R/popest.R
In USFWS/AKaerial: Analysis of Alaska Region Aerial Survey Data
Defines functions
popest
Documented in
popest
#' Title
#' @param dat A data frame containing transect-level sums of observed singles and pairs, see details.
#' @param vcfData A data frame containing visibility correction factors by strata, see details.
#' @param M A named vector giving the total number of possible transects in each strata. Names must match strata names in data.
#' @param prob A logical indicating where to do the calculation based on detection probability or it reciprocal, vcf.
#' @return A list containing population estimates of 'indicated breeding birds', the 'index" estimates (without detection correction), standard errors of each, and a ggplot objects of each by year.
#' @details
#'
#' @examples
popest <- function(dat=NA, vcfData=NA, Mdat=NA, prob=FALSE){
#Code to calculate detection-corrected population estimates from stratified plot data
# using a ratio estimator. The approach is from Fieberg and Giudice (2008, JWM 72:837),
# hereafter F&G.
#The key parts are equations A3 and A4 of F&G. A3 is explained in F&G;
# A4 is explained best in the text book of Thompson (2012, Sampling).
#Key difference from above citation is the use of the ratio estimator
#for the mean number observed across transects. This was not in F&G or Thompson.
#Code originally written by Erik Osnas, December 2018. Modified in January & February 2019.
#Modified 20190925 for data request from Kylee Durham.
# write as function that accepts dataframe of transect level summaries, and output VCF corrected population estimates.
#Copyright: GNU General Public License v3.0
library(ggplot2)
library(dplyr)
dat <- left_join(dat, vcfData[,1:2])
sing <- dat[dat$Obs_Type=="single", ]
names(sing)[6] <- "Singles"
pair <- dat[dat$Obs_Type=="pair", ]
names(pair)[6] <- "Pairs"
dat2 <- data.frame(sing[,-c(4,6)], Singles=sing[,"Singles"], Pairs=pair[,"Pairs"])
#flkdrakes are treated as singles.
#singles, pairs, and flkdrake of <=4 are doubled for indicated breeding birds
fdop1 <- dat[ (dat$Obs_Type=="flkdrake" & dat$Num==1), ]
names(fdop1)[11] <- "fdop1"
fdop1 <- fdop1[,-4]
fdop2 <- dat[ (dat$Obs_Type=="flkdrake" & dat$Num==2), ]
names(fdop2)[11] <- "fdop2"
fdop2 <- fdop2[,-4]
fdopsf3 <- dat[ (dat$Obs_Type=="flkdrake" & dat$Num==3), ]
names(fdopsf3)[11] <- "fdopsf3"
fdopsf3 <- fdopsf3[,-4]
fdopsf4 <- dat[ (dat$Obs_Type=="flkdrake" & dat$Num==4), ]
names(fdopsf4)[11] <- "fdopsf4"
fdopsf4 <- fdopsf4[,-4]
dat2 <- left_join(dat2, fdop1[,c(1:4,10)])
dat2$fdop1 <- ifelse(is.na(dat2$fdop1),0,dat2$fdop1)
dat2 <- left_join(dat2, fdop2[,c(1:4,10)])
dat2$fdop2 <- ifelse(is.na(dat2$fdop2),0,dat2$fdop2)
dat2 <- left_join(dat2, fdopsf3[,c(1:4,10)])
dat2$fdopsf3 <- ifelse(is.na(dat2$fdopsf3),0,dat2$fdopsf3)
dat2 <- left_join(dat2, fdopsf4[,c(1:4,10)])
dat2$fdopsf4 <- ifelse(is.na(dat2$fdopsf4),0,dat2$fdopsf4)
dat2$Singles <- dat2$Singles + dat2$fdop1 + 2*dat2$fdop2 + 3*dat2$fdopsf3 + 4*dat2$fdopsf4
dat2$Pairs <- dat2$Pairs
dat <- dat2
rm(dat2)
########################
#find mean by statum and year
msing <- aggregate(dat$Singles, by=list(strata=dat$strata, Year=dat$Year), mean)
mpair <- aggregate(dat$Pairs, by=list(strata=dat$strata, Year=dat$Year), mean)
mx <- aggregate(dat$obs.area, by=list(strata=dat$strata, Year=dat$Year), mean) #, na.rm=TRUE
Area <- aggregate(dat$strata.area, by=list(strata=dat$strata, Year=dat$Year), mean)
Year=unique(dat$Year)
mvcf <- aggregate(dat$vcf, by=list(strata=dat$strata, Year=dat$Year), mean)
#calculate "index" estimate, sum singles and pairs over strata by year
#Area*msing/mx is the ratio estimate of the population total of observed singles and pairs
# Yibb is twice the number of singles and pairs
Ysing <- aggregate(Area$x*msing$x/mx$x, list(Year=mx$Year), sum)
Ypair <- aggregate(Area$x*mpair$x/mx$x, list(Year=mx$Year), sum)
Yibb <- 2*(Ysing$x + Ypair$x)
#Find population estimate by applying VCF correction
#For each strata ain each year, multiply by VCF before summing over strata
Nsing <- aggregate(mvcf$x*Area$x*msing$x/mx$x,
list(Year=mx$Year), sum)
Npair <- aggregate(mvcf$x*Area$x*mpair$x/mx$x,
list(Year=mx$Year), sum)
Nibb <- 2*(Nsing$x + Npair$x)
###############################################################################
#Find variance of estimates
#put this all together in a list so we can interate through by strata and year
#to find variance and covariance between counts and transect areas
ss <- list(Year=msing$Year, strata=msing$strata, m = c(), M=c(), Area=Area$x, xVar=c(), sCov = c(),
sVar=c(), pVar=c(), pCov=c(), msing=msing$x, mpair=mpair$x, mx=mx$x, vcf=c())
for(i in 1:length(msing$x)){
tempdat <- dat[c(dat$Year== msing$Year[i] & dat$strata==msing$strata[i]),
c("obs.area", "Singles")]
ss$m[i] <- dim(tempdat)[1]
ss$M[i] <- Mdat[Mdat$Year== msing$Year[i] & Mdat$strata==msing$strata[i], "M"]
temp <- cov(as.matrix(tempdat), use="na.or.complete")
ss$xVar[i] <- temp[1,1]
ss$sCov[i] <- temp[1,2]
ss$sVar[i] <- temp[2,2]
temp <- cov(as.matrix(dat[c(dat$Year== msing$Year[i] & dat$strata==msing$strata[i]),
c("obs.area", "Pairs")]), use="na.or.complete")
ss$pCov[i] <- temp[1,2]
ss$pVar[i] <- temp[2,2]
ss$vcf[i] = vcfData[vcfData$strata==msing$strata[i],"vcf"]
ss$vcfse[i] = vcfData[vcfData$strata==msing$strata[i],"se"]
}
#########################
#Calculate index variance
#first sample variance of the mean number observed per transect.
#The sample variance is eq. 7.5 from Thompson 2012.
ss$VarIndexSing <- with(ss, {
(1 - m/M)*(sVar + (msing/mx)^2*xVar - 2*(msing/mx)*sCov)/m })
ss$VarIndexPair <- with(ss, {
(1 - m/M)*(pVar + (mpair/mx)^2*xVar - 2*(mpair/mx)*pCov)/m })
#expand to all strata area; sum singles and pairs; convert to IBB
ss$VarExpandIBB <- 4 * ss$M^2 * (ss$VarIndexSing + ss$VarIndexPair)
VarIndex <- aggregate(VarExpandIBB ~ Year, data=as.data.frame(ss), FUN=sum)[,2]
#End index variance
#########################
#########################
#Calculate variance in population estimate (that is, "index" corrected by VCF).
#First find the variance in the mean observed across transects.
#This has a component due to sample variance (VarIndexSing/Pair) and due to binomial variance
# (Area/M)*(msing/mx)*(vcf-1)/(vcf*M).
#This is A4 from F&G, where F&G's y_bar is replaced with (Area/M)*(msing/mx),
# the ratio estimate of the transects mean.
#The reciprocal of detection (VCF) is used.
ss$VarYSingHat <- with(ss, {
VarIndexSing + (Area/M)*(msing/mx)*(vcf-1)/(vcf*M) })
ss$VarYPairHat <- with(ss, {
VarIndexPair + (Area/M)*(mpair/mx)*(vcf-1)/(vcf*M) })
#Now sum over strata as in F&G equation A3
#VCF is shared for low, LowE, LowN, and LowS; independent estimates for other strata
#VCF is also shared between singles and pairs.
#need to transform data to long format (couldn't think of a better way)
Singles <- as.data.frame(ss)[,c("Year", "strata", "m", "M", "Area", "xVar", "sCov", "sVar", "msing", "mx", "vcf", "vcfse", "VarYSingHat")]
Pairs <- as.data.frame(ss)[,c("Year", "strata", "m", "M", "Area", "xVar", "pCov", "pVar", "mpair", "mx", "vcf", "vcfse", "VarYPairHat")]
#rename VarYSingHat and VarYPairHat to VarYHat so we can stack
names(Singles) <- names(Pairs) <-
c("Year", "strata", "m", "M", "Area", "xVar", "yxCov", "yVar", "my", "mx", "vcf", "vcfse", "VarYHat")
ssLong <- rbind(Singles, Pairs)
#write as function todo F&G A3 calculation
#summing over strata that share a VCF (here density strata and singles and pairs)
VarFG <- function(data=NA, VisCorFact=NA, VisCorFactSE=NA){
with(data, {
SumAy <- aggregate( Area*(my/mx), by=list(Year), sum) #Area*(my/mx) is the ratio estimate of the mean
SumA2vy <- aggregate( VarYHat*(M^2), by=list(Year), sum) #Area*(my/mx) = (my/mx)*(Area/M)*M = y_bar*M
data.frame(Year=SumAy[,1], Var=SumAy$x^2 * (VisCorFactSE^2) + (VisCorFact^2) * SumA2vy$x - (VisCorFactSE^2)*SumA2vy$x)
})
}
#select the four low density strata and common VCF
VarLow <- VarFG(data=as.list(as.data.frame(ssLong)[ssLong$strata%in%c("Low", "LowN", "LowS", "LowE"),]),
VisCorFact=vcfData[vcfData$strata=="Low", "vcf"],
VisCorFactSE = vcfData[vcfData$strata=="Low", "se"])
names(VarLow)[2] <- "VarLow"
#Medium strata
VarMed <- VarFG(data=as.list(as.data.frame(ssLong)[ssLong$strata=="Medium",]),
VisCorFact=vcfData[vcfData$strata=="Medium", "vcf"],
VisCorFactSE = vcfData[vcfData$strata=="Medium", "se"])
names(VarMed)[2] <- "VarMed"
#High strata
VarHigh <- VarFG(data=as.list(as.data.frame(ssLong)[ssLong$strata=="High",]),
VisCorFact=vcfData[vcfData$strata=="High", "vcf"],
VisCorFactSE = vcfData[vcfData$strata=="High", "se"])
names(VarHigh)[2] <- "VarHigh"
#sum and convert to IBB
VarData <- left_join(left_join(left_join(data.frame(Year), VarLow), VarMed), VarHigh)
VarTotal <- 4*rowSums(VarData[,-1], na.rm=TRUE)
#End population variance
#########################
#########################
#########################
#Plot the results
jig <- 0.1
plotData1 <- data.frame(Year=Year+jig, type=rep("Density-adjusted VCF", length(Year)),
Ibb=Nibb,
lower=ifelse(Nibb-2*sqrt(VarTotal)<0,0,Nibb-2*sqrt(VarTotal)),
upper=Nibb+2*sqrt(VarTotal))
plotData2 <- data.frame(Year=Year-jig, type=rep("Index", length(Year)),
Ibb=Yibb,
lower=ifelse(Yibb-2*sqrt(VarIndex)<0,0,Yibb-2*sqrt(VarIndex)),
upper=Yibb+2*sqrt(VarIndex))
plotData <- rbind(plotData1, plotData2)
gplot <- ggplot() +
geom_point(data = plotData, aes(x=Year, y=Ibb, color=type), size=2) +
labs(title="", x="Year", y = "Indicated Breeding Birds") +
scale_x_continuous(breaks = c(seq(1986, 2020, by=2)), limits = c(1986,2020)) +
# scale_y_continuous(breaks = c(seq(0,30000,2000)), limits = c(0,30000)) +
geom_linerange(data=plotData, aes(x=Year, ymin=lower, ymax=upper, color=type), size=1.1) +
scale_colour_manual(values=c("black", "gray50")) +
theme(legend.position=c(0.2, 0.75), legend.title = element_blank())
return(list(popest=data.frame(Year=Year, Nibb=Nibb, seNibb=sqrt(VarTotal), Index=Yibb, seIndex=sqrt(VarIndex)),
plot=gplot)
)
}
USFWS/AKaerial documentation built on April 3, 2025, 4:06 p.m.
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Defines functions popest
Documented in popest
#' Title
#' @param dat A data frame containing transect-level sums of observed singles and pairs, see details.
#' @param vcfData A data frame containing visibility correction factors by strata, see details.
#' @param M A named vector giving the total number of possible transects in each strata. Names must match strata names in data.
#' @param prob A logical indicating where to do the calculation based on detection probability or it reciprocal, vcf.
#' @return A list containing population estimates of 'indicated breeding birds', the 'index" estimates (without detection correction), standard errors of each, and a ggplot objects of each by year.
#' @details
#'
#' @examples
popest <- function(dat=NA, vcfData=NA, Mdat=NA, prob=FALSE){
#Code to calculate detection-corrected population estimates from stratified plot data
# using a ratio estimator. The approach is from Fieberg and Giudice (2008, JWM 72:837),
# hereafter F&G.
#The key parts are equations A3 and A4 of F&G. A3 is explained in F&G;
# A4 is explained best in the text book of Thompson (2012, Sampling).
#Key difference from above citation is the use of the ratio estimator
#for the mean number observed across transects. This was not in F&G or Thompson.
#Code originally written by Erik Osnas, December 2018. Modified in January & February 2019.
#Modified 20190925 for data request from Kylee Durham.
# write as function that accepts dataframe of transect level summaries, and output VCF corrected population estimates.
#Copyright: GNU General Public License v3.0
library(ggplot2)
library(dplyr)
dat <- left_join(dat, vcfData[,1:2])
sing <- dat[dat$Obs_Type=="single", ]
names(sing)[6] <- "Singles"
pair <- dat[dat$Obs_Type=="pair", ]
names(pair)[6] <- "Pairs"
dat2 <- data.frame(sing[,-c(4,6)], Singles=sing[,"Singles"], Pairs=pair[,"Pairs"])
#flkdrakes are treated as singles.
#singles, pairs, and flkdrake of <=4 are doubled for indicated breeding birds
fdop1 <- dat[ (dat$Obs_Type=="flkdrake" & dat$Num==1), ]
names(fdop1)[11] <- "fdop1"
fdop1 <- fdop1[,-4]
fdop2 <- dat[ (dat$Obs_Type=="flkdrake" & dat$Num==2), ]
names(fdop2)[11] <- "fdop2"
fdop2 <- fdop2[,-4]
fdopsf3 <- dat[ (dat$Obs_Type=="flkdrake" & dat$Num==3), ]
names(fdopsf3)[11] <- "fdopsf3"
fdopsf3 <- fdopsf3[,-4]
fdopsf4 <- dat[ (dat$Obs_Type=="flkdrake" & dat$Num==4), ]
names(fdopsf4)[11] <- "fdopsf4"
fdopsf4 <- fdopsf4[,-4]
dat2 <- left_join(dat2, fdop1[,c(1:4,10)])
dat2$fdop1 <- ifelse(is.na(dat2$fdop1),0,dat2$fdop1)
dat2 <- left_join(dat2, fdop2[,c(1:4,10)])
dat2$fdop2 <- ifelse(is.na(dat2$fdop2),0,dat2$fdop2)
dat2 <- left_join(dat2, fdopsf3[,c(1:4,10)])
dat2$fdopsf3 <- ifelse(is.na(dat2$fdopsf3),0,dat2$fdopsf3)
dat2 <- left_join(dat2, fdopsf4[,c(1:4,10)])
dat2$fdopsf4 <- ifelse(is.na(dat2$fdopsf4),0,dat2$fdopsf4)
dat2$Singles <- dat2$Singles + dat2$fdop1 + 2*dat2$fdop2 + 3*dat2$fdopsf3 + 4*dat2$fdopsf4
dat2$Pairs <- dat2$Pairs
dat <- dat2
rm(dat2)
########################
#find mean by statum and year
msing <- aggregate(dat$Singles, by=list(strata=dat$strata, Year=dat$Year), mean)
mpair <- aggregate(dat$Pairs, by=list(strata=dat$strata, Year=dat$Year), mean)
mx <- aggregate(dat$obs.area, by=list(strata=dat$strata, Year=dat$Year), mean) #, na.rm=TRUE
Area <- aggregate(dat$strata.area, by=list(strata=dat$strata, Year=dat$Year), mean)
Year=unique(dat$Year)
mvcf <- aggregate(dat$vcf, by=list(strata=dat$strata, Year=dat$Year), mean)
#calculate "index" estimate, sum singles and pairs over strata by year
#Area*msing/mx is the ratio estimate of the population total of observed singles and pairs
# Yibb is twice the number of singles and pairs
Ysing <- aggregate(Area$x*msing$x/mx$x, list(Year=mx$Year), sum)
Ypair <- aggregate(Area$x*mpair$x/mx$x, list(Year=mx$Year), sum)
Yibb <- 2*(Ysing$x + Ypair$x)
#Find population estimate by applying VCF correction
#For each strata ain each year, multiply by VCF before summing over strata
Nsing <- aggregate(mvcf$x*Area$x*msing$x/mx$x,
list(Year=mx$Year), sum)
Npair <- aggregate(mvcf$x*Area$x*mpair$x/mx$x,
list(Year=mx$Year), sum)
Nibb <- 2*(Nsing$x + Npair$x)
###############################################################################
#Find variance of estimates
#put this all together in a list so we can interate through by strata and year
#to find variance and covariance between counts and transect areas
ss <- list(Year=msing$Year, strata=msing$strata, m = c(), M=c(), Area=Area$x, xVar=c(), sCov = c(),
sVar=c(), pVar=c(), pCov=c(), msing=msing$x, mpair=mpair$x, mx=mx$x, vcf=c())
for(i in 1:length(msing$x)){
tempdat <- dat[c(dat$Year== msing$Year[i] & dat$strata==msing$strata[i]),
c("obs.area", "Singles")]
ss$m[i] <- dim(tempdat)[1]
ss$M[i] <- Mdat[Mdat$Year== msing$Year[i] & Mdat$strata==msing$strata[i], "M"]
temp <- cov(as.matrix(tempdat), use="na.or.complete")
ss$xVar[i] <- temp[1,1]
ss$sCov[i] <- temp[1,2]
ss$sVar[i] <- temp[2,2]
temp <- cov(as.matrix(dat[c(dat$Year== msing$Year[i] & dat$strata==msing$strata[i]),
c("obs.area", "Pairs")]), use="na.or.complete")
ss$pCov[i] <- temp[1,2]
ss$pVar[i] <- temp[2,2]
ss$vcf[i] = vcfData[vcfData$strata==msing$strata[i],"vcf"]
ss$vcfse[i] = vcfData[vcfData$strata==msing$strata[i],"se"]
}
#########################
#Calculate index variance
#first sample variance of the mean number observed per transect.
#The sample variance is eq. 7.5 from Thompson 2012.
ss$VarIndexSing <- with(ss, {
(1 - m/M)*(sVar + (msing/mx)^2*xVar - 2*(msing/mx)*sCov)/m })
ss$VarIndexPair <- with(ss, {
(1 - m/M)*(pVar + (mpair/mx)^2*xVar - 2*(mpair/mx)*pCov)/m })
#expand to all strata area; sum singles and pairs; convert to IBB
ss$VarExpandIBB <- 4 * ss$M^2 * (ss$VarIndexSing + ss$VarIndexPair)
VarIndex <- aggregate(VarExpandIBB ~ Year, data=as.data.frame(ss), FUN=sum)[,2]
#End index variance
#########################
#########################
#Calculate variance in population estimate (that is, "index" corrected by VCF).
#First find the variance in the mean observed across transects.
#This has a component due to sample variance (VarIndexSing/Pair) and due to binomial variance
# (Area/M)*(msing/mx)*(vcf-1)/(vcf*M).
#This is A4 from F&G, where F&G's y_bar is replaced with (Area/M)*(msing/mx),
# the ratio estimate of the transects mean.
#The reciprocal of detection (VCF) is used.
ss$VarYSingHat <- with(ss, {
VarIndexSing + (Area/M)*(msing/mx)*(vcf-1)/(vcf*M) })
ss$VarYPairHat <- with(ss, {
VarIndexPair + (Area/M)*(mpair/mx)*(vcf-1)/(vcf*M) })
#Now sum over strata as in F&G equation A3
#VCF is shared for low, LowE, LowN, and LowS; independent estimates for other strata
#VCF is also shared between singles and pairs.
#need to transform data to long format (couldn't think of a better way)
Singles <- as.data.frame(ss)[,c("Year", "strata", "m", "M", "Area", "xVar", "sCov", "sVar", "msing", "mx", "vcf", "vcfse", "VarYSingHat")]
Pairs <- as.data.frame(ss)[,c("Year", "strata", "m", "M", "Area", "xVar", "pCov", "pVar", "mpair", "mx", "vcf", "vcfse", "VarYPairHat")]
#rename VarYSingHat and VarYPairHat to VarYHat so we can stack
names(Singles) <- names(Pairs) <-
c("Year", "strata", "m", "M", "Area", "xVar", "yxCov", "yVar", "my", "mx", "vcf", "vcfse", "VarYHat")
ssLong <- rbind(Singles, Pairs)
#write as function todo F&G A3 calculation
#summing over strata that share a VCF (here density strata and singles and pairs)
VarFG <- function(data=NA, VisCorFact=NA, VisCorFactSE=NA){
with(data, {
SumAy <- aggregate( Area*(my/mx), by=list(Year), sum) #Area*(my/mx) is the ratio estimate of the mean
SumA2vy <- aggregate( VarYHat*(M^2), by=list(Year), sum) #Area*(my/mx) = (my/mx)*(Area/M)*M = y_bar*M
data.frame(Year=SumAy[,1], Var=SumAy$x^2 * (VisCorFactSE^2) + (VisCorFact^2) * SumA2vy$x - (VisCorFactSE^2)*SumA2vy$x)
})
}
#select the four low density strata and common VCF
VarLow <- VarFG(data=as.list(as.data.frame(ssLong)[ssLong$strata%in%c("Low", "LowN", "LowS", "LowE"),]),
VisCorFact=vcfData[vcfData$strata=="Low", "vcf"],
VisCorFactSE = vcfData[vcfData$strata=="Low", "se"])
names(VarLow)[2] <- "VarLow"
#Medium strata
VarMed <- VarFG(data=as.list(as.data.frame(ssLong)[ssLong$strata=="Medium",]),
VisCorFact=vcfData[vcfData$strata=="Medium", "vcf"],
VisCorFactSE = vcfData[vcfData$strata=="Medium", "se"])
names(VarMed)[2] <- "VarMed"
#High strata
VarHigh <- VarFG(data=as.list(as.data.frame(ssLong)[ssLong$strata=="High",]),
VisCorFact=vcfData[vcfData$strata=="High", "vcf"],
VisCorFactSE = vcfData[vcfData$strata=="High", "se"])
names(VarHigh)[2] <- "VarHigh"
#sum and convert to IBB
VarData <- left_join(left_join(left_join(data.frame(Year), VarLow), VarMed), VarHigh)
VarTotal <- 4*rowSums(VarData[,-1], na.rm=TRUE)
#End population variance
#########################
#########################
#########################
#Plot the results
jig <- 0.1
plotData1 <- data.frame(Year=Year+jig, type=rep("Density-adjusted VCF", length(Year)),
Ibb=Nibb,
lower=ifelse(Nibb-2*sqrt(VarTotal)<0,0,Nibb-2*sqrt(VarTotal)),
upper=Nibb+2*sqrt(VarTotal))
plotData2 <- data.frame(Year=Year-jig, type=rep("Index", length(Year)),
Ibb=Yibb,
lower=ifelse(Yibb-2*sqrt(VarIndex)<0,0,Yibb-2*sqrt(VarIndex)),
upper=Yibb+2*sqrt(VarIndex))
plotData <- rbind(plotData1, plotData2)
gplot <- ggplot() +
geom_point(data = plotData, aes(x=Year, y=Ibb, color=type), size=2) +
labs(title="", x="Year", y = "Indicated Breeding Birds") +
scale_x_continuous(breaks = c(seq(1986, 2020, by=2)), limits = c(1986,2020)) +
# scale_y_continuous(breaks = c(seq(0,30000,2000)), limits = c(0,30000)) +
geom_linerange(data=plotData, aes(x=Year, ymin=lower, ymax=upper, color=type), size=1.1) +
scale_colour_manual(values=c("black", "gray50")) +
theme(legend.position=c(0.2, 0.75), legend.title = element_blank())
return(list(popest=data.frame(Year=Year, Nibb=Nibb, seNibb=sqrt(VarTotal), Index=Yibb, seIndex=sqrt(VarIndex)),
plot=gplot)
)
}
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