_hat <- cumsum(ct_hat)                          # Schnute (1983) equation 1.1
  T_hat <- Ti_hat[k]
  # negative log-likelihood (Schnute (1983) G(Z) and H(Z) (no K)
  #   from equation 2.6). The [catch>0] solves issues with
  #   when a catch=0
  N*log(N)-T*log(T)-(N-T)*log(N-T_hat)-lchoose(N,T)+sum(catch[catch>0]*log(catch[catch>0]/ct_hat[catch>0]))
}

iMoran <- function(catch,conf.level,Tmult) {
  ## Follows methodology described at the bottom of page 2157 in Schnute (1983)
  ##   Note sum(Ti[-k]) in Schnute is X here
  ##   Note q, Tk in Schnute are p, T here
  # Intermediate Calculations
  int <- iRemovalKTX(catch)
  k <- int[["k"]]
  T <- int[["T"]]
  X <- int[["X"]]
  # A check
  if (k<3) STOP("The Moran method requires at least three samples.")
  if (k==3 & catch[3]==0) {
    WARN("The Moran method will fail when the catch in the last of three samples is 0.")
    est <- c(No=NA,No.LCI=NA,No.UCI=NA,p=NA)
    tmp <- list(objective=NA)
    tmpci <- list(LCImsg=NA,UCImsg=NA)
  } else {
    # optimize for N
    tmp <- stats::optimize(iLHMoran,c(T,ceiling(Tmult*T)),
                           catch=catch,k=k,T=T,XX=X)
    N0 <- tmp$minimum
    p <- T/(k*N0-X)
    # compute confidence intervals for No
    tmpci <- iRemovalLHCI("Moran",catch,conf.level,k,T,X,tmp$objective,Tmult)
    est <- c(No=N0,No.LCI=tmpci$CI[[1]],No.UCI=tmpci$CI[[2]],p=p)
  }
  # return list
  list(est=est,catch=catch,min.nlogLH=tmp$objective,Tmult=Tmult,
       LCImsg=tmpci$LCImsg,UCImsg=tmpci$UCImsg,
       lbl="Moran (1951) K-Pass Removal Method")
}

#=============================================================
# INTERNAL -- Calculate Schnute estimates
#=============================================================
## Schnute negative log-likelihood function
iLHSchnute <- function(N,catch,k,T,XX) {
  ##   Note sum(Ti[-k]) in Schnute is XX (to play well with apply()) here
  ##   Note sum(Ti[-k]-catch[1]) in Schnute is XX-(k-1)*catch[1] here
  ##   Note q, Tk in Schnute are p, T here
  # Estimated p's
  p1_hat <- catch[1]/N                                            # Schnute (1983) equation 3.7
  p_hat <- (T-catch[1])/((k-1)*(N-catch[1])-(XX-(k-1)*catch[1]))  # Schnute (1983) equation 3.8
  # Predicted catches and total catches
  ct1_hat <- N*p1_hat                                             # Schnute (1983) equation 1.12
  i <- seq(2,k)
  ct_hat <- N*p_hat*(1-p1_hat)*(1-p_hat)^(i-2)                    # Schnute (1983) equation 1.12
  ct_hat <- c(ct1_hat,ct_hat)
  Ti_hat <- cumsum(ct_hat)
  T_hat <- Ti_hat[k]
  # log-likelihood (Schnute (1983) G(Z) and H(Z) (no K) from equation 2.6)
  #   the [catch>0] solves issues with when a catch=0
  N*log(N)-T*log(T)-(N-T)*log(N-T_hat)-lchoose(N,T)+sum(catch[catch>0]*log(catch[catch>0]/ct_hat[catch>0]))
}

iSchnute <- function(catch,conf.level,Tmult) {
  ## Follows methodology described in Schnute (1983)
  ##   Note sum(Ti[-k]) in Schnute is X here
  ##   Note sum(Ti[-k]-catch[1]) in Schnute is X-(k-1)*catch[1] here
  ##   Note q, Tk in Schnute are p, T here
  # Intermediate Calculations
  int <- iRemovalKTX(catch)
  k <- int[["k"]]
  T <- int[["T"]]
  X <- int[["X"]]
  # A check
  if (k<3) STOP("The Schnute method requires at least three samples.")
  if (k==3 & catch[3]==0) {
    WARN("The Schnute method will fail when the catch in the last of three samples is 0.")
    est <- c(No=NA,No.LCI=NA,No.UCI=NA,p=NA,p1=NA)
    tmp <- list(objective=NA)
    tmpci <- list(LCImsg=NA,UCImsg=NA)
  } else {
    # optimize for N
    tmp <- stats::optimize(iLHSchnute,c(T,ceiling(Tmult*T)),catch=catch,k=k,T=T,XX=X)
    N0 <- tmp$minimum
    p1 <- catch[1]/N0
    p <- (T-catch[1])/((k-1)*(N0-catch[1])-(X-(k-1)*catch[1]))
    # compute confidence intervals for No
    tmpci <- iRemovalLHCI("Schnute",catch,conf.level,k=k,T=T,X=X,tmp$objective,Tmult)
    est <- c(No=N0,No.LCI=tmpci$CI[[1]],No.UCI=tmpci$CI[[2]],p=p,p1=p1)
  }
  # return list
  list(est=est,catch=catch,min.nlogLH=tmp$objective,Tmult=Tmult,
       LCImsg=tmpci$LCImsg,UCImsg=tmpci$UCImsg,
       lbl="Schnute (1983) K-Pass Removal Method w/ Non-constant Initial Catchability")
}

#=============================================================
# INTERNAL -- Calculate Zippin estimates
#=============================================================
iZippinNoVar <- function(N0,p,k) {
  q <- 1-p
  # equation 7.23 (top) in Seber (2002) p.312 (note that k=s)
  (N0*(1-q^k)*(q^k))/(((1-q^k)^2)-((p*k)^2)*q^(k-1))
}

iZippinpVar <- function(N0,p,k) {
  q <- 1-p
  # equation 7.23 (top) in Seber (2002) p.312 (note that k=s)
  (((q*p)^2)*(1-q^k))/(N0*(q*((1-q^k)^2)-((p*k)^2)*(q^k)))
}

iZippin <- function(catch,conf.level) {
  # Get intermediate calculations
  int <- iRemovalKTX(catch)
  k <- int[["k"]]
  T <- int[["T"]]
  X <- int[["X"]]
  # Method does not work with all zero catches
  if (sum(catch)==0) {
    WARN("No catches on any pass results in Zippin model failure.")
    # empty vector of estimates
    tmp <- rep(NA,8)
  } else if (X <= (((T-1)*(k-1))/2)-1) {
    # This condition is from equation 6 in Carle & Strub (1978)
    WARN("Catch data results in Zippin model failure.")
    # empty vector of estimates
    tmp <- rep(NA,8)
  } else {
    # Uses modified equation 3 from Carle & Strub (1978) in the while statement
    N0 <- T
    while ((N0+0.5)*((k*N0-X-T)^k) >= (N0-T+0.5)*((k*N0-X)^k) ) { N0 <- N0+1 }
    # capture probability formula from Zippin (1956) according to Sweka (2006)
    p <- T/(k*N0-X)
    # compute variances
    p.var <- iZippinpVar(N0,p,k)
    N0.var <- iZippinNoVar(N0,p,k)
    # compute CIs
    N0.ci <- iRemovalNCI(N0,sqrt(N0.var),conf.level)
    p.ci <- iRemovalNCI(p,sqrt(p.var),conf.level)
    # vector of estimates
    tmp <- c(N0,sqrt(N0.var),N0.ci,p,sqrt(p.var),p.ci)
  }
  # names to vector of estimates
  names(tmp) <- c("No","No.se","No.LCI","No.UCI","p","p.se","p.LCI","p.UCI")
  # return list
  list(est=tmp,int=int,catch=catch,lbl="Zippin (1956,1958) K-Pass Removal Method")
}

#=============================================================
# INTERNAL -- Calculate Carle-Strub estimates
#=============================================================
iCarleStrub <- function(catch,conf.level,alpha,beta,CS.se=c("Zippin","alternative")) {
  # Some checks
  CS.se <- match.arg(CS.se)
  if (alpha<=0 | beta<=0) STOP("'alpha' and 'beta' must be positive.")
  # Get intermediate calculations
  int <- iRemovalKTX(catch)
  k <- int[["k"]]
  T <- int[["T"]]
  X <- int[["X"]]
  i <- 1:k
  # Uses equation 7 from Carle & Strub (1978) in the while statement
  N0 <- T
  while (((N0+1)/(N0-T+1))*prod((k*N0-X-T+beta+k-i)/(k*N0-X+alpha+beta+k-i)) >= 1.0) { N0 <- N0+1 }
  p <- T/(k*N0-X)
  p.var <- iZippinpVar(N0,p,k)
  if (CS.se=="Zippin") {
    # PE variance ... Seber notes that this works under certain conditions
    N0.var <- iZippinNoVar(N0,p,k)
  } else {
    # Have yet to find a solid reference for this
    N0.var <- (N0*(N0-T)*T)/((T^2)-N0*(N0-T)*(((k*p)^2)/(1-p)))
  }
  # compute CIs
  N0.ci <- iRemovalNCI(N0,sqrt(N0.var),conf.level)
  p.ci <- iRemovalNCI(p,sqrt(p.var),conf.level)
  # vector of estimates
  tmp <- c(N0,sqrt(N0.var),N0.ci,p,sqrt(p.var),p.ci)
  names(tmp) <- c("No","No.se","No.LCI","No.UCI","p","p.se","p.LCI","p.UCI")
  # return list
  list(est=tmp,int=int,CS.prior=c(alpha=alpha,beta=beta),CS.se=CS.se,
       catch=catch,lbl="Carle & Strub (1978) K-Pass Removal Method")
}

#=============================================================
# INTERNAL -- Calculate Seber 3-pass estimates
#=============================================================
iSeber3 <- function(catch,conf.level) {
  # Get intermediate calculations
  int <- iRemovalKTX(catch)
  k <- int[["k"]]
  T <- int[["T"]]
  X <- int[["X"]]
  # Some checks
  if (k!=3) {
    STOP("Seber (2002) 3-pass method can only be used with three samples.")
  } else if (catch[3] >= catch[1]) {
    WARN("Catch data results in model failure for Seber (2002) 3-pass method.")
    # empty vector of estimates
    tmp <- rep(NA,8)
  } else {
    # PE estimate (equation 7.24 (top) (p.315) of Seber (2002) ... note that T=Y)
    #   also matches Cowx (1983) equation 4
    N0 <- (6*X^2 - 3*X*T - T^2 + T*sqrt(T^2 + 6*X*T - 3*X^2))/(18*(X-T))
    # capture probability estimate (equation 7.24 (bottom) (p.315) of Seber (2002) ... note that T=Y)
    p <- (3*X - T - sqrt(T^2 + 6*X*T - 3*X^2))/(2*X)
    p.var <- iZippinpVar(N0,p,k)
    N0.var <- iZippinNoVar(N0,p,k)
    # compute CIs
    N0.ci <- iRemovalNCI(N0,sqrt(N0.var),conf.level)
    p.ci <- iRemovalNCI(p,sqrt(p.var),conf.level)
    # vector of estimates
    tmp <- c(N0,sqrt(N0.var),N0.ci,p,sqrt(p.var),p.ci)
  }
  # names to vector of estimates
  names(tmp) <- c("No","No.se","No.LCI","No.UCI","p","p.se","p.LCI","p.UCI")
  # return list
  list(est=tmp,int=int,catch=catch,lbl="Seber (2002) 3-Pass Removal Method")
}

#=============================================================
# INTERNAL -- Calculate Seber 2-pass estimates
#=============================================================
iSeber2 <- function(catch,conf.level) {
  if (length(catch)!=2) {
    STOP("Seber (2002) 2-Pass method can only be used with two samples.")
  } else if (catch[2] >= catch[1]) {
    WARN("Catch data results in model failure for Seber (2002) 2-Pass method.")
    # empty vector of estimates
    tmp <- rep(NA,8)
  } else {
    # PE estimate from middle of page 312 in Seber (2002)
    N0 <- catch[1]^2/(catch[1]-catch[2])
    # PE variance from equation 7.30 in Seber (2002)
    N0.var <- ((catch[1]^2)*(catch[2]^2)*(catch[1]+catch[2]))/((catch[1]-catch[2])^4)
    # Capture probability from middle of page 312 in Seber (2002)
    p <- 1-catch[2]/catch[1]
    # Capture probability variance from equation 7.31 in Seber (2002)
    p.var <- catch[2]*(catch[1]+catch[2])/(catch[1]^3)
    # compute CIs
    N0.ci <- iRemovalNCI(N0,sqrt(N0.var),conf.level)
    p.ci <- iRemovalNCI(p,sqrt(p.var),conf.level)
    # vector of estimates
    tmp <- c(N0,sqrt(N0.var),N0.ci,p,sqrt(p.var),p.ci)
  }
  # names to vector of estimates
  names(tmp) <- c("No","No.se","No.LCI","No.UCI","p","p.se","p.LCI","p.UCI")
  # return list
  list(est=tmp,catch=catch,lbl="Seber (2002) 2-Pass Removal Method")
}

#=============================================================
# INTERNAL -- Calculate Robson-Regier 2-pass estimates
#=============================================================
iRobsonRegier2 <- function(catch,conf.level) {
  if (length(catch)!=2) {
    STOP("Robson & Regier (1968) 2-pass method can only be used with two samples.")
  } else if (catch[2] >= catch[1]) {
    WARN("Catch data results in model failure for Robson & Regier (1968) 2-pass method.")
    # return empty vector
    tmp <- rep(NA,8)
  } else {
    N0 <- (catch[1]^2-catch[2])/(catch[1]-catch[2])
    N0.var <- ((catch[1]^2)*(catch[2]^2)*(catch[1]+catch[2]))/((catch[1]-catch[2])^4)
    p <- 1-(catch[2]/(catch[1]+1))
    p.var <- catch[2]*(catch[1]+catch[2])/(catch[1]^3)
    # compute CIs
    N0.ci <- iRemovalNCI(N0,sqrt(N0.var),conf.level)
    p.ci <- iRemovalNCI(p,sqrt(p.var),conf.level)
    # vector of estimates
    tmp <- c(N0,sqrt(N0.var),N0.ci,p,sqrt(p.var),p.ci)
  }
  # names to vector of estimates
  names(tmp) <- c("No","No.se","No.LCI","No.UCI","p","p.se","p.LCI","p.UCI")
  # return list
  list(est=tmp,catch=catch,lbl="Robson & Regier (1968) 2-Pass Removal Method")
}

#=============================================================
# INTERNAL -- Calculate Burnham k-pass estimates
#
# Note in notes below that V&P refers to Van Deventer and Platts (1983)
#=============================================================
iBurnham <- function(catch,conf.level,Tmult,CIMicroFish){
  # Get intermediate calculations
  int <- iRemovalKTX(catch)
  k <- int[["k"]] # T in V&P
  T <- int[["T"]] # S in V&P
  i <- 1:k
  # An additional intermediate value from step 2 equation (page 352 in V&P)
  C <- sum(catch*i)

  # Create data.frame for iterative search that finds the No and p
  #   First, check that Burnham is not going to break
  #     the non descending removal pattern must be checked for last.
  if (T==0) {
    WARN("No catches on any pass results in model failure for Burnham method.")
    tmp <- rep(NA,8)
  } else if (T==1) {
    WARN("Total catch of one fish results in model failure for Burnham method.")
    tmp <- rep(NA,8)
  } else if (T==catch[1]) {
    WARN("All fish captured on first pass results in model failure for Burnham method.")
    tmp <- rep(NA,8)
  } else {
    # Note: MicroFish uses Tmult=5 (Van Deventer 1989), whereas FSA::removal defaults
    #       to Tmult=3. This formula is a little convoluted but it forces the iteration
    #       to stop on Tmult+1. We have to explore through Tmult+1 to discover if
    #       Tmult is the population estimate.
    search.df <- data.frame(I=0:(((T*Tmult)-T)+1))
    search.df$N <- T+search.df$I        # Equation in step 3a (page 352 in V&P)
    search.df$P <-T/(C+k*search.df$I)   # Equation in step 3b (page 352 in V&P)
    for (i in 1:length(search.df[,1])) {
      # An exception to the equation in step 3c for the initial row in search.df
      #   from page 352 in V&P. Note that search.df$I=0 when the loop counter 'i'=1.
      if (i==1) search.df$H[i] <- 0+log(1+T/(search.df$I[i]+1))
      # Equivalent to the equation in step 3c from page 352 in V&P
      if (i>1) search.df$H[i] <- search.df$H[i-1]+log(1+T/(search.df$I[i-1]+1))
    }
    # The likelihood function: Equation 3d from page 352 in V&P
    search.df$L <- search.df$H+T*log(search.df$P)+(C-T+(k*search.df$I))*log(1-search.df$P)
    # Find the iteration (row) that has the maximum likelihood
    max.like <- grep(max(search.df$L),search.df$L)
    N0 <- T+search.df$I[max.like]
    p <- search.df$P[max.like]
    N0.var <- ((N0*(1-p)^k)*(1-(1-p)^k))/(((1-(1-p)^k)^2)-((k*p)^2)*((1-p)^(k-1)))
    p.var <- ((p/N0)^2)*((N0.var)/((1-p)^(k-1)))
    # Compute CIs ... Note that (N0-1) is how Ken Burnham recommends calculating
    #   df (Jack Van Deventer, personal correspondance)
    t.statistic <- stats::qt((1-conf.level)/2,N0-1)
    # If asked round t for CI calcs according to MicroFish way
    if (CIMicroFish) t.statistic <- round(t.statistic,digits=ifelse(N0<=100,3,2))
    N0.ci <- N0+c(1,-1)*sqrt(N0.var)*t.statistic
    # If asked for MicroFish confidence intervals, round N0.ci to avoid fractional fish
    if (CIMicroFish) N0.ci <- round(N0.ci,digits=0)
    p.ci <- p+c(1,-1)*sqrt(p.var)*t.statistic
    # Organize the results into a vector
    tmp <- c(N0,sqrt(N0.var),N0.ci,p,sqrt(p.var),p.ci)
    # Check if N0=Tmult+1. If so, the Burnham approach searched from T to Tmult
    #   without finding a population estimate.
    if (N0==(Tmult*T)+1) {
      WARN("The Burnham method failed to find estimates for population size and capture\n",
           " probability. It may help to increase the Tmult option to a value greater\n",
           " than the default (3), but that is unlikely to make a difference. This problem\n",
           " is characteristic of a non-depleting catch pattern.")
      tmp <- rep(NA,8)   # empty the vector of estimates
    }
  }
  # Name the values in the results vector and return them
  names(tmp) <- c("No","No.se","No.LCI","No.UCI","p","p.se","p.LCI","p.UCI")
  list(est=tmp,int=c(int,C=C),catch=catch,
       lbl="Burnham K-Pass Removal Method (Van Deventer and Platts 1983)")
}

#' @rdname removal
#' @export
summary.removal <- function(object,parm=c("No","p","p1"),digits=getOption("digits"),verbose=FALSE,...) {
  parm <- match.arg(parm,several.ok=TRUE)
  # send warning if chose 'p1' parameter but not Schnute method
  #   but don't warn if all parameters are chosen
  #   but stop if only p1 was chosen
  if (("p1" %in% parm) & object$method!="Schnute") {
    msg <- paste("'p1' parameter not relevant for the ",object$method," method.")
    if (length(parm)==1) STOP(msg)
    if (length(parm)<3) WARN(msg)
    parm <- parm[-which(parm=="p1")]
  }
  if (verbose) {
    if (object$method %in% c("Moran","Schnute")) message("The ",object$lbl," was used (SEs not computed).")
    else message("The ",object$lbl," was used.")
  }
  if (object$method %in% c("Zippin","CarleStrub","Seber3","Seber2","RobsonRegier2","Burnham")) {
    res <- matrix(object$est[c("No","No.se","p","p.se")],nrow=2,byrow=TRUE)
    colnames(res) <- c("Estimate","Std. Error")
    rownames(res) <- c("No","p")
  } else if (object$method=="Moran") {
    res <- matrix(object$est[c("No","p")],nrow=2)
    colnames(res) <- c("Estimate")
    rownames(res) <- c("No","p")
  } else {
    res <- matrix(object$est[c("No","p","p1")],nrow=3)
    colnames(res) <- c("Estimate")
    rownames(res) <- c("No","p","p1")
  }
  res <- res[which(rownames(res) %in% parm),,drop=FALSE]
  round(res,digits)
}

#' @rdname removal
#' @export
confint.removal <- function(object,parm=c("No","p"),
                            level=conf.level,conf.level=NULL,
                            digits=getOption("digits"),verbose=FALSE,...) {
  if (!is.null(level)) WARN("The confidence level is not set here, it is set with 'conf.level=' in 'removal()'.")
  parm <- match.arg(parm,several.ok=TRUE)
  if (object$method %in% c("Zippin","CarleStrub","Seber3","Seber2","RobsonRegier2","Burnham")) {
    res <- matrix(object$est[c("No.LCI","No.UCI","p.LCI","p.UCI")],nrow=2,byrow=TRUE)
    rownames(res) <- c("No","p")
    res <- res[which(rownames(res) %in% parm),,drop=FALSE]
  } else {
    ## Handle some messaging
    if (object$method %in% c("Moran","Schnute")) {
      # warn about no CIs for p with Moran and Schnute but only if p is the only parm chosen
      if ("p" %in% parm) {
        if (length(parm)==1) STOP("Confidence intervals for 'p' cannot be computed with ",object$method," method.")
        parm <- "No"
      }
      # print messages about CI fails if they exist
      if (!is.na(object$LCImsg) & verbose) message(object$LCImsg)
      if (!is.na(object$UCImsg) & verbose) message(object$UCImsg)
    }
    res <- matrix(object$est[c("No.LCI","No.UCI")],nrow=1)
    rownames(res) <- c("No")
  }
  colnames(res) <- iCILabel(object$conf.level)
  round(res,digits)
}