removal: 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.

Usage

removal(catch, method = c("CarleStrub", "Zippin", "Seber3", "Seber2", "RobsonRegier2", "Moran", "Schnute"), alpha = 1, beta = 1, CS.se = c("Zippin", "alternative"), conf.level = 0.95, just.ests = FALSE, Tmult = 3)
"summary"(object, parm = c("No", "p", "p1"), digits = getOption("digits"), verbose = FALSE, ...)
"confint"(object, parm = c("No", "p"), level = conf.level, conf.level = NULL, digits = getOption("digits"), verbose = FALSE, ...)

Arguments

catch

A numerical vector of catch at each pass.

method

A single string that identifies the removal method to use. See details.

alpha

A single numeric value for the alpha parameter in the CarleStrub method (default is 1).

beta

A single numeric value for the beta parameter in the CarleStrub method (default is 1).

CS.se

A single string that identifies whether the SE in the CarleStrub method should be computed according to Seber or Zippin.

conf.level

A single number representing the level of confidence to use for constructing confidence intervals. This is sent in the main removal function rather than confint.

just.ests

A logical that indicates whether just the estimates (=TRUE) or the return list (=FALSE; default; see below) is returned.

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 method. Large values are much slower to compute, but too low of a value can result in missing the best estimate. A warning is issued if too low of a value is suspected.

object

An object saved from removal().

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.

digits

A single numeric that controls the number of decimals in the output from summary and confint.

verbose

A logical that indicates whether descriptive labels should be printed from summary and if certain warnings are shown with confint.

level

Note used, but her for compatability with generic confint function.

...

Additional arguments for methods.

Value

A vector that contains the estimates and standard errors for No and p if just.ests=TRUE or (default) a list with at least the following items:

catch The original vector of observed catches.
method The method used (provided by the user).
lbl A descriptive label for the method used.
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

min.nlogLH The minimum value of the negative log-likelihood function.
Tmult The Tmult value sent by the user.

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 esitmate 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 resultsin Cowx (1983) The Zippin method results do not match the examples in Seber (2002) or Cowx (1983) because 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.

IFAR Chapter

10-Abundance from Depletion Data.

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 seven methods chosen with method=. The possible methods are:

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).
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.
method="Seber3": The special case for k=3 estimator shown in equation 7.24 of Seber(2002).
method="Seber2": The special case for k=2 estimator shown on page 312 of Seber(2002).
method="RobsonRegier2": The special case for k=2 estimator shown by Robson and Regier (1968).
method="Moran": The likelihood method of Moran (1951) as implemented by Schnute (1983).
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..

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 last 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..

In the Carle Strub method, if the resultant No estimate is equal to the sum of the catches (T) then the estimate of No that is returned will be the sum of the catches. In this instance, and if the “Seber” method of computing the standard error is used, then the SE will not be estimable and the confidence intervals can not be constructed.

References

Ogle, D.H. 2016. 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.]

Examples

Run this code

## 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)


## 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

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