Curtis and Qua Creeks Productivity Analysis 2016

Data Collection

The data were collected by Poisson Consulting and the Salmo Streamkeepers. The invertebrate data were processed by Westcott Environmental Services and the water chemistry and periphyton data by CARO Analytic. The data were provided in the form of an Excel database.

Data Preparation

The data were manipulated and plotted using R version 3.3.2 (R Core Team 2015).

Statistical Analysis

Hierarchical Bayesian models were fitted to the electrofishing data using R version 3.3.2 (R Core Team 2015) and JAGS 4.2.0 (Plummer 2015) which interfaced with each other via jaggernaut 2.5.0 (Thorley 2013). For additional information on hierarchical Bayesian modelling in the BUGS language, of which JAGS uses a dialect, the reader is referred to Kery and Schaub (2011, 41–44).

Unless indicated otherwise, the models used prior distributions that were vague in the sense that they did not affect the posterior distributions (Kery and Schaub 2011, 36). The posterior distributions were estimated from a minimum of 1,000 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of three chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that Rhat (Kery and Schaub 2011, 40) was less than 1.1 for each of the parameters in the model (Kery and Schaub 2011, 61). Where relevant, model adequacy was confirmed by examination of residual plots.

The posterior distributions of the fixed (Kery and Schaub 2011, 75) parameters are summarised in terms of a point estimate (mean), lower and upper 95% credible limits (2.5th and 97.5th percentiles), the standard deviation (SD), percent relative error (half the 95% credible interval as a percent of the point estimate) and significance (Kery and Schaub 2011, 37, 42).

Variable selection was achieved by dropping fixed (Kery and Schaub 2011, 77–82) variables with two-sided p-values \(\geq\) 0.05 (Kery and Schaub 2011, 37, 42) and random variables with percent relative errors \(\geq\) 80%. The Deviance Information Criterion (DIC) was not used because it is of questionable validity when applied to hierarchical models (Kery and Schaub 2011, 469).

The results are displayed graphically by plotting the modeled relationships between particular variables and the response with 95% credible intervals (CRIs) with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their mean and first level values respectively while random variables are held constant at their typical values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). Where informative the influence of particular variables is expressed in terms of the effect size (i.e., percent change in the response variable) with 95% CRIs (Bradford, Korman, and Higgins 2005).

Model Descriptions

Model Code

The JAGS model code, which uses a series of naming conventions, is presented below.

model{
  bEfficiency ~ dnorm(0, 5^-2)
  bDensity ~ dnorm(0, 5^-2)

  bEfficiencyYear[1] <- 0
  for (i in 2:nYear) {
    bEfficiencyYear[i] ~ dnorm(0, 5^-2)
  }

  bDensityCreek[1] <- 0
  for (i in 2:nCreek) {
    bDensityCreek[i] ~ dnorm(0, 5^-2)
  }

  sDensitySite ~ dunif(0, 5)
  for (i in 1:nSite) {
    bDensitySite[i] ~ dnorm(0, sDensitySite^-2)
  }

  for (i in 1:length(Length)) {
    logit(eEfficiency[i]) <- bEfficiency + bEfficiencyYear[Year[i]]
    log(eDensity[i]) <- bDensity + bDensitySite[Site[i]]  + bDensityCreek[Creek[i]]

    eAbundance[i] ~ dpois(eDensity[i] * Length[i])
    eRemaining[i,1] <- eAbundance[i]

    for(j in 1:nPass) {
      Catch[i,j] ~ dbin(eEfficiency[i], eRemaining[i,j])
      eRemaining[i,j+1] <- eRemaining[i,j] - Catch[i,j]
    }
  }
}

Model Parameters

The posterior distributions for the fixed (Kery and Schaub 2011 p. 75) parameters in each model are summarised below.

Figures

Temperature

Periphyton

Benthic Invertebrates

Rainbow Trout Length-Weight

Rainbow Trout Density

Chemical

Tables