Lower Columbia River Sculpin and Dace Stranding Analysis 2013

Data Collection

Following reduction events, crews are dispatched to salvage stranded fish from areas of concern. At each site the crew record the number of pools, the number of pools surveyed and the number of fish counted by species, genus, family or all species in the surveyed pools. Since 2011, the crew have also lengthed a subsample of individuals.

Data Preparation

The fish stranding data were provided by Golder Associates in the form of an Access database. The discharge data were queried from a BC Hydro database maintained by Poisson Consulting.

The data were prepared for analysis using R version 3.1.0 (Team, 2013).

Statistical Analysis

Hierarchical Bayesian models were fitted to the stranding data using R version 3.1.0 (Team, 2013) and JAGS 3.4.0 (Plummer, 2012) which interfaced with each other via jaggernaut 1.8.1 (Thorley, 2014). 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) pages 41-44.

Unless specified, the models assumed vague (low information) prior distributions (Kéry and Schaub, 2011, p. 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 (Kéry and Schaub, 2011, pp. 38-40). Model convergence was confirmed by ensuring that Rhat (Kéry and Schaub, 2011, p. 40) was less than 1.1 for each of the parameters in the model (Kéry and Schaub, 2011, p. 61). Model adequacy was confirmed by examination of residual plots.

The posterior distributions of the fixed (Kéry and Schaub, 2011, p. 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 (Kéry and Schaub, 2011, p. 37,42).

Variable selection was achieved by dropping uninformative explanatory variables where a variable was considered to be uninformative if its percent relative error was \(\geq\) 100%. In the case of fixed effects this is approximately equivalent to dropping insignificant variables, i.e., those with a significance \(\geq\) 0.05.

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) (Kéry and Schaub, 2011, pp. 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 et al. 2005). Plots were produced using the ggplot2 R package (Wickham, 2009).

Pool Stranding

The probability of stranding a threshold number of fish was analysed using a hierarchical Bayesian general linear mixed model (GLMM) (Kéry and Schaub, 2011, pp. 73-74).

Key assumptions of the GLMM include:

• The probability of stranding varies with the magnitude of the reduction, the stage, whether a site has been recontoured and the day of the year as a second order polynomial.
• The probability of stranding can vary randomly with the site and year.
• Stranding events are Bernoulli distributed

Preliminary analysis indicated that the data were not overdispersed. Preliminary analysis also indicated that the rate of reduction, the delay between the reduction and salvage and the diel period were not informative predictors of the probability of stranding.

Model Code

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

model{

  bStranding ~ dnorm(0, 5^-2)
  bStrandingMagnitude ~ dnorm(0, 2^-2)

  bStrandingStage ~ dnorm(0, 2^-2)

  bStrandingDayte ~ dnorm(0, 2^-2)
  bStrandingDayte2 ~ dnorm(0, 2^-2)

  bStrandingRecontoured[1] <- 0 
  for (i in 2:nRecontoured) {
    bStrandingRecontoured[i] ~ dnorm(0, 2^-2)
  }

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

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

  for(i in 1:length(Stranding)) {

    logit(eStranding[i]) <- bStranding + bStrandingMagnitude * Magnitude[i] + bStrandingStage * Stage[i] + bStrandingDayte * Dayte[i] + bStrandingDayte2 * Dayte[i]^2 + bStrandingRecontoured[Recontoured[i]] + bStrandingSite[Site[i]] + bStrandingYear[Year[i]]

    Stranding[i] ~ dbern(eStranding[i])
  }
}

Model Parameters

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

Figures

Fork Length

Pool Stranding - Sculpin - One Or More

Pool Stranding - Sculpin - Ten Or More

Pool Stranding - Dace - One Or More

Pool Stranding - Dace - Ten Or More