# Mica Tailrace Fish Indexing Study 2013

The main intepretive report by CCRIFC and Poisson Consulting, which was prepared for BC Hydro, is available from BC Hydro.

The suggested citation for this online appendix is:

Thorley, J.L. (2015) Mica Tailrace Fish Indexing Analysis 2013. A Poisson Consulting Analysis Report. URL: http://www.poissonconsulting.ca/f/186605684.

## Background

The Mica Tailrace Fish Indexing Study is a four 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.

## Methods

### Data

The data were provided by the Canadian Columbia River Inter-Tribal Fishery Commission (CCRIFC) in the form of an Access database. All data manipulation was performed using R version 3.0.2 (Team, 2013).

#### Fish Indexing Data

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

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

### Statistical Analysis

Hierarchical Bayesian models were fitted to the data using R version 3.0.2 (Team, 2013) and JAGS 3.3.0 (Plummer, 2012) which interfaced with each other via jaggernaut 1.7 (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). When possible model adequacy was confirmed by examination of residual plots.

The posterior distributions of the fixed (Kéry and Schaub, 2011, p. 75) parameters are summarised below 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).

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% credible intervals (Bradford et al. 2005). Through the report bull trout data and estimates are plotted in black while rainbow trout are plotted in red. Plots were produced using the ggplot2 R package (Wickham, 2009).

### 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 (Kéry, 2010; Kéry and Schaub, 2011, pp. 168-170,180 and 55-56). Lineal densities are by kilometre of river (as opposed to kilometre of bank).

Key assumptions of the relative abundance model include:

• Lineal density varies 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.

#### Species Evenness

The shannon index of evenness ($$E$$) was calculated from the relative abundance analyses for the adult salmonids using the following formula where $$S$$ is the number of species (in this case four) and $$p_i$$ is the proportion of the sum of the relative abundances belonging to the ith species.

$E = \frac{-\sum p_i \log(p_i)}{ln(S)}$

### Model Code

The first three tables describe the JAGS distributions, functions and operators used in the models. For additional information on the JAGS dialect of the BUGS language see the JAGS User Manual (Plummer, 2012).

#### JAGS Distributions

Distribution Description
dgamma(shape, rate) Gamma distribution
dlnorm(mu, sd^-2) Log-normal distribution
dnorm(mu, sd^-2) Normal distribution
dpois(lambda) Poisson distribution
dunif(a, b) Uniform distribution

#### JAGS Functions

Function Description
length(x) Length of vector x
log(x) Natural logarithm of x

#### JAGS Operators

Operator Description
<- Deterministic relationship
~ Stochastic relationship
1:n Vector of integers from 1 to n
a[1:n] Subset of first n values in a
for (i in 1:n) {...} Repeat for 1 to n times incrementing i each time
x^y Power where x is raised to the power of y

Variable and parameter definitions and JAGS model code for the analyses are presented below.

The model code adopts the following naming conventions:

• Data variables are named using upper camel case, i.e., site length is SiteLength.
• The number of levels of a discrete data variable Factor is referenced by nFactor.
• Estimated parameters are named using upper camel case prefixed by a lower case character, i.e., bDensityRegime.
• The SD of a vector of estimated random effects bRandom is indicated by sRandom.
• Unless stated otherwise all effects are linear.

### Body Condition

Variable/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

#### Body Condition - Model 1

model{

bWeightAlpha ~ dnorm(5, 5^-2)
bWeightBeta ~ dnorm(3, 5^-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 ~ dunif(0, 5)
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)
}
}

### Relative Abundance

Variable/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

#### Relative Abundance - Model 1

model{

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

bDensity ~ dnorm(0, 5^-2)

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

sDispersion ~ dunif(0, 5)
for (i in 1:length(Year)) {

log(eEfficiency[i]) <- bEfficiencyVisitType[VisitType[i]]

log(eDensity[i]) <- bDensity
+ 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])
}
}

## Parameter Estimates

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

### Body Condition - Bull Trout

Parameter Estimate Lower Upper SD Error Significance
bWeightAlpha 6.6663 6.62297 6.7071 0.02126 1 0.0000
bWeightAlphaYear[2] 0.1110 0.02351 0.1898 0.04266 75 0.0107
bWeightAlphaYear[3] 0.1870 0.05888 0.3140 0.06467 68 0.0040
bWeightBeta 3.0549 2.92265 3.1858 0.06722 4 0.0000
bWeightBetaYear[2] 0.1476 -0.11745 0.4231 0.14284 183 0.3067
bWeightBetaYear[3] 0.1628 -0.24565 0.5931 0.20786 258 0.4440
sWeight 0.1660 0.14380 0.1928 0.01255 15 0.0000
Rhat Iterations
1.05 1000

### Body Condition - Kokanee

Parameter Estimate Lower Upper SD Error Significance
bWeightAlpha 3.7258 3.0774 4.4352 0.347120 18 0
bWeightAlphaYear[2] 0.9481 0.2396 1.5962 0.349190 72 0
bWeightAlphaYear[3] 1.0612 0.3473 1.7041 0.349320 64 0
bWeightBeta 2.3539 1.8540 2.8981 0.273610 22 0
bWeightBetaYear[2] 0.9756 0.4257 1.4835 0.275000 54 0
bWeightBetaYear[3] 1.0646 0.5188 1.5771 0.277360 50 0
sWeight 0.1989 0.1808 0.2177 0.009798 9 0
Rhat Iterations
2.48 8000

### Body Condition - Mountain Whitefish

Parameter Estimate Lower Upper SD Error Significance
bWeightAlpha 5.30073 5.28951 5.31227 0.005996 0 0.0000
bWeightAlphaYear[2] -0.02928 -0.05159 -0.00839 0.011308 74 0.0093
bWeightAlphaYear[3] 0.03649 0.01065 0.06238 0.013181 71 0.0040
bWeightBeta 3.09222 3.04021 3.14160 0.026296 2 0.0000
bWeightBetaYear[2] -0.08493 -0.23357 0.05644 0.073012 171 0.2280
bWeightBetaYear[3] 0.16362 0.07622 0.25032 0.046251 53 0.0000
sWeight 0.11210 0.10560 0.11907 0.003466 6 0.0000
Rhat Iterations
1.01 1000

### Body Condition - Sculpin

Parameter Estimate Lower Upper SD Error Significance
bWeightAlpha 0.60805 0.44454 0.7623 0.08052 26 0.0000
bWeightAlphaYear[2] -0.08973 -0.34894 0.1815 0.13498 296 0.5133
bWeightAlphaYear[3] 0.02680 -0.41144 0.4415 0.21587 1591 0.8880
bWeightBeta 2.01672 1.42554 2.5854 0.29212 29 0.0000
bWeightBetaYear[2] 0.74049 0.06063 1.4480 0.35404 94 0.0320
bWeightBetaYear[3] -0.05337 -1.63674 1.5873 0.81467 3020 0.9320
sWeight 0.41920 0.34300 0.5119 0.04351 20 0.0000
Rhat Iterations
1.01 2000

### Relative Abundance - Bull Trout - Adult

Parameter Estimate Lower Upper SD Error Significance
bDensity -4.1543 -5.1693 -3.25991 0.4803 23 0.0000
bDensityYear[2] -0.3369 -1.1760 0.57849 0.4493 260 0.4491
bDensityYear[3] -0.7237 -1.6146 0.27099 0.4818 130 0.1257
bEfficiencyVisitType[2] -0.8194 -1.5697 -0.08067 0.3883 91 0.0399
sDispersion 0.6178 0.3101 0.98243 0.1679 54 0.0000
Rhat Iterations
1.04 10000

### Relative Abundance - Kokanee - Adult

Parameter Estimate Lower Upper SD Error Significance
bDensity -4.0264 -4.8106 -2.8808 0.4722 24 0.0000
bDensityYear[2] 2.1117 1.1227 3.1958 0.5078 49 0.0000
bEfficiencyVisitType[2] -0.8493 -1.8183 0.1401 0.5014 115 0.1101
sDispersion 1.0093 0.7081 1.3934 0.1855 34 0.0000
Rhat Iterations
1.04 20000

### Relative Abundance - Mountain Whitefish - Adult

Parameter Estimate Lower Upper SD Error Significance
bDensity -1.1009 -1.7849 -0.4046 0.35218 63 0.0019
bDensityYear[2] -0.4333 -1.0253 0.1731 0.30816 138 0.1595
bDensityYear[3] -0.5189 -1.1470 0.1345 0.32841 123 0.1026
bEfficiencyVisitType[2] -1.5020 -2.0758 -0.9049 0.29297 39 0.0000
sDispersion 0.6197 0.4653 0.8134 0.08966 28 0.0000
Rhat Iterations
1.03 40000

### Relative Abundance - Mountain Whitefish - Juvenile

Parameter Estimate Lower Upper SD Error Significance
bDensity -4.20980 -6.343 -1.9485 1.1265 52 0.0000
bDensityYear[2] -0.06503 -2.405 2.1444 1.1877 3498 0.9700
bDensityYear[3] 0.11825 -1.921 2.1892 1.0555 1738 0.9314
bEfficiencyVisitType[2] -2.43930 -4.201 -0.8194 0.8690 69 0.0058
sDispersion 1.77201 1.154 2.5978 0.3799 41 0.0000
Rhat Iterations
1.03 20000

### Relative Abundance - Rainbow Trout - Adult

Parameter Estimate Lower Upper SD Error Significance
bDensity -4.9413 -7.4493 -2.6395 1.2654 49 0.0000
bDensityYear[2] 0.6820 -1.6920 3.1542 1.2903 355 0.6327
bDensityYear[3] -2.2282 -5.0568 0.8387 1.5161 132 0.1437
bEfficiencyVisitType[2] -3.9713 -5.9724 -1.8908 1.0204 51 0.0000
sDispersion 0.9266 0.2585 2.2490 0.5351 107 0.0000
Rhat Iterations
1.07 10000

## Acknowledgements

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