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

depth/dayte

depth/dayte-year

Hydroacoustic Surveys - July - -33 Decibels

db/July/33/year

db/July/33/transect

Hydroacoustic Surveys - July - -32 Decibels

db/July/32/year

db/July/32/transect

Hydroacoustic Surveys - September - -33 Decibels

db/September/33/year

db/September/33/transect

Hydroacoustic Surveys - September - -32 Decibels

db/September/32/year

db/September/32/transect

Depth Distributions

combined/depth-db

Acknowledgements

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

References


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