Lower Duncan River Rainbow Trout Spawning Analysis 2017

The suggested citation for this analytic report is:

Thorley, J.L. & Dalgarno, S.I.J. (2018) Lower Duncan River Rainbow Trout Spawning Analysis 2017. A Poisson Consulting Analysis Report. URL: http://www.poissonconsulting.ca/f/897442882.


The Lardeau River (LAR) flows 45 km from Trout Lake to join the Lower Duncan River approximately 1 km downstream of Duncan Dam (DDM). The Duncan River downstream of the Lardeau River (DRL) then extends a further 11 km downstream to Kootenay Lake. The study area for this analysis is the Duncan Above Lardeau station (DAL), located between the end of the DDM discharge channel and the confluence with the Lardeau River.

Rainbow Trout spawning below DDM has been monitored since 2005 by periodic redd surveys. Since 2011, the river stage at DAL has been recorded.

The main objectives of the current analyses were to:

  1. estimate the abundance of Gerrard Rainbow Trout redds in the tailout area below Duncan Dam;
  2. estimate the spawn timing for the fish utilizing the tailout area;
  3. estimate the number of redds dewatered.

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 \(\mathrm{sd}/\mathrm{mean}\) 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

Redd Surveys

The redd survey counts under surveyable conditions (visibility > 0.6 m) were analysed using a hierarchical Bayesian Area-Under-the-Curve (AUC) model (Hilborn, Bue, and Sharr 1999; Su, Adkison, and Van Alen 2001).

Key assumptions of this model include:

Redd Dewatering

The actual DAL stage heights were compared to the recorded redd elevations. The relative redd elevations were calculated by subtracting the depth of each redd from the stage height at the time the redd depth was recorded.

Key assumptions of this calculation include:

Model Templates

Redd Surveys

    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.



Redd Surveys

Table 1. Parameter descriptions.

Parameter Description
bAbundance Intercept for eAbundance
bAbundanceYear[i] Effect of ith Year on bAbundance
bDuration Intercept for eDuration
bDurationYear[i] Effect of ith Year on bDuration
bTiming Intercept for eTiming
bTimingYear[i] Effect of ith Year on bTiming
Doy[i] Day of the year of ith survey
eAbundance[i] Expected total cumulative redd count in the ith Year
eDuration[i] Expected SD for the duration of spawning for ith survey
eRedds[i] Expected cumulative redd count on the ith survey
eTiming[i] Expected peak timing of the ith survey
Redds[i] Cumulative redd count on the ith survey
sAbundanceYear SD of bAbundanceYear
sDurationYear SD of bDurationYear
sRedds SD of residual variation about eRedds
sTimingYear SD of bTimingYear
Year[i] Year of ith survey

Table 2. Model coefficients.

term estimate sd zscore lower upper pvalue
bAbundance 4.097766 0.2384242 17.192075 3.6147698 4.559933 7e-04
bDuration 13.175263 2.9076779 4.662641 9.1325248 20.786795 7e-04
bTiming 109.783425 4.5357696 24.200132 100.9568936 119.407397 7e-04
sAbundanceYear 0.709550 0.1895057 3.900325 0.4518339 1.192033 7e-04
sDurationYear 5.389518 2.7654778 2.160457 2.2474155 13.413146 7e-04
sRedds 3.879193 0.4147368 9.440992 3.2403727 4.819841 7e-04
sTimingYear 12.263854 3.6444787 3.496405 7.0625535 21.219208 7e-04

Table 3. Model summary.

n K nchains niters nthin ess rhat converged
83 7 3 500 500 390 1.01 TRUE

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

term estimate std.error statistic p.value
Spawners 0.0406722 0.004943 8.22821 2.8e-06


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

Year Total Dewatered Percent
2011 136 1 1
2012 78 0 0
2013 140 0 0
2014 116 0 0
2015 36 7 19
2016 19 2 11
2017 26 1 4



Figure 1. Mean hourly discharge from Duncan Dam (DDM) and mean daily discharge in the Lardeau River at Marblehead (LRM) by date and year.
Figure 2. Hourly stage height at Duncan abover Lardeau (DAL) by date and year.
Figure 3. Hourly stage height at Duncan above Lardeau (DAL) by date and year with the relative elevations of observed redds.

Redd Surveys

Figure 4. The timing of surveys by date and year.
Figure 5. Predicted cumulative redd count in the tailout by date, year and visibility.
Figure 6. Predicted redd abundance in the tailout by year (with 95% CIs).
Figure 7. Predicted redd abundance in the tailout by spawner abundance at Gerrard. The estimated linear regression is plotted (with 95% CIs).
Figure 8. Predicted spawn timing by year (with 95% CIs).


The organisations and individuals whose contributions have made this analysis report possible include:


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Hilborn, Ray, Brian G Bue, and Samuel Sharr. 1999. “Estimating Spawning Escapements from Periodic Counts: A Comparison of Methods.” Canadian Journal of Fisheries and Aquatic Sciences 56 (5): 888–96. https://doi.org/10.1139/f99-013.

Irvine, J.R. 1978. “The Gerrard Rainbow Trout of Kootenay Lake, British Columbia - A Discussion of Their Life History with Management, Research and Enhancement Recommendations.” Fisheries Management Report 72. Victoria, BC: Fish; Wildlife Branch.

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McElreath, Richard. 2016. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Chapman & Hall/CRC Texts in Statistical Science Series 122. Boca Raton: CRC Press/Taylor & Francis Group.

Plummer, Martyn. 2015. “JAGS Version 4.0.1 User Manual.” http://sourceforge.net/projects/mcmc-jags/files/Manuals/4.x/.

R Core Team. 2017. “R: A Language and Environment for Statistical Computing.” Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Su, Zhenming, Milo D. Adkison, and Benjamin W. Van Alen. 2001. “A Hierarchical Bayesian Model for Estimating Historical Salmon Escapement and Escapement Timing.” Canadian Journal of Fisheries and Aquatic Sciences 58 (8): 1648–62. https://doi.org/10.1139/cjfas-58-8-1648.