# Mica Dam Expansion Water Temperature and Fish Indexing Study 2019

The suggested citation for this analytic appendix is:

Thorley, J.L. & Amies-Galonski, E. (2020) Mica Dam Expansion Water Temperature and Fish Indexing Study 2019. A Poisson Consulting Analysis Appendix. URL: http://www.poissonconsulting.ca/f/1134568871.

## Background

The Mica Tailrace Fish Indexing Study is a multi-year program to estimate the effects of the addition of two new turbines (Mica 5 and 6) on the ichyofauna and thermal regime in the 2.5 km of the Columbia River downstream of Mica Dam. A single year of fish indexing data (2008) was also available from a previous program. As per the Terms of Reference (TOR) the relative abundance, condition and spatial distribution of the fish populations was assessed. In addition, changes in the species evenness were also estimated.

Mica 5 became operational on January 28th 2015 and Mica 6 became operational on December 22nd 2015.

### Data Preparation

The fish and temperature data were provided by the Ktunaxa Nation. The discharge and elevation data were queried from the Columbia Basin Hydrological Database.

The data were cleaned and tidied using R version 4.0.1 (R Core Team 2015).

### Length Cutoffs

Individuals were classified as fry (age-0), juvenile (age-1 and older subadults) or adult (sexually mature) based on the length cut-offs in Table 1.

### 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 were vague in the sense that they 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 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).

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.

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 4.0.1 (R Core Team 2018) and the mbr family of packages.

### Body Condition

The annual variation in condition (body weight when accounting for body length) was estimated from the boat and backpack electrofishing captures using a mass-length model (He et al. 2008).

Key assumptions of the condition model include:

• Weight varies with body length as an allometric relationship, i.e., $$W = \alpha L^{\beta}$$.
• $$\alpha$$ varies with year.
• $$\beta$$ varies with year.
• The residual variation in weight is log-normally distributed.

Preliminary analyses indicated that site and day of the year were not informative predictors of condition.

### Relative Abundance

The annual variation in relative abundance was estimated from the boat count and catch data using an over-dispersed Poisson model (Kery and Schaub 2011). Lineal densities are by kilometre of river (as opposed to kilometre of bank).

Key assumptions of the relative abundance model include:

• Lineal density varies by period.
• Lineal density varies randomly with year.
• Lineal catch density is a fixed proportion of lineal count density.
• Expected counts (and catches) are the product of the count (catch) density and the length of river (half the length of bank) sampled.
• Observed counts (and catches) are described by a Poisson-gamma distribution.

Preliminary analyses indicated that site was not an informative predictor of lineal density.

The model does not distinguish between the abundance and observer efficiency, i.e., it estimates the count which is the product of the two. As such it is necessary to assume that changes in observer efficiency by year are negligible in order to interpret the estimates as relative abundance.

### Water Temperature

Climatic variation can cause large differences in annual temperatures. Consequently, we explored the data for an effect of the additional turbines on the difference in the water temperature between the right versus left bank and when moving downstream. All apparent systematic differences were within the accuracy of the temperature loggers ($$\pm 0.2^{\circ}\text{C}$$).

### Model Templates

#### Condition

.model{
bWeightAlpha ~ dnorm(5, 2^-2)
bWeightBeta ~ dnorm(3, 2^-2)
bWeightAlphaYear[1] <- 0
for(i in 2:nYear) {
bWeightAlphaYear[i] ~ dnorm(0, 2^-2)
}
bWeightBetaYear[1] <- 0
for(i in 2:nYear) {
bWeightBetaYear[i] ~ dnorm(0, 2^-2)
}
sWeight ~ dnorm(0, 2^-2) T(0,)
for (i in 1:length(Weight)) {
eWeightAlpha[i] <- bWeightAlpha + bWeightAlphaYear[Year[i]]
eWeightBeta[i] <- bWeightBeta + bWeightBetaYear[Year[i]]
log(eWeight[i]) <- eWeightAlpha[i] + eWeightBeta[i] * Length[i]
Weight[i] ~ dlnorm(log(eWeight[i]), sWeight^-2)
}

Block 1.

#### Relative Abundance

.model{
bEfficiency <- 1
bEfficiencyVisitType[1] <- 0
for (i in 2:nVisitType) {
bEfficiencyVisitType[i] ~ dunif(0, 1)
}
bDensity ~ dnorm(0, 5^-2)
bDensityPeriod[1] <- 0
for(i in 2:nPeriod) {
bDensityPeriod[i] ~ dnorm(0, 2^-2)
}
sDensityYear ~ dnorm(0, 2^-2) T(0, )
for(i in 1:nYear) {
bDensityYear[i] ~ dnorm(0, sDensityYear^-2)
}
sDispersion ~ dnorm(0, 2^-2) T(0, )
for (i in 1:length(Year)) {

eEfficiency[i] <- bEfficiency - bEfficiencyVisitType[VisitType[i]]
log(eDensity[i]) <- bDensity + bDensityPeriod[Period[i]] + bDensityYear[Year[i]]
eAbundance[i] <- eDensity[i] * SiteLength[i] / 2
eDispersion[i] ~ dgamma(1 / sDispersion^2, 1 / sDispersion^2)
Count[i] ~ dpois(eAbundance[i] * eEfficiency[i] * eDispersion[i])
}

Block 2.

## Results

### Tables

Table 1. Stage Length Cutoffs by Species.

Species Fry Juvenile
Bull Trout 120 400
Mountain Whitefish 120 175
Rainbow Trout 120 250
Kokanee 100 250

#### Condition

Table 2. Parameter descriptions.

Parameter Description
bWeightAlpha Intercept for eAlpha
bWeightAlphaYear[i] Effect of ith Year on eAlpha
bWeightBeta Intercept for eBeta
bWeightBetaYear[i] Effect of ith Year on eBeta
eAlpha[i] Predicted allometric intercept (on centred log length) for ith fish
eBeta[i] Predicted allometric slope for ith fish
eWeight[i] Predicted weight of ith fish
Length[i] Centred log Length of ith fish
sWeight SD of residual variation in log(Weight)
Weight[i] Weight of ith fish
Year[i] Year of capture of of ith fish
##### Bull Trout

Table 3. Model coefficients.

term estimate sd zscore lower upper pvalue
bWeightAlpha 6.7449388 0.0247983 272.0063933 6.6966093 6.7927562 0.0006662
bWeightAlphaYear[2] 0.1537945 0.0472707 3.2677351 0.0624549 0.2518433 0.0006662
bWeightAlphaYear[3] 0.1913102 0.0713777 2.6761293 0.0517245 0.3267519 0.0139907
bWeightAlphaYear[4] 0.0031772 0.0867638 0.0305312 -0.1664233 0.1567937 0.9733511
bWeightBeta 3.0575844 0.0762974 40.0780567 2.9110947 3.2025522 0.0006662
bWeightBetaYear[2] 0.1474677 0.1585681 0.9521226 -0.1521114 0.4512170 0.3457695
bWeightBetaYear[3] 0.1461466 0.2390108 0.6152690 -0.3140118 0.6426729 0.5363091
bWeightBetaYear[4] 0.1514321 0.3616289 0.4521290 -0.5229309 0.8884279 0.6548967
sWeight 0.1899506 0.0139870 13.6456842 0.1658411 0.2193084 0.0006662

Table 4. Model summary.

n K nchains niters nthin ess rhat converged
106 9 3 500 1000 1240 1.002 TRUE
##### Mountain Whitefish

Table 5. Model coefficients.

term estimate sd zscore lower upper pvalue
bWeightAlpha 5.2961725 0.0059544 889.456113 5.2842468 5.3082016 0.0006662
bWeightAlphaYear[2] -0.0277357 0.0117387 -2.374274 -0.0510029 -0.0046830 0.0166556
bWeightAlphaYear[3] 0.0375127 0.0127376 2.921893 0.0121240 0.0619230 0.0059960
bWeightAlphaYear[4] 0.0172917 0.0123370 1.399918 -0.0063828 0.0406689 0.1778814
bWeightBeta 3.0904454 0.0254063 121.669058 3.0416907 3.1419747 0.0006662
bWeightBetaYear[2] -0.0910429 0.0696022 -1.299649 -0.2219548 0.0462926 0.1992005
bWeightBetaYear[3] 0.1669278 0.0460264 3.624426 0.0779921 0.2590992 0.0006662
bWeightBetaYear[4] 0.1010421 0.0932185 1.058267 -0.0943012 0.2740300 0.2804797
sWeight 0.1091740 0.0030894 35.357695 0.1035001 0.1156018 0.0006662

Table 6. Model summary.

n K nchains niters nthin ess rhat converged
666 9 3 500 1000 1436 1 TRUE
##### Kokanee

Table 7. Model coefficients.

term estimate sd zscore lower upper pvalue
bWeightAlpha 4.1183601 0.5538914 7.451938 3.0454152 5.1944731 0.0006662
bWeightAlphaYear[2] 0.5702805 0.5552647 1.004447 -0.5200763 1.6465351 0.3391073
bWeightAlphaYear[3] 0.6871742 0.5540270 1.215960 -0.4114166 1.7519429 0.2391739
bWeightAlphaYear[4] 0.5984600 0.5561922 1.064848 -0.4794231 1.6964302 0.2951366
bWeightBeta 2.6612270 0.4324766 6.167712 1.8108572 3.5160638 0.0006662
bWeightBetaYear[2] 0.6675097 0.4338728 1.524901 -0.2002297 1.5181464 0.1379081
bWeightBetaYear[3] 0.7487141 0.4403075 1.699720 -0.1168977 1.6070710 0.1032645
bWeightBetaYear[4] 0.8154131 0.4425914 1.818776 -0.0646487 1.6832351 0.0712858
sWeight 0.2401528 0.0117704 20.448440 0.2188555 0.2648829 0.0006662

Table 8. Model summary.

n K nchains niters nthin ess rhat converged
224 9 3 500 1000 348 1.002 TRUE

#### Relative Abundance

Table 9. Parameter descriptions.

Parameter Description
bDensity Intercept for log(eDensity)
bDensityYear[i] Effect of ith Year on log(eDensity)
bEfficiencyVisitType[i] Value of log(eEfficiency) for ith VisitType
Count[i] Number of fish counted or captured on ith site visit
eAbundance[i] Predicted relative abundance for ith site visit
eDensity[i] Predicted relative lineal density for ith site visit
eDispersion[i] Predicted over-dispersion for ith site visit
eEfficiency[i] Predicted efficiency relative to counting for ith site visit
sDispersion SD of eDispersion
SiteLength[i] Length of bank surveyed on ith site visit
VisitType[i] Type of ith site visit, i.e., count versus catch
Year[i] Year of ith site visit
##### Bull Trout

Table 10. Model coefficients.

term estimate sd zscore lower upper pvalue
bDensity -4.5483913 0.3275357 -13.8314625 -5.1263613 -3.8404662 0.0006662
bDensityPeriod[2] 0.2374427 0.4562055 0.4749045 -0.7523516 1.0582628 0.5389740
bDensityYear[1] 0.0434851 0.2777674 0.3065370 -0.4161822 0.7544030 0.7228514
bDensityYear[2] 0.0024447 0.2778457 0.0084901 -0.6117651 0.5784625 0.9853431
bDensityYear[3] -0.0781649 0.3071506 -0.5057760 -0.9298301 0.3152702 0.5909394
bDensityYear[4] 0.0088800 0.3102782 0.1018950 -0.5685696 0.7494457 0.9160560
bDensityYear[5] -0.0034100 0.3197221 -0.0566432 -0.7464800 0.6721996 0.9586942
bEfficiencyVisitType[2] 0.4074703 0.1600380 2.5065399 0.0620530 0.6805954 0.0006662
sDensityYear 0.2298812 0.3165827 0.9837614 0.0110697 1.0198487 0.0006662
sDispersion 0.6699084 0.1253256 5.3942209 0.4514440 0.9393357 0.0006662

Table 11. Model summary.

n K nchains niters nthin ess rhat converged
47 10 3 500 300 1275 1.001 TRUE
##### Mountain Whitefish

Table 12. Model coefficients.

term estimate sd zscore lower upper pvalue
bDensity -3.9998553 0.6871282 -5.7399114 -5.1511511 -2.4978027 0.0019987
bDensityPeriod[2] 0.4268979 0.8479944 0.4798337 -1.2914679 2.0425909 0.5762825
bDensityYear[1] -0.0073196 0.6416822 -0.0953173 -1.5692711 1.2080869 0.9746835
bDensityYear[2] -0.0642652 0.6142377 -0.2271524 -1.6528382 0.9890030 0.8321119
bDensityYear[3] -0.0155940 0.6210001 -0.0984553 -1.4948206 1.1158375 0.9347102
bDensityYear[4] -0.2525526 0.7023522 -0.5547476 -2.0781039 0.7669402 0.5296469
bDensityYear[5] 0.3003693 0.7209217 0.6002241 -0.8024000 2.0763404 0.5083278
bEfficiencyVisitType[2] 0.9336702 0.0652125 14.0521196 0.7608646 0.9827258 0.0006662
sDensityYear 0.6012339 0.5681324 1.2832400 0.0274513 2.1549447 0.0006662
sDispersion 1.4462137 0.2430129 6.0276759 1.0426073 1.9873752 0.0006662

Table 13. Model summary.

n K nchains niters nthin ess rhat converged
47 10 3 500 300 871 1.003 TRUE

Table 14. Model coefficients.

term estimate sd zscore lower upper pvalue
bDensity -1.2081265 0.4729651 -2.4990591 -1.9980132 -0.2001549 0.0233178
bDensityPeriod[2] -0.0010120 0.6868463 -0.0526711 -1.5154086 1.2675274 0.9986676
bDensityYear[1] 0.1710568 0.4484460 0.5043951 -0.6510449 1.2310244 0.5629580
bDensityYear[2] -0.0890931 0.4265791 -0.2839872 -1.0715728 0.6931258 0.7521652
bDensityYear[3] -0.1307897 0.4559763 -0.3856545 -1.1574369 0.7151188 0.7268488
bDensityYear[4] 0.3903445 0.5622053 0.8114263 -0.5389353 1.8028275 0.3644237
bDensityYear[5] -0.3652971 0.5798686 -0.7599976 -1.7969391 0.5782719 0.3764157
bEfficiencyVisitType[2] 0.7920571 0.0757220 10.2928288 0.5893424 0.8896675 0.0006662
sDensityYear 0.5837329 0.4369668 1.5495358 0.0764352 1.8190535 0.0006662
sDispersion 0.8079563 0.0889596 9.1376155 0.6580172 1.0056799 0.0006662

Table 15. Model summary.

n K nchains niters nthin ess rhat converged
47 10 3 500 300 226 1.016 TRUE
##### Rainbow Trout

Table 16. Model coefficients.

term estimate sd zscore lower upper pvalue
bDensity -5.2409836 0.9294088 -5.6698277 -7.1337351 -3.5049568 0.0006662
bDensityPeriod[2] -0.8827318 1.1774497 -0.7279018 -3.0024553 1.6377676 0.4270486
bDensityYear[1] 0.2251140 1.1755007 0.2203285 -2.1209930 2.6264584 0.8121252
bDensityYear[2] 0.9714359 0.9350035 1.0558831 -0.7758965 2.9116305 0.2644903
bDensityYear[3] -1.1767884 1.0787730 -1.2194615 -3.6498930 0.4372475 0.1632245
bDensityYear[4] -0.0719036 1.0643474 -0.1310672 -2.5849614 1.8924375 0.9320453
bDensityYear[5] -0.2923180 1.1039778 -0.3583067 -2.8157443 1.6691879 0.6948701
bEfficiencyVisitType[2] 0.9819937 0.0287034 33.9186002 0.8962625 0.9982746 0.0006662
sDensityYear 1.3086579 0.7446227 1.9292106 0.3619239 3.3089916 0.0006662
sDispersion 0.6939334 0.3209207 2.3315279 0.2616054 1.4923112 0.0006662

Table 17. Model summary.

n K nchains niters nthin ess rhat converged
47 10 3 500 300 1326 1.003 TRUE
##### Kokanee

Table 18. Model coefficients.

term estimate sd zscore lower upper pvalue
bDensity -2.6582197 1.0064719 -2.5979870 -4.4954955 -0.3376729 0.0366422
bDensityPeriod[2] -0.5358329 1.1167043 -0.4663438 -2.8458772 1.6139544 0.5936043
bDensityYear[1] -0.9169311 1.0064755 -1.0125952 -3.2711543 0.7346803 0.2245170
bDensityYear[2] 0.8553464 0.9959584 0.8460199 -1.2940564 2.7062580 0.3057961
bDensityYear[3] 0.0329223 0.9904930 -0.0244477 -2.2107747 1.8996879 0.9720187
bDensityYear[4] -0.1216088 1.0062846 -0.1714605 -2.4059291 1.7498253 0.8800799
bEfficiencyVisitType[2] 0.6767797 0.1447751 4.4805587 0.2570010 0.8543102 0.0006662
sDensityYear 1.1835941 0.7487135 1.8104093 0.4240694 3.2462607 0.0006662
sDispersion 0.9973497 0.1321665 7.6025711 0.7737543 1.2995119 0.0006662

Table 19. Model summary.

n K nchains niters nthin ess rhat converged
37 9 3 500 300 190 1.043 TRUE

## Acknowledgements

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

• BC Hydro
• Trish Joyce
• Jason Watson
• Guy Martel
• Peter McCann
• Fred Katunar
• Alf Leake
• Karen Bray
• Ktunaxa Nation
• Katrina Caley
• Joanne Fisher
• Jim Clarricoates
• Jon Bisset
• Will Warnock
• Bill Green
• Poisson Consulting
• Robyn Irvine
• Seb Dalgarno
• Applied Aquatic Research
• Tom Boag
• Ministry of Forests, Lands and Natural Resource Operations
• Albert Chirico
• Mark Thomas
• Charlotte Houston

## References

Bradford, Michael J, Josh Korman, and Paul S Higgins. 2005. “Using Confidence Intervals to Estimate the Response of Salmon Populations (Oncorhynchus Spp.) to Experimental Habitat Alterations.” Canadian Journal of Fisheries and Aquatic Sciences 62 (12): 2716–26. https://doi.org/10.1139/f05-179.

Brooks, Steve, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng, eds. 2011. Handbook for Markov Chain Monte Carlo. Boca Raton: Taylor & Francis.

He, Ji X., James R. Bence, James E. Johnson, David F. Clapp, and Mark P. Ebener. 2008. “Modeling Variation in Mass-Length Relations and Condition Indices of Lake Trout and Chinook Salmon in Lake Huron: A Hierarchical Bayesian Approach.” Transactions of the American Fisheries Society 137 (3): 801–17. https://doi.org/10.1577/T07-012.1.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Boston: Academic Press. http://www.vogelwarte.ch/bpa.html.

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. 2015. “R: A Language and Environment for Statistical Computing.” Vienna, Austria: R Foundation for Statistical Computing. http://www.R-project.org/.

———. 2018. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.