Kettle and Granby Rivers Adult Rainbow Trout Density Analysis 2017

Data Preparation

The data were provided in the form of an Access database. The data were rearchived in an SQLite database and prepared for analysis using R version 3.5.0 (R Core Team 2018).

Statistical Analysis

Model parameters were estimated using Bayesian methods. The Bayesian estimates were produced using JAGS (Plummer 2015). For additional information on Bayesian estimation the reader is referred to McElreath (2016).

Unless indicated otherwise, the Bayesian analyses used normal and uniform prior distributions that did not constrain the posteriors (Kery and Schaub 2011, 36). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that \(\hat{R} \leq 1.05\) (Kery and Schaub 2011, 40) and \(\textrm{ESS} \geq 150\) for each of the monitored parameters (Kery and Schaub 2011, 61). Where \(\hat{R}\) is the potential scale reduction factor and \(\textrm{ESS}\) is the effective sample size (Brooks et al. 2011).

The parameters are summarised in terms of the point estimate, standard deviation (sd), the z-score, lower and upper 95% confidence/credible limits (CLs) and the p-value (Kery and Schaub 2011, 37, 42). The estimate is the median (50th percentile) of the MCMC samples, the z-score is \(\mathrm{mean}/\mathrm{sd}\) and the 95% CLs are the 2.5th and 97.5th percentiles. A p-value of 0.05 indicates that the lower or upper 95% CL is 0.

Where relevant, model adequacy was confirmed by examination of residual plots for the full model(s).

The results are displayed graphically by plotting the modeled relationships between particular variables and the response(s) 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). When informative the influence of particular variables is expressed in terms of the effect size (i.e., percent change in the response variable) with 95% confidence/credible intervals (CIs, Bradford, Korman, and Higgins 2005).

The analyses were implemented using R version 3.5.0 (R Core Team 2018) and the mbr family of packages.

Model Descriptions

Density

The lineal density and observer efficiency for Rainbow \(\geq\) 30 cm in length were estimated from the snorkel count and mark-resighting data for the Kettle River using an overdispersed Poisson mark-recapture model.

Key assumptions of the density model include:

• Density varies randomly by year and site.
• The number of resighted individuals is described by a binomial distribution where the probability of resighting is the observer efficiency.
• The expected number of observed individuals is the product of the density, site length, proportion of the site surveyed and the observer efficiency.
• The number of observed individuals is described by a gamma-Poisson distribution.

Model Templates

Density

model {
  bEfficiency ~ dunif(0, 1)

  bDensity ~ dnorm(5, 2^-2)

  sDensityAnnual ~ dunif(0, 1)
  for (yr in 1:nAnnual) {
    bDensityAnnual[yr] ~ dnorm(0, sDensityAnnual^-2)
  }

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

  sDispersion ~ dunif(0, 1)
  for (i in 1:length(Count)) {

    eEfficiency[i] <- bEfficiency
    eResightingEfficiency[i] <- min(eEfficiency[i], Marked[i])

    eAdjustedMarked[i] <- max(Marked[i], 1)
    Resighted[i] ~ dbin(eResightingEfficiency[i] * ProportionSurveyed[i], eAdjustedMarked[i])

    log(eDensity[i]) <- bDensity + bDensityAnnual[Annual[i]] + bDensitySite[Site[i]]
    eAbundance[i] <- eDensity[i] * SiteLength[i]
    eDispersion[i] ~ dgamma(1 / sDispersion^2, 1 / sDispersion^2)
    Count[i] ~ dpois(eAbundance[i] * eEfficiency[i] * ProportionSurveyed[i] * eDispersion[i])
  }
..

Template 1. The model description.

Tables

Density

Table 1. Parameter descriptions.

Table 2. Model coefficients.

Table 3. Model summary.

Figures

Density