# Kootenay Lake Large Piscivorous Trout Hydroacoustic Analysis 2013

**The main intepretive report by Redfish Consulting, which was prepared for the Fish and Wildlife Compensation Program, Habitat Conservation Trust Foundation and the Ministry of Forests, Lands and Natural Resource Operations, is available from EcoCat.**

The suggested citation for this online appendix is:

Thorley J.L. and Hogan P.M. (2014) Kootenay Lake Large Piscivorous Trout Hydroacoustic Analysis 2013. A Poisson Consulting Ltd. Analysis. URL: http://www.poissonconsulting.ca/analyses/2014/01/13/kotl-hydroacoustic-13.

## Background

Hierarchical Bayesian models were fitted to the hydroacoustic density data and acoustic tag depth detection for Kootenay Lake using using R version 3.0.2 (Team, 2013 ) and JAGS 3.3.0 (Plummer, 2012 ) which interfaced with each other via the jaggernaut (Thorley, 2014 ) R package. For additional information on hierarchical Bayesian modelling in the BUGS language, of which JAGS uses a dialect, the reader is referred to Kery and Schuab (2011) pages 41-44.

The hydroacoustic data was provided by the Ministry of Forest, Lands and Natural Resource Operations (MFLNRO) and the acoustic tag depth data by the Kootenay Lake Exploitation Study (Andrusak and Thorley, 2013 ).

The source code is available from GitHub.

## Methods

Unless specified, the models assumed vague (low information) prior distributions (Kery 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 (Kery and Schaub, 2011 , pp. 38-40). Model convergence was confirmed by ensuring that Rhat (Kery and Schaub, 2011 , p. 40) was less than 1.1 for each of the parameters in the model (Kery and Schaub, 2011 , p. 61).

The posterior distributions of the **fixed** (Kery and Schaub 2011 p. 75) parameters are summarised below in terms of a *point* estimate (mean), *lower* and *upper* 95% credibility limits (2.5th and 97.5th percentiles), the standard deviation (*SD*), percent relative *error* (half the 95% credibility interval as a percent of the point estimate) and *significance* (Kery 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) 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 , 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 ). Plots were produced using the ggplot2 R package (Wickham, 2009 ).

### Depth Tags

The depth tag data from the Kooteny Lake Exploitation Study were analysed using a Bayesian polynomial model.

Key assumptions of the depth tag model include: - Relative use varies as a third-order polynomial of the standardised depth. - Relative use is log-normally distributed.

Only detections in less than 50 m of water during the hours of darkness were included in the model. The number of fish and number of detections are tabulated by month and species below.

Month | Species | Fish | Detections |
---|---|---|---|

July | Bull Trout | 2 | 179 |

July | Rainbow Trout | 4 | 889 |

September | Bull Trout | 2 | 338 |

September | Rainbow Trout | 4 | 890 |

### Hydroacoustic Surveys

The hydroacoustic survey data were analysed using a hierarchical Bayesian zero-inflated (Kery and Schaub, 2011 , pp. 401-402) log-normal polynomial model.

Key assumptions of the hydroacoustic model include: - Positive densities (one or more fish detected) vary with year. - Positive densities vary randomly with respect to transect. - Zero-inflation varies as a third order polynomial of the standardized depth. - Density is zero-inflated log-normally distributed.

Only detections in less than 50 m of water were included in the model. The surveys were conducted during the hours of darkness. The data consisted of detection densities by decibel cutoff from 18 transects spanning 4 years.

## Model Code

The JAGS model code, which uses a series of naming conventions, is presented below.

### Depth Tags

Variable/Parameter | Description |
---|---|

`bDepth0` |
Intercept of `eLogUse` |

`bDepth1` |
Linear effect of `Depth` on `eLogUse` |

`bDepth2` |
Quadratic effect of `Depth` on `eLogUse` |

`bDepth3` |
Cubic effect of `Depth` on `eLogUse` |

`Depth[i]` |
Standardised depth of ith depth bin |

`eLogUse[i]` |
Expected log relative use at ith depth bin |

`sUse` |
SD of the residual log-normal relative use |

`Use[i]` |
Observed relative use at ith depth bin |

#### Depth Tags - Model 1

```
model {
sUse ~ dunif(0, 5)
bDepth0 ~ dnorm(0, 5^-2)
bDepth1 ~ dnorm(0, 5^-2)
bDepth2 ~ dnorm(0, 5^-2)
bDepth3 ~ dnorm(0, 5^-2)
for (i in 1:length(Depth)) {
eLogUse[i] <- bDepth0 + bDepth1 * Depth[i]
+ bDepth2 * Depth[i]^2 + bDepth3 * Depth[i]^3
Use[i] ~ dlnorm(eLogUse[i], sUse^-2)
}
}
```

### Hydroacoustic Surveys

Variable/Parameter | Description |
---|---|

`bDepth` |
Linear effect of `Depth` on `logit(eSuitability)` |

`bDepth2` |
Quadratic effect of `Depth` on `logit(eSuitability)` |

`bDepth3` |
Cubic effect of `Depth` on `logit(eSuitability)` |

`bIntercept` |
Intercept of `eLogDensity` |

`bSuitable` |
Intercept of `logit(eSuitability)` |

`bTransect[tr]` |
Linear effect of trth transect on `eLogDensity` |

`bYear[yr]` |
Linear effect of yrth year on `eLogDensity` |

`Density[i]` |
Observed density on ith transect depth survey |

`Depth[i]` |
Standardised depth of ith transect depth survey |

`eLogDensity[i]` |
Expected log density on ith transect depth survey |

`eSuitability[i]` |
Expected probability of positive density on ith transect depth survey |

`sDensity` |
SD of the residual log-normal density |

`sTransect` |
SD of `bTransect` |

#### Hydroacoustic Surveys - Model 1

```
model {
bIntercept ~ dnorm(0, 2^-2)
bSuitable ~ dnorm(0, 2^-2)
sDensity ~ dunif(0, 2)
bYear[1] <- 0
for(yr in 2:nYear) {
bYear[yr] ~ dnorm(0, 2^-2)
}
bDepth ~ dnorm(0, 2^-2)
bDepth2 ~ dnorm(0, 2^-2)
bDepth3 ~ dnorm(0, 2^-2)
sTransect ~ dunif(0, 2)
for(tr in 1:nTransect) {
bTransect[tr] ~ dnorm(0, sTransect^-2)
}
for (i in 1:length(Depth)) {
eLogDensity[i] <- bIntercept + bYear[Year[i]] + bTransect[Transect[i]]
logit(eSuitable[i]) <- bSuitable + bDepth * Depth[i]
+ bDepth2 * Depth[i]^2 + bDepth3 * Depth[i]^3
dFish[i] ~ dbern(eSuitable[i])
dLogDensity[i] <- ifelse(dFish[i], eLogDensity[i], log(0.00001))
Density[i] ~ dlnorm(dLogDensity[i], sDensity^-2)
}
}
```

## Parameter Estimates

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

### Depth Tags - July

Parameter | Estimate | Lower | Upper | SD | Error | Significance |
---|---|---|---|---|---|---|

bDepth0 | -4.7957 | -5.0634 | -4.5291 | 0.13705 | 6 | 0.0000 |

bDepth1 | -2.6458 | -3.0815 | -2.2328 | 0.22277 | 16 | 0.0000 |

bDepth2 | -0.0119 | -0.2331 | 0.2193 | 0.11249 | 1901 | 0.9022 |

bDepth3 | 0.6926 | 0.4807 | 0.9211 | 0.11232 | 32 | 0.0000 |

sUse | 0.5703 | 0.4531 | 0.7362 | 0.07226 | 25 | 0.0000 |

### Depth Tags - September

Parameter | Estimate | Lower | Upper | SD | Error | Significance |
---|---|---|---|---|---|---|

bDepth0 | -5.1710 | -5.4910 | -4.85531 | 0.1626 | 6 | 0.0000 |

bDepth1 | 0.0808 | -0.4601 | 0.58420 | 0.2662 | 646 | 0.7525 |

bDepth2 | 0.5632 | 0.3234 | 0.80308 | 0.1216 | 43 | 0.0000 |

bDepth3 | -0.3498 | -0.6151 | -0.07659 | 0.1350 | 77 | 0.0180 |

sUse | 0.7476 | 0.6110 | 0.91796 | 0.0789 | 21 | 0.0000 |

### Hydroacoustic Surveys - July - -33 Decibels

Parameter | Estimate | Lower | Upper | SD | Error | Significance |
---|---|---|---|---|---|---|

bDepth | -1.8033 | -2.52230 | -1.0699 | 0.370650 | 40 | 0.0000 |

bDepth2 | -2.1369 | -2.81402 | -1.5521 | 0.323180 | 30 | 0.0000 |

bDepth3 | 0.7775 | 0.03492 | 1.4304 | 0.344780 | 90 | 0.0379 |

bIntercept | -0.7290 | -1.09462 | -0.3612 | 0.182280 | 50 | 0.0000 |

bSuitable | -0.9951 | -1.34131 | -0.7030 | 0.164530 | 32 | 0.0000 |

bYear[2] | 0.7341 | 0.58320 | 0.8828 | 0.077983 | 20 | 0.0000 |

bYear[3] | 1.3768 | 1.23617 | 1.5233 | 0.075969 | 10 | 0.0000 |

bYear[4] | 1.0164 | 0.73771 | 1.3046 | 0.148530 | 28 | 0.0000 |

sDensity | 0.2331 | 0.22256 | 0.2448 | 0.005738 | 5 | 0.0000 |

sTransect | 0.6931 | 0.46332 | 1.0175 | 0.145530 | 40 | 0.0000 |

### Hydroacoustic Surveys - July - -32 Decibels

Parameter | Estimate | Lower | Upper | SD | Error | Significance |
---|---|---|---|---|---|---|

bDepth | -2.01608 | -3.37031 | -0.75880 | 0.660740 | 65 | 0.0000 |

bDepth2 | -1.66246 | -2.98013 | -0.72550 | 0.553370 | 68 | 0.0000 |

bDepth3 | 0.51071 | -0.57459 | 1.45855 | 0.520380 | 199 | 0.3072 |

bIntercept | -0.31265 | -0.84010 | 0.26592 | 0.308940 | 177 | 0.3498 |

bSuitable | -2.84400 | -3.51732 | -2.28938 | 0.307000 | 22 | 0.0000 |

bYear[2] | 0.58501 | 0.47008 | 0.70451 | 0.059740 | 20 | 0.0000 |

bYear[3] | 0.54197 | 0.42181 | 0.66805 | 0.061617 | 23 | 0.0000 |

bYear[4] | 0.55857 | 0.31891 | 0.79469 | 0.118180 | 43 | 0.0000 |

sDensity | 0.08402 | 0.08007 | 0.08839 | 0.002148 | 5 | 0.0000 |

sTransect | 0.96926 | 0.60838 | 1.56124 | 0.239070 | 49 | 0.0000 |

### Hydroacoustic Surveys - September - -33 Decibels

Parameter | Estimate | Lower | Upper | SD | Error | Significance |
---|---|---|---|---|---|---|

bDepth | 0.53101 | -0.2222 | 1.3505 | 0.410850 | 148 | 0.1936 |

bDepth2 | -3.44685 | -4.3541 | -2.6280 | 0.439780 | 25 | 0.0000 |

bDepth3 | -0.63124 | -1.7505 | 0.4658 | 0.574580 | 176 | 0.2874 |

bIntercept | -0.60551 | -0.8043 | -0.4124 | 0.101940 | 32 | 0.0000 |

bSuitable | -0.48413 | -0.7801 | -0.1949 | 0.147290 | 60 | 0.0060 |

bYear[2] | 0.08771 | -0.0595 | 0.2243 | 0.069976 | 162 | 0.2116 |

bYear[3] | 1.51410 | 1.3505 | 1.6640 | 0.081539 | 10 | 0.0000 |

bYear[4] | 0.59134 | 0.3473 | 0.8333 | 0.124540 | 41 | 0.0000 |

sDensity | 0.24535 | 0.2341 | 0.2580 | 0.006253 | 5 | 0.0000 |

sTransect | 0.36754 | 0.2404 | 0.5499 | 0.077297 | 42 | 0.0000 |

### Hydroacoustic Surveys - September - -32 Decibels

Parameter | Estimate | Lower | Upper | SD | Error | Significance |
---|---|---|---|---|---|---|

bDepth | -0.43852 | -1.60497 | 0.61084 | 0.565930 | 253 | 0.4290 |

bDepth2 | -3.81101 | -5.38164 | -2.49603 | 0.743030 | 38 | 0.0000 |

bDepth3 | -0.53423 | -2.12642 | 1.09748 | 0.831900 | 302 | 0.4908 |

bIntercept | -0.09416 | -0.35094 | 0.16003 | 0.125920 | 271 | 0.4309 |

bSuitable | -1.42946 | -1.84198 | -1.06056 | 0.200880 | 27 | 0.0000 |

bYear[2] | -0.69404 | -0.76953 | -0.61378 | 0.039188 | 11 | 0.0000 |

bYear[3] | 0.28385 | 0.19054 | 0.37437 | 0.048683 | 32 | 0.0000 |

bYear[4] | 0.09459 | -0.00848 | 0.20825 | 0.054347 | 115 | 0.0773 |

sDensity | 0.07548 | 0.07191 | 0.07941 | 0.001931 | 5 | 0.0000 |

sTransect | 0.46874 | 0.32598 | 0.68769 | 0.092371 | 39 | 0.0000 |

## Figures

### Depth Tags

### Hydroacoustic Surveys - July - -33 Decibels

### Hydroacoustic Surveys - July - -32 Decibels

### Hydroacoustic Surveys - September - -33 Decibels

### Hydroacoustic Surveys - September - -32 Decibels

### Depth Distributions

## Acknowledgements

This analysis was made possible through the support of the following organisations:

- Habitat Conservation Trust Foundation (HCTF) and the anglers, hunters, trappers and guides who contribute to the Trust
- Ministry of Forests, Lands and Natural Resource Operations (MFLNRO)
- Redfish Consulting

## References

- Greg Andrusak, Joseph Thorley, (2013) Kootenay Lake Exploitation Study: Fishing and Natural Mortality of Large Rainbow Trout and Bull Trout: 2013 Annual Report.
- Michael Bradford, Josh Korman, Paul 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) 2716-2726 10.1139/f05-179 - Hadley Wickham, (2009) ggplot2: elegant graphics for data analysis. http://had.co.nz/ggplot2/book
- Joseph Thorley, (2014) jaggernaut: An R package to facilitate Bayesian analyses using JAGS (Just Another Gibbs Sampler). https://github.com/joethorley/jaggernaut
- Marc Kery, Michael Schaub, (2011) Bayesian population analysis using {WinBUGS} : a hierarchical perspective.
- Martyn Plummer, (2012) {JAGS} Version 3.3.0 User Manual. http://sourceforge.net/projects/mcmc-jags/files/Manuals/3.x/
- R Team, (2013) R: A Language and Environment for Statistical Computing. http://www.R-project.org