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Last active Jul 31, 2017

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The R codes to implement the algorithms and figures in Costa, Lopes & Singer (2015).

Description

These are the R codes to implement the algorithms and figures in Costa, Lopes & Singer (2015).

Reference

License

The MIT License (MIT)

Copyright (c) 2016-2017 Eliardo G. Costa & Julio M. Singer

Permission is hereby granted, free of charge, to any person obtaining a copy of
this software and associated documentation files (the "Software"), to deal in
the Software without restriction, including without limitation the rights to
use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
the Software, and to permit persons to whom the Software is furnished to do so,
subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS
FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR
COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER
IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
#-----------------
# Power function
#-----------------
par(cex.main = 1.8, cex.lab = 2)
d <- 8
g <- 0.3
plot(function(x) d*g*x^(g - 1), 0, 5, ylim = c(0, 10), ylab = "concentration (org/mL)",
xlab = "deballasting volume (t)",
main = expression(lambda(t)==paste(delta, gamma, t^{gamma-1})), axes = FALSE)
box()
d <- 1
g <- 2
plot(function(x) d*g*x^(g - 1), 0, 5, ylab = "", xlab = "", add = TRUE)
d <- 5
g <- 1
plot(function(x) d*g*x^(g - 1), 0, 5, ylab = "", xlab = "", add = TRUE)
d <- 0.05
g <- 3
plot(function(x) d*g*x^(g - 1), 0, 5, ylab = "", xlab = "", add = TRUE)
d <- 2
g <- 1.5
plot(function(x) d*g*x^(g - 1), 0, 5, ylab = "", xlab = "", add = TRUE)
text(4.4, 5.4, expression(list(delta==5, gamma==1)), cex = 1.4)
text(0.9, 9.5, expression(list(delta==8, gamma<1)), cex = 1.4)
text(3.6, 9, expression(list(delta==1, gamma==2)), cex = 1.4)
text(3.5, 3.2, expression(list(delta==0.05, gamma>2)), cex = 1.4)
text(4.1, 6.7, expression(list(delta==2, paste("1 ", "< ", gamma, " <", " 2"))),
cex = 1.4, srt = 20)
#--------------------------------#
# Algorithm for Poisson model
#--------------------------------#
alpha <- 0.10
beta <- 0.10
eps <- 0.01 # epsilon
v <- 0.01 # v_0
lambdaA <- 12
c <- qpois(alpha, lambda = v*10, lower.tail = FALSE)
power<- 1 - ppois(c, lambda = v*lambdaA)
while (power < 1 - beta) {
v <- v + eps
c <- qpois(alpha, lambda = v*10, lower.tail = FALSE)
power <- 1 - ppois(c, lambda = v*lambdaA)
}
round(v, 2) # sample volume
c # compliance threshold
round(power, 2) # detection power for lambdaA=12
# optional commands
# detection power for lambdaA= 11.5, 12.5 and 13
lambdaA <- 11.5
round(1 - ppois(c, lambda = v*lambdaA), 2)
lambdaA <- 12.5
round(1 - ppois(c, lambda = v*lambdaA), 2)
lambdaA <- 13
round(1 - ppois(c, lambda = v*lambdaA), 2)
#------------------------------------------#
# Algorithm for stratified Poisson model
#------------------------------------------#
H <- 4 # number of strata
e <- 1 # maximum estimation error
Vh <- c(135, 75, 40, 20) # volume by stratum
alpha.h <- c(0.02, 0.01, 0.01, 0.01)# alpha by stratum
sum(alpha.h) # alpha
a.h <- c(1, 1, 1, 1) # minimum value for the concentration by stratum
b.h <- c(25, 40, 30, 60) # maximum value for the concentration by stratum
V <- sum(Vh) # total volume
Wh <- Vh/V # weights
Wh
eh <- e/(Wh*H) # maximum estimation error by stratum
eh
# computation of the sample sizes n_1, n_2, n_3 and n_4
source("https://raw.githubusercontent.com/eliardocosta/samplesize/master/sample.sizeP.R")
sample.sizeP(d = alpha.h[1], a = a.h[1], b = b.h[1], ea = eh[1])
sample.sizeP(d = alpha.h[2], a = a.h[2], b = b.h[2], ea = eh[2])
sample.sizeP(d = alpha.h[3], a = a.h[3], b = b.h[3], ea = eh[3])
sample.sizeP(d = alpha.h[4], a = a.h[4], b = b.h[4], ea = eh[4])
#-------------------------------------------#
# Algorithm for the Poisson process model
#-------------------------------------------#
# Code for fitting model
# $\lambda(t|\delta,\gamma)=\delta\gamma t^{\gamma-1}$.
V <- 270 # total volume
t <- seq(5, V-5, length = 30) # deballasting volumes
X <- c(0, 1, 6, 4, 8, 9, 9, 13, 12, 17, 18, 19,
14, 21, 30, 30, 25, 28, 24, 31, 31, 24, 45,
38, 31, 36, 40, 46, 53, 45) # organism counts
lambda <- function(d, g, t) { # concentration function
d*g*t^(g - 1)
}
d <- 0.05 # delta parameter
g <- 2.1 # gamma parameter
round(d*V^(g - 1), 3) # true concentration
# (-1)*log-likelihood function
logver <- function(d, g) {
d*g*sum(t^(g - 1)) - log(d*g)*sum(X) - (g - 1)*sum(X*log(t)) + sum(log(factorial(X)))
}
if (is.element("stats4", installed.packages()[,1]) == FALSE) {
install.packages("stats4")
library(stats4)
} else {
library(stats4)
}
fit <- mle(logver, start = list(d = 0.1, g = 0.1)) # fitting the model
summary(fit, digits = 2)
d.est <- coef(fit)[1] # delta estimate
d.est
g.est <- coef(fit)[2] # gamma estimate
g.est
round(sqrt(vcov(fit)), 2) # standard errors
round(confint(fit), 2) # confidence intervals (95%)
# concentration estimate
lambda.est <- d.est*V^(g.est - 1)
round(lambda.est, 2)
# variance for lambda estimator via the delta method
var.lambda <- function(d, g, fit) {
t(c(V^(g - 1),d*log(V)*V^(g - 1)))%*%vcov(fit)%*%c(V^(g - 1),d*log(V)*V^(g - 1))
}
# std error for lambda estimate
se.lambda <- sqrt(var.lambda(d.est, g.est, fit))
round(se.lambda, 2)
# 95% confidence interval for lambda
round(lambda.est + c(-1, 1)*qnorm(1 - 0.025)*se.lambda, 2)
# Z statistics
round((lambda.est - 10)/se.lambda, 2)
# Simulation of the data depicted in Table 1.
X <- numeric()
con <- numeric()
j <- 1
for (i in t) {
con[j] <- lambda(d, g, i)
set.seed(2014 + j)
X[j] <- rpois(1, con[j])
j <- j + 1
}
X # simulated data
# Code for fitting model
# $\lambda(t|\delta,\gamma,\eta)=\eta+[(t-\gamma)/\delta]^2$
V <- 270 # total volume
t <- seq(5, V - 5, length = 30) # deballasting volumes
X <- c(14, 16, 26, 16, 20, 18, 15, 18, 13, 17, 15, 14,
9, 12, 18, 16, 12, 13, 9, 14, 13, 8, 22, 17, 11, 15,
17, 22, 27, 21) # organism counts
lambda <- function(d, e, g, t) { # concentration function
e + ((t - g)/d)^2
}
d <- 40 # delta parameter
e <- 12.5 # eta parameter
g <- 130 # gamma parameter
round((e*V + (V - g)^3/(3*d^2) + g^3/(3*d^2))/V, 2) # true concentration
# (-1)*log-likelihood function
logver <- function(d, e, g) {
length(X)*e + sum(((t - g)/d)^2) - sum(X*log(e + ((t - g)/d)^2)) + sum(log(factorial(X)))
}
fit <- mle(logver, start = list(d = 76, e = min(X), g=t[which.min(X)])) # fitting the model
summary(fit, digits = 3)
d.est <- coef(fit)[1] # delta estimate
d.est
e.est <- coef(fit)[2] # eta estimate
e.est
g.est <- coef(fit)[3] # gamma estimate
g.est
round(sqrt(vcov(fit)), 2) # standard errors
round(confint(fit), 2) # confidence intervals (95%)
# concentration estimate
lambda.est <- (e.est*V + (V - g.est)^3/(3*d.est^2) + g.est^3/(3*d.est^2))/V
round(lambda.est, 2)
# variance for lambda estimator via delta method
var.lambda <- function(d, e, g, fit) {
t(c(-2*((V - g)^3 + g^3)/(3*V*d^3), ((V - g)^3 + g^3)/(3*V*d^2),
(g^2 - (V - g)^2)/(V*d^2)))%*%vcov(fit)%*%c(-2*((V - g)^3 + g^3)/(3*V*d^3),
((V - g)^3 + g^3)/(3*V*d^2), (g^2 - (V - g)^2)/(V*d^2))
}
# std error for lambda estimate
se.lambda <- sqrt(var.lambda(d.est, e.est, g.est, fit))
round(se.lambda, 2)
# 95% confidence interval for lambda
round(lambda.est + c(-1, 1)*qnorm(1 - 0.025)*se.lambda, 2)
# Z statistics
round((lambda.est - 10)/se.lambda, 2)
# Simulation of the data depicted in Table 2.
X <- numeric()
con <- numeric()
j <- 1
for (i in t) {
con[j] <- lambda(d, e, g, i)
set.seed(2014 + j)
X[j] <- rpois(1, con[j])
j <- j + 1
}
X # simulated data
#---------------------------------------------#
# Algorithm for the negative binomial model
#---------------------------------------------#
alpha <- 0.10
beta <- 0.10
phi <- 10
w <- 0.001
lambdaA <- 12
n <- 1
c <- qnbinom(alpha, mu = n*w*10, size = n*phi, lower.tail = FALSE)
power <- 1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi)
while (power < 1 - beta) {
n <- n + 1
c <- qnbinom(alpha, mu = n*w*10, size = n*phi, lower.tail = FALSE)
power <- 1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi)
}
round(n*w, 2) # sample volume
c # compliance threshold
round(power, 2) # detection power for lambdaA=12
# optional commands
# detection powers for lambdaA= 11.5, 12.5 and 13
lambdaA <- 11.5
round(1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi), 2)
lambdaA <- 12.5
round(1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi), 2)
lambdaA <- 13
round(1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi), 2)
#---------------------------------------------------------#
# R codes for the analysis of the Gollasch & David data
#---------------------------------------------------------#
# Poisson model
alpha <- 0.05
beta <- 0.10
w <- 0.27 # volume sampled by Gollasch&David
n <- 9 # number of replicates
lambdaA <- 12
c <- qpois(alpha, lambda = n*w*10, lower.tail = FALSE)
c
power <- 1 - ppois(c, lambda = n*w*lambdaA)
round(power, 2)
while (power < 1 - beta) {
n <- n + 1
c <- qpois(alpha, lambda = n*w*10, lower.tail = FALSE)
power <- 1 - ppois(c, lambda = n*w*lambdaA)
}
n # number of replicates
c # compliance threshold
round(n*w, 2) # sample volume
round(power, 2) # detection power for lambdaA=12
# optional commands
# detection powers for lambdaA= 11.5, 12.5 and 13
lambdaA <- 11.5
round(1 - ppois(c, lambda = n*w*lambdaA), 2)
lambdaA <- 12.5
round(1 - ppois(c, lambda = n*w*lambdaA), 2)
lambdaA <- 13
round(1 - ppois(c, lambda = n*w*lambdaA), 2)
# estimating the parameter phi of the negative binomial model
if (is.element("MASS", installed.packages()[,1])==FALSE) {
install.packages("MASS")
library(MASS)
} else {
library(MASS)
}
case1 <- c(4, 2, 1, 2, 2, 2, 6, 5, 3) # uptake
mod1 <- fitdistr(case1, "negative binomial")
mod1
confint(mod1)
case2 <- c(8, 6, 2, 6, 3, 4, 29, 17, 25) # discharge
mod2 <- fitdistr(case2, "negative binomial")
mod2
confint(mod2)
# Negative binomial model
alpha <- 0.05
beta <- 0.10
phi <- 1.66 # estimate of phi
w <- 0.27 # volume sampled by Gollasch & David
n <- 9 # number of replicates
lambdaA <- 12
c <- qnbinom(alpha, mu = n*w*10, size = n*phi, lower.tail = FALSE)
c
power <- 1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi)
round(power, 2)
while (power < 1 - beta) {
n <- n + 1
c <- qnbinom(alpha, mu = n*w*10, size = n*phi, lower.tail = FALSE)
power <- 1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi)
}
n # number of replicates
c # compliance threshold
round(n*w, 2) # sample volume
round(power, 2) # detection power for lambdaA=12
# optional commands
# detection powers for lambdaA= 11.5, 12.5 and 13
lambdaA <- 11.5
round(1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi), 2)
lambdaA <- 12.5
round(1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi), 2)
lambdaA <- 13
round(1 - pnbinom(c, mu = n*w*lambdaA, size = n*phi), 2)
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