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S2. Statistical Appendix.
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Several distinct statistical procedures are described here. All statistical analyses were conducted
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in R (version 2.11.1; [1]). Bayesian statistics were conducted using the BRugs package, which
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interfaces R to OpenBUGS [2]. All approaches follow methods described by McCarthy [3].
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This appendix is presented in three parts: A description of the model selection procedure (A.), a
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description of the procedure used to generate Figures 4 and 5 (B.) and a description of the
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process used to generate the mean and 95% credible intervals presented in Figure 3 (C.).
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A. Model selection
In this model selection procedure, a comma-separated values (.csv) file of the data (which
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corresponds to a spreadsheet in the Data Appendix) is attached so that R can 'see' the individual
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columns as variables. In the example code below, AllAg (all forms of agricultural land cover) is
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natural-log converted prior to analysis. As a result, in this example the DIC and parameter
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estimates are made for a model logarithmically relating agricultural land cover to the consumer
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tissue δ15N (FF15N). The general format for the regression equation was:
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Consumer tissue δ15N = WatershedProperty(β) + Intercept
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library("BRugs")
## This loads the BRugs package
Year1RiverFF <- read.csv("C:/Users/jhlarson/Desktop/USGS Science/Stable
Isotopes 2011/Analysis/R Model Selection/Year1RiverFF.csv") ## Loading the
dataset
attach(Year1RiverFF)
## attaching an imported dataset
LogAllAg <- log(AllAg+1)
## Regression Model - This creates a function
regressionmodel <- function(){
a~dnorm(0,1.0E-6) ## Non-informative prior
b~dnorm(0,1.0E-6) ## Non-informative prior
prec~dgamma(0.001,0.001) ## Non-informative
that BRugs can use in OpenBUGS
y-intercept
slope
model precision
sy2 <- pow(sd(y[]),2)
R2B <- 1 - 1/(prec*sy2)
## Bayesian R2
for (i in 1:11) ## 1:N, where N is the number of observations
{
mean [i] <- a+b*x[i]
y[i] ~dnorm(mean[i],prec)
}
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}
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used for any 1-parameter model. However, for 2-parameter models, this procedure was slightly
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modified by the addition of a new model:
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regressionmodelfile <- file.path(tempdir(),"regressionmodel.txt")
model <- writeModel(regressionmodel,regressionmodelfile)
inits <- "C:\\Users\\jhlarson\\Desktop\\USGS Science\\OpenBugs
Code\\Regression\\Initials.txt"
## location of the initials file: a=0, b=0, prec=100
## Test for AllAg.
x <- LogAllAg
## for convenience variable is converted to x to match model
y <- FF15N
## for convenience variable is converted to y to match model
bdata <-bugsData(c("y","x"),,digits=5) ## this places the data into a form
OpenBUGS can read
modelCheck(regressionmodelfile) ## Tells OpenBUGS to check the model
modelData(bdata)
## Tells OpenBUGS to load the data
modelCompile(numChains=1)
## Compiles the model with number of chains
modelInits(inits,)
## Loads the initials
modelUpdate(50000)
## Updates the model 50000 times as a burn-in
samplesSet(c("b","R2B","a","prec"))## Tells OpenBUGS to keep data on these
variables
dicSet()
## Tells OpenBUGS to keep data on DIC
modelUpdate(50000)
## Updates model 50000 times to collect data
YR1R15NLogAllAg <- samplesStats("*")
## Store the variable estimates in
a designated file
YR1R15NLogAllAgDIC <- dicStats()
## Store the DIC estimates in a
designated file
This basic approach was repeated for every model tested. The model above could be
## Two-parameter Regression Model
regressionmodel <- function(){
a~dnorm(0,1.0E-6)
b1~dnorm(0,1.0E-6)
b2~dnorm(0,1.0E-6)
prec~dgamma(0.001,0.001)
sy2 <- pow(sd(y[]),2)
R2B <- 1 - 1/(prec*sy2)
for (i in 1:11)
{
mean [i] <- a+b1*x1[i]+b2*x2[i]
y[i] ~dnorm(mean[i],prec)
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}
}
regressionmodelfile <- file.path(tempdir(),"regressionmodel.txt")
model <- writeModel(regressionmodel,regressionmodelfile)
inits <- "C:\\Users\\jhlarson\\Desktop\\USGS Science\\OpenBugs
Code\\Regression\\Initials2variables.txt"
x1 <- Wdep
x2 <- LogAllAg
y <- FF15N
bdata <-bugsData(c("y","x1","x2"),,digits=5)
In this example, initials were set at a=0, b1=0, b2=0 and prec=100. Other aspects of the
procedure were identical, except that parameters a,b1,b2, R2B and prec were monitored.
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The above models have non-informative prior distributions, and these models were used
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on the data from Larson et al. [4] to generate informative prior distributions that could be used in
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the analysis of the new data. This re-analysis of the earlier data is summarized in Appendix
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Table 1. These distributions were used as informative priors in the analysis of the new data. In
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the example below, data from Appendix Table 1 is used to create prior distributions for model
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parameters and model precision.
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attach(RiverFF)
LogAllAg <- log(AllAg+1)
## Regression Model
regressionmodel <- function(){
a~dnorm(5.213,0.5259)
b~dnorm(1.497,5.4641)
prec~dgamma(13.164,4.5087)
sy2 <- pow(sd(y[]),2)
R2B <- 1 - 1/(prec*sy2)
for (i in 1:22)
{
mean [i] <- a+b*x[i]
y[i] ~dnorm(mean[i],prec)
}
}
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regressionmodelfile <- file.path(tempdir(),"regressionmodel.txt")
model <- writeModel(regressionmodel,regressionmodelfile)
inits <- "C:\\Users\\jhlarson\\Desktop\\USGS Science\\OpenBugs
Code\\Regression\\Initials.txt"
## Test for AllAg.
x <- LogAllAg
y <- FF15N
bdata <-bugsData(c("y","x"),,digits=5)
modelCheck(regressionmodelfile)
modelData(bdata)
modelCompile(numChains=1)
modelInits(inits,)
modelUpdate(50000)
samplesSet(c("a","b","R2B","prec"))
dicSet()
## Tells OpenBUGS to keep data on DIC
modelUpdate(50000)
R15NLogAllAg <- samplesStats("*")
R15NLogAllAgDIC <- dicStats()
## Store the DIC estimates in a
designated file
B. Visualizing the model and 95% credible intervals
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Creating a visual representation of the model plus 95% credible intervals can be done by
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creating predictions from the model across the parameter space and displaying those predictions
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in a graphic. Although not necessarily the best mechanism to do this, we made these estimates in
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R using the BRugs, then exported the resulting estimates to Excel to build a figure. In this
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example, we calculated predictions for the range of possible values following a procedure
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suggested by McCarthy [3].
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RiverFF <- read.csv("C:/Users/jhlarson/Desktop/USGS Science/Stable Isotopes
2011/Analysis/R Model Selection/RiverFF.csv")
attach(RiverFF)
LogAllAg <- log(AllAg+1)
## Regression Model
regressionmodel <- function(){
a~dnorm(5.213,0.5259)
b~dnorm(1.497,5.4641)
prec~dgamma(13.164,4.5087)
prediction0.15<-a+b*0.15
## This generates the prediction for a particular value
prediction0.25<-a+b*0.25
prediction0.35<-a+b*0.35
prediction0.5<-a+b*0.5
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prediction1<-a+b*1
prediction1.6<-a+b*1.6
prediction1.8<-a+b*1.8
prediction2<-a+b*2
prediction2.2<-a+b*2.2
prediction2.4<-a+b*2.4
prediction2.6<-a+b*2.6
prediction3<-a+b*3
prediction3.1<-a+b*3.1
prediction3.2<-a+b*3.2
prediction3.3<-a+b*3.3
prediction3.4<-a+b*3.4
prediction3.5<-a+b*3.5
prediction3.6<-a+b*3.6
prediction3.7<-a+b*3.7
prediction3.8<-a+b*3.8
prediction3.9<-a+b*3.9
prediction4<-a+b*4
prediction4.1<-a+b*4.1
prediction4.2<-a+b*4.2
prediction4.3<-a+b*4.3
prediction4.4<-a+b*4.4
prediction4.45<-a+b*4.45
prediction4.5<-a+b*4.5
prediction4.55<-a+b*4.55
prediction4.6<-a+b*4.6
prediction4.65<-a+b*4.65
prediction5<-a+b*5
sy2 <- pow(sd(y[]),2)
for (i in 1:22)
{
mean [i] <- a+b*x[i]
y[i] ~dnorm(mean[i],prec)
}
}
regressionmodelfile <- file.path(tempdir(),"regressionmodel.txt")
model <- writeModel(regressionmodel,regressionmodelfile)
inits <- "C:\\Users\\jhlarson\\Desktop\\USGS Science\\OpenBugs
Code\\Regression\\Initials.txt"
## Test for AllAg.
x <- LogAllAg
y <- FF15N
bdata <-bugsData(c("y","x"),,digits=5)
modelCheck(regressionmodelfile)
modelData(bdata)
modelCompile(numChains=1)
modelInits(inits,)
modelUpdate(50000)
samplesSet(c("a","b","prediction0.15","prediction0.25","prediction0.35","pred
iction0.5","prediction1","prediction1.6","prediction1.8","prediction2","predi
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ction2.2","prediction2.4","prediction2.6","prediction3","prediction3.1","pred
iction3.2","prediction3.3","prediction3.4","prediction3.5","prediction3.6","p
rediction3.7","prediction3.8","prediction3.9","prediction4","prediction4.1","
prediction4.2","prediction4.3","prediction4.4","prediction4.45","prediction4.
5","prediction4.55","prediction4.6","prediction4.65","prediction5"))
modelUpdate(50000)
LogAgPredictions <- samplesStats("*")
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C. Descriptive statistics
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Estimating a mean using a Bayesian approach includes estimation of 95% credible
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intervals [3]. This makes for a simple test of statistically significant differences: If intervals
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overlap, then the means are not different. The following code was used to estimate mean and
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95% credible intervals for consumer tissue δ15N.
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## First the data file is loaded.
RiverFF <- read.csv("C:/Users/jhlarson/Desktop/USGS Science/Stable Isotopes
2011/Analysis/R Model Selection/RiverFF.csv")
attach(RiverFF) ## This allows R to read columns as variables
## This defines the function for OpenBUGS
regressionmodel <- function(){
## the naming of this function is arbitrary
for (i in 1:22){
x[i] ~ dnorm (mu[1], tau[1])
}
mu[1] ~ dnorm (0, 0.0001) ## non-informative prior distributions were used
tau[1] ~ dgamma (0.001, 0.001)
}
regressionmodelfile <- file.path(tempdir(),"regressionmodel.txt")
model <- writeModel(regressionmodel,regressionmodelfile)
inits <- "C:\\Users\\jhlarson\\Desktop\\USGS Science\\OpenBugs
Code\\1meansinitials.txt" ## this is the location of the initials file
x <- FF15N ## Transforming the variable to the same term used in the
function
bdata <-bugsData(c("x"),,digits=5) ## This prepares the data in the format
OpenBUGS uses
modelCheck(regressionmodelfile)
modelData(bdata)
modelCompile(numChains=1)
modelInits(inits,)
modelUpdate(50000)
## These are the same as described above.
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samplesSet(c("mu","tau"))
modelUpdate(50000)
MeanFF15N<- samplesStats("*")
## The same process is repeated for the RM sites below
RMFF <- read.csv("C:/Users/jhlarson/Desktop/USGS Science/Stable Isotopes
2011/Analysis/RM Model Selection/RMFF.csv")
attach(RMFF)
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1.
R Development Core Team (2010) R: A language and environment for statistical
computing.
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2.
Openbugs T, Best N, Lunn D (2007) The BRugs Package.
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3.
McCarthy M (2007) Bayesian methods for ecology. New York, New York, USA:
Cambridge University Press. p.
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4.
Larson JH, Richardson WB, Vallazza JM, Nelson JC (2012) An exploratory investigation
of the landscape-lake interface: Land cover controls over consumer N and C isotopic
composition in Lake Michigan rivermouths. Journal of Great Lakes Research 38: 610–
619.
## Regression Model
regressionmodel <- function(){
for (i in 1:21){
x[i] ~ dnorm (mu[1], tau[1])
}
mu[1] ~ dnorm (0, 0.0001)
tau[1] ~ dgamma (0.001, 0.001)
}
regressionmodelfile <- file.path(tempdir(),"regressionmodel.txt")
model <- writeModel(regressionmodel,regressionmodelfile)
inits <- "C:\\Users\\jhlarson\\Desktop\\USGS Science\\OpenBugs
Code\\1meansinitials.txt"
x <- FF15N
bdata <-bugsData(c("x"),,digits=5)
modelCheck(regressionmodelfile)
modelData(bdata)
modelCompile(numChains=1)
modelInits(inits,)
modelUpdate(50000)
samplesSet(c("mu","tau"))
modelUpdate(50000)
RMMeanFF15N<- samplesStats("*")
References
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Larson et al. 8 of 8