Lower Duncan River Rainbow Trout Spawning Analysis 2017

Data Preparation

The data were provided by BC Hydro, Ministry of Forests, Lands and Natural Resource Operations, Water Survey of Canada and Mountain Waters Research. The data were prepared for analysis using R version 3.4.4 (R Core Team 2017). The total spawner abundance at Gerrard was assumed to be the observed peak spawner count multiplied 3.08 (Irvine 1978).

Statistical Analysis

Model parameters were estimated using Bayesian methods (McElreath 2016).

Unless indicated otherwise, the Bayesian analyses used uninformative normal prior distributions (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 \(\hat{R} \leq 1.1\) (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.

The parameters are summarised in terms of the point estimate, standard deviation (sd), the z-score, lower and upper 95% 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 3.4.4 (R Core Team 2017), JAGS (Plummer 2015), and the mbr family of packages.

Model Descriptions

Model Templates

Redd Surveys

model{
    bAbundance ~ dnorm(0, 5^-2)
    bTiming ~ dunif(70, 140)
    bDuration ~ dunif(7, 28)

    sAbundanceYear ~ dunif(0, 5)
    sTimingYear ~ dunif(0, 40)
    sDurationYear ~ dunif(0, 20)
    for(i in 1:nYear){
      bTimingYear[i] ~ dnorm (0, sTimingYear^-2)
      bAbundanceYear[i] ~ dnorm (0, sAbundanceYear^-2)
      bDurationYear[i] ~ dnorm (0, sDurationYear^-2)
    }

    sRedds ~ dunif(0, 20)
    for(i in 1:nObs){
      log(eAbundance[i]) <- bAbundance + bAbundanceYear[Year[i]]
      eTiming[i] <- bTiming + bTimingYear[Year[i]]
      eDuration[i] <- bDuration + bDurationYear[Year[i]]

      eRedds[i] <- phi((Doy[i] - eTiming[i]) / eDuration[i]) * eAbundance[i]
      Redds[i] ~ dnorm(eRedds[i], sRedds^-2)
    }
  }

Template 1. The model description.

Tables

Redd Surveys

Table 1. Parameter descriptions.

Table 2. Model coefficients.

Table 3. Model summary.

Table 4. Coefficient table for the linear regression of redd abundance in the tailout on spawner abundance at Gerrard.

Dewatered

Table 5. The total number of redds counted and the number dewatered based on the stage elevation at DAL.

Figures

Discharge

Redd Surveys