# Elk River WCT Abundance Analysis 2022

The suggested citation for this analytic appendix is:

*Hill, N.E., Pearson, A.D., & Thorley, J.L. (2023) Elk River WCT
Abundance Analysis 2022. A Poisson Consulting Analysis Appendix. URL:
https://www.poissonconsulting.ca/f/876409566.*

## Background

The Elk River is one of eight upper Kootenay River tributaries (Bull,
Michel, Skookumchuck, St. Mary, upper Kootenay, White and Wigwam) that
are designated as Class II due to the importance of their recreational
fisheries for Westslope Cutthroat Trout (*Oncorhynchus clarkii lewisi*).
The population abundance of Westslope Cutthroat Trout in the Elk River
is uncertain. To improve population estimates a mark-recapture analysis
using Passive Integrated Transponder (PIT) tags deployed by both boat
electrofishing and guides was recommended (J. L. Thorley 2021).

After a successful pilot study in 2021, data collection via boat electrofishing and guides continued in 2022. The goal of the present analysis is to use the additional data to get an improved estimate of Westslope Cutthroat Trout population abundance in the Elk River.

### Data Preparation

The data were provided by Nupqu Limited Partnership. The data were prepared for analysis using R version 4.2.2 (R Core Team 2022) and organized in a SQLite database. All river distances are based on the BC Freshwater Atlas layer.

Key assumptions of the data preparation included:

- PIT Tag numbers were recorded correctly in instances where the data was not verifiable by the PIT Reader log.
- PIT Reader log was taken to be correct in instances where the recorded PIT Tag code did not match the PIT Reader log.
- 2021 data were submitted in Mountain time; 2022 data were submitted in GMT-6.
- Fork lengths were measured accurately.
- GPS coordinates of outing start/end points were extended if they did not overlaps locations of fish captured on a given outing using the coordinates of a previous outing.

### Statistical Analysis

Model parameters were estimated using Bayesian methods. The estimates were produced using JAGS (Plummer 2003) and STAN (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2020).

Unless stated otherwise, the Bayesian analyses used weakly informative normal and half-normal prior distributions (Gelman, Simpson, and Betancourt 2017). 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).

Model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the proportion of zeros in the data and the first four central moments (mean, variance, skewness and kurtosis) in the deviance residuals were compared to the expected values by simulating new data based on the posterior distribution and assumed sampling distribution and calculating the deviance residuals.

Where computationally practical, the sensitivity of the posteriors to the choice of prior distributions was evaluated by increasing the standard deviations of all normal, half-normal and log-normal priors by an order of magnitude and then using \(\hat{R}\) to evaluate whether the samples were drawn from the same posterior distribution (Joseph L. Thorley and Andrusak 2017).

The parameters are summarised in terms of the point *estimate*, *lower*
and *upper* 95% compatibility limits (Rafi and Greenland 2020) and the
surprisal *s-value* (Greenland 2019). The estimate is the median
(50th percentile) of the MCMC samples while the 95% CLs are the 2.5th
and 97.5th percentiles. The s-value indicates how surprising it would be
to discover that the true value of the parameter is in the opposite
direction to the estimate (Greenland 2019). An s-value of \(>\)
4.32 bits, which is equivalent to a p-value \(<\) 0.05
(Kery and Schaub 2011; Greenland and Poole 2013), indicates that the
surprise would be equivalent to throwing at least 4.3 heads in a row.

Model selection was based the heuristic of directional certainty (Kery and Schaub 2011). Fixed effects were included if their s-value was \(>\) 4.32 bits (Kery and Schaub 2011). Based on a similar argument, random effects were included if their standard deviation had lower 95% CLs \(>\) 5% of the estimate.

The results are displayed graphically by plotting the modeled relationships between individual variables and the response 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 average values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). Primary explanatory variables are evaluated based on their estimated effect sizes with 95% CLs (Bradford, Korman, and Higgins 2005)

The analyses were implemented using R version 4.2.2
(R Core Team 2022) and the
`mbr`

family of packages.

### Model Descriptions

#### Mark-Recapture

The data were analysed using a hierarchical Bayesian mark-recapture abundance model. Each zone was split into two kilometres sites except sites at the end of the zone were added to the previous site if the length was less than one kilometre. Captures from sites that had less then 10% of their length surveyed were not included in the analysis.

Key assumptions of the mark-recapture model include:

- Lineal density varies randomly by year and zone within year.
- There is no movement of marked fish between sites within a sampling season.
- There is no mortality of fish within a sampling season.
- The probability of capturing a marked or unmarked fish is the same.
- All recaptured fish are correctly identified as being marked and there is no tag loss.
- Capture efficiency varies by electrofishing effort.
- Capture efficiency varies randomly by site visit.
- The number of recaptures (of fish marked at that site in that year) and the total number of fish caught are each binomially distributed.

### Model Templates

#### Mark-Recapture

```
.model{
bDensity ~ dnorm(6, 2^-2)
bEfficiency ~ dnorm(-5, 2^-2)
bEfficiencySecondsPerMetre ~ dnorm(0, 1^-2)
bTheta ~ dexp(10)
bDensityAnnual[1] <- 0
for (i in 2:nannual) {
bDensityAnnual[i] ~ dnorm(0, 2^-2)
}
sDensityAnnualZoneID ~ dnorm(0, 1^-2) T(0,)
for (i in 1:nannual) {
for (j in 1:nzone_id) {
bDensityAnnualZoneID[i, j] ~ dnorm(0, sDensityAnnualZoneID^-2)
}
}
bEfficiencyAnnual[1] <- 0
for (i in 2:nannual) {
bEfficiencyAnnual[i] ~ dnorm(0, 2^-2)
}
for (i in 1:nObs) {
log(eDensity[i]) <- bDensity + bDensityAnnual[annual[i]] + bDensityAnnualZoneID[annual[i], zone_id[i]]
logit(eProb[i]) <- bEfficiency + bEfficiencySecondsPerMetre * seconds_per_metre[i] + bEfficiencyAnnual[annual[i]]
eEfficiency[i] ~ dbeta((2 * eProb[i]) / bTheta, (2 * (1 - eProb[i])) / bTheta)
recaptures[i] ~ dbin(eEfficiency[i], marked[i])
bAbundance[i] ~ dpois(eDensity[i] * site_length[i] * survey_proportion[i])
count[i] ~ dbin(eEfficiency[i], bAbundance[i])
}
```

Block 1. Model description.

## Results

### Tables

#### Mark-Recapture

Table 1. Parameter descriptions.

Parameter | Description |
---|---|

`annual[i]` |
Year of the `i` ^{th} survey visit |

`bAbundance[i]` |
Expected abundance during `i` ^{th} site visit |

`bDensityAnnualZoneID[i, j]` |
Effect of the `i` ^{th} year and the `j` ^{th} zone on
`bDensity` |

`bDensityAnnual[i]` |
Effect of `i` ^{th} year on `bDensity` |

`bDensity` |
Intercept of `log(eDensity)` |

`bEfficiencyAnnual[i]` |
Effect of `i` ^{th} year on `logit(eProb)` |

`bEfficiencySecondsPerMetre` |
Effect of electrofishing seconds per metre
travelled on `eEfficiency` |

`bEfficiency` |
Intercept for `logit(eEfficiency)` |

`bTheta` |
Variation in the random effect of site visit on
`eEfficiency` |

`count[i]` |
The number of fish captured during the `i` ^{th}
site visit |

`eDensity[i]` |
Expected fish density during the `i` ^{th} site
visit |

`eEfficiency[i]` |
Expected capture efficiency including the random effect of site visit |

`eProb[i]` |
Expected capture efficiency during the `i` ^{th}
site visit |

`marked[i]` |
Number of fish marked prior to the `i` ^{th} site
visit |

`recaptures[i]` |
Number of marked fish observed during the `i` ^{th}
site visit |

`seconds_per_metre[i]` |
Number of electrofishing seconds per metre on the
`i` ^{th} site visit |

`site_length[i]` |
Length of the site visited for the `i` ^{th} site
visit |

`survey_proportion[i]` |
Proportion of site surveyed during the `i` ^{th}
site visit |

`zone_id[i]` |
Zone of the `i` ^{th} survey visit |

Table 2. Model coefficients.

term | estimate | lower | upper | svalue |
---|---|---|---|---|

bDensity | 5.88000 | 5.0300 | 6.7000 | 10.6000 |

bDensityAnnual[2] | -0.00432 | -0.8500 | 1.0900 | 0.0174 |

bEfficiency | -5.44000 | -6.0100 | -4.8700 | 10.6000 |

bEfficiencyAnnual[2] | 1.03000 | 0.5010 | 1.5600 | 10.6000 |

bEfficiencySecondsPerMetre | 0.20400 | 0.0507 | 0.3540 | 6.4600 |

bTheta | 0.01910 | 0.0122 | 0.0292 | 10.6000 |

sDensityAnnualZoneID | 0.40000 | 0.1020 | 0.9990 | 10.6000 |

Table 3. Model convergence.

n | K | nchains | niters | nthin | ess | rhat | converged |
---|---|---|---|---|---|---|---|

188 | 7 | 3 | 500 | 5000 | 156 | 1.01 | TRUE |

Table 4. The estimated abundance of WCT in zones 2 to 6 on Elk River, by year (with 95% CIs).

annual | estimate | lower | upper |
---|---|---|---|

2021 | 35125.60 | 19755.97 | 67838.93 |

2022 | 32167.51 | 23676.28 | 46452.53 |

Table 5. Model posterior predictive checks.

moment | observed | median | lower | upper | svalue |
---|---|---|---|---|---|

zeros | 0.3031915 | 0.2340426 | 0.1595745 | 0.3138298 | 3.8934968 |

mean | -1.0171159 | -0.9137361 | -1.0081736 | -0.8258678 | 4.9971194 |

variance | 0.3130478 | 0.3714921 | 0.2921255 | 0.4683314 | 2.6999592 |

skewness | -0.6425487 | -0.6136020 | -1.0146960 | -0.3480178 | 0.2152017 |

kurtosis | 0.2399603 | -0.0390868 | -0.6950496 | 1.7591280 | 0.7109303 |

Table 6. Model sensitivity.

all | analysis | sensitivity | bound |
---|---|---|---|

all | 1.01 | 1.027 | 1.018 |

### Figures

#### Maps

#### Size

#### Captures

#### Mark-Recapture

#### Outing

#### Codes

## Recommendations

Recommendations include:

- Measure and record fork length to 1 mm.
- Use a unique naming convention for photo file names, as a
combination of the capture date-time, reader ID, PIT Tag code and
photo type (R = reader; F = fish left; D = damage)
- YYYY-MM-DD-hhmmss_readerID_PITTagCode_PhotoType
- e.g., 2020-07-23-131042_6_900088000907916_R

- Develop a growth model that incorporates measurement error.

## Acknowledgements

The organizations and individuals whose contributions have made this analytic appendix possible include:

- Nupqu Limited Partnership
- Mark Fjeld
- Dominique Nicholas
- Rafael Lugo

- Teck Coal Ltd.
- Bronwen Lewis
- Dorian Turner

- BC Government
- Matt Neufeld
- Will Warnock

## References

*Canadian Journal of Fisheries and Aquatic Sciences*62 (12): 2716–26. https://doi.org/10.1139/f05-179.

*Handbook for Markov Chain Monte Carlo*. Boca Raton: Taylor & Francis.

*Stan*: A Probabilistic Programming Language.”

*Journal of Statistical Software*76 (1). https://doi.org/10.18637/jss.v076.i01.

*Entropy*19 (10): 555. https://doi.org/10.3390/e19100555.

*p*-Values Behave Exactly as They Should: Some Misleading Criticisms of

*p*-Values and Their Resolution With

*s*-Values.”

*The American Statistician*73 (sup1): 106–14. https://doi.org/10.1080/00031305.2018.1529625.

*Epidemiology*24 (1): 62–68. https://doi.org/10.1097/EDE.0b013e3182785741.

*Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective*. Boston: Academic Press. http://www.vogelwarte.ch/bpa.html.

*Statistical Rethinking: A Bayesian Course with Examples in R and Stan*. 2nd ed. CRC Texts in Statistical Science. Boca Raton: Taylor; Francis, CRC Press.

*Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003)*, edited by Kurt Hornik, Friedrich Leisch, and Achim Zeileis. Vienna, Austria.

*BMC Medical Research Methodology*20 (1): 244. https://doi.org/10.1186/s12874-020-01105-9.

*PeerJ*5 (January): e2874. https://doi.org/10.7717/peerj.2874.