Elk River

Data Preparation

The data were provided by Nupqu Limited Partnership. The data were prepared for analysis using R version 4.2.2 (R Core Team 2022) and organized in a SQLite database. All river distances are based on the BC Freshwater Atlas layer.

Key assumptions of the data preparation included:

• PIT Tag numbers were recorded correctly in instances where the data was not verifiable by the PIT Reader log.
• PIT Reader log was taken to be correct in instances where the recorded PIT Tag code did not match the PIT Reader log.
• 2021 data were submitted in Mountain time; 2022 data were submitted in GMT-6.
• Fork lengths were measured accurately.
• GPS coordinates of outing start/end points were extended if they did not overlaps locations of fish captured on a given outing using the coordinates of a previous outing.

Statistical Analysis

Model parameters were estimated using Bayesian methods. The estimates were produced using JAGS (Plummer 2003) and STAN (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2020).

Unless stated otherwise, the Bayesian analyses used weakly informative normal and half-normal prior distributions (Gelman, Simpson, and Betancourt 2017). 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 the potential scale reduction factor \(\hat{R} \leq 1.05\) (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) \(\textrm{ESS} \geq 150\) for each of the monitored parameters (Kery and Schaub 2011, 61).

Model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the proportion of zeros in the data and the first four central moments (mean, variance, skewness and kurtosis) in the deviance residuals were compared to the expected values by simulating new data based on the posterior distribution and assumed sampling distribution and calculating the deviance residuals.

Where computationally practical, the sensitivity of the posteriors to the choice of prior distributions was evaluated by increasing the standard deviations of all normal, half-normal and log-normal priors by an order of magnitude and then using \(\hat{R}\) to evaluate whether the samples were drawn from the same posterior distribution (Joseph L. Thorley and Andrusak 2017).

The parameters are summarised in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020) and the surprisal s-value (Greenland 2019). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of \(>\) 4.32 bits, which is equivalent to a p-value \(<\) 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row.

Model selection was based the heuristic of directional certainty (Kery and Schaub 2011). Fixed effects were included if their s-value was \(>\) 4.32 bits (Kery and Schaub 2011). Based on a similar argument, random effects were included if their standard deviation had lower 95% CLs \(>\) 5% of the estimate.

The results are displayed graphically by plotting the modeled relationships between individual variables and the response 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 average values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). Primary explanatory variables are evaluated based on their estimated effect sizes with 95% CLs (Bradford, Korman, and Higgins 2005)

The analyses were implemented using R version 4.2.2 (R Core Team 2022) and the mbr family of packages.

Model Descriptions

Model Templates

.model{
  bDensity ~ dnorm(6, 2^-2)
  bEfficiency ~ dnorm(-5, 2^-2)
  bEfficiencySecondsPerMetre ~ dnorm(0, 1^-2)
  bTheta ~ dexp(10)
  bDensityAnnual[1] <- 0
  for (i in 2:nannual) {
    bDensityAnnual[i] ~ dnorm(0, 2^-2)
  }
  sDensityAnnualZoneID ~ dnorm(0, 1^-2) T(0,)
  for (i in 1:nannual) {
    for (j in 1:nzone_id) {
      bDensityAnnualZoneID[i, j] ~ dnorm(0, sDensityAnnualZoneID^-2)
    }
  }
  bEfficiencyAnnual[1] <- 0
  for (i in 2:nannual) {
    bEfficiencyAnnual[i] ~ dnorm(0, 2^-2)
  }
  for (i in 1:nObs) {
    log(eDensity[i]) <- bDensity + bDensityAnnual[annual[i]] + bDensityAnnualZoneID[annual[i], zone_id[i]]
    logit(eProb[i]) <- bEfficiency + bEfficiencySecondsPerMetre * seconds_per_metre[i] + bEfficiencyAnnual[annual[i]]
    eEfficiency[i] ~ dbeta((2 * eProb[i]) / bTheta, (2 * (1 - eProb[i])) / bTheta)  
    recaptures[i] ~ dbin(eEfficiency[i], marked[i])
    bAbundance[i] ~ dpois(eDensity[i] * site_length[i] * survey_proportion[i])
    count[i] ~ dbin(eEfficiency[i], bAbundance[i])
  }