Llgaay Gwii sdiihlda (Restoring Balance project) 2014-2019

Fri, 16 Feb 2024 00:00:00 +0000

The suggested citation for this analytic appendix is:

Thorley, J.L. & Irvine, R.L. (2024) Llgaay Gwii sdiihlda (Restoring Balance project) 2014-2019. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/994043377.

An associated paper is available from publications.

Background

To restore balance, a deer removal program was implemented on various islands in Gwaii Haanas.

The primary goal of the current analyses is to answer the following questions:

What was the relative efficiency of the removal methods?

What would the cost of removing the remaining deer have been?

Data Preparation

The data were provided by Parks Canada in the form of Excel spreadsheets and prepared for analysis using R version 4.3.2 (R Core Team 2020).

Statistical Analysis

Model parameters were estimated using Bayesian methods. The estimates were produced using JAGS (Plummer 2003). 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 et al. 2017). The posterior distributions were estimated from 4000 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 4 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that the potential scale reduction factor $\hat{R} \leq 1.01$ (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) $\textrm{ESS} \geq 1000$ for each of the monitored parameters (Kery and Schaub 2011, 61).

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 $>$ 4.3 bits, which is equivalent to a significant 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 on Leave-one-out cross-validation (LOO-CV) as implemented using the Pareto-smoothed importance sampling (PSIS) algorithm (Vehtari et al. 2017). LOO-CV is asymptotically equal to the Widely Applicable Information Criterion Watanabe (2013). Model weight ($w_i$) was based on $\exp(-0.5 \Delta_i)$ as proposed by Akaike (1978) for Akaike Information Criterion (AIC) and by Watanabe (2010) for WAIC where $\Delta_i$ is the absolute difference in the out-of-sample predictive density of the $i$th model relative to the best model (Burnham and Anderson 2002). Primary explanatory variables were evaluated based on their estimated effect sizes with 95% CLs (Bradford et al. 2005)

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. In this context each s-value indicates how surprising it would be to discover that the actual data was generated using the same distribution as the simulated data.

The sensitivity of the parameters to the choice of prior distributions was evaluated by doubling the standard deviations of all the normal and half-normal priors and then using $\hat{R}$ to evaluate whether the samples were drawn from the same posterior distribution (Thorley and Andrusak 2017).

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).

The analyses were implemented using R version 4.3.2 (R Core Team 2022) and the mbr family of packages.

Model Descriptions

Model

The data were analysed using a power function (Ward et al. 2013) for the relationship between efficiency and density

\[\text{Efficiency} = \alpha \cdot {\text{Density}}^{\beta} \]

Four models were considered which varied in whether they included variation in $\alpha$ and $\beta$ by method.

Key assumptions of all the models include:

There is no migration among the islands during the course of the study.
The relationship between efficiency and density is described by a power function.
The expected number of deer removed by island, method and day is the product of the efficiency and effort.
The residual variation in the number deer removed by island, method and day is described by an overdispersed Poisson distribution (negative binomial).

The full model (variation in $\alpha$ and $\beta$ by method = ambm) is defined algebraically as follows:

\[\text{Efficiency}_{m,i,d} = \alpha_{m} \cdot {\text{Density}_{i,d}}^{\beta_{m}} \] where $\text{Efficiency}_{m,i,d}$ is the efficiency of the $m$th method on the $i$th island during the $d$th day, $\alpha_{m}$ is the efficiency of the $m$th method at a density of 1 deer per km², $\text{Density}_{i,d}$ is the deer density (deer/km²) on the $i$th island at the start of the $d$th day and $\beta_{m}$ is the scaling constant.

The allometric coefficient and the scaling exponent in the power function are given by

\[\log(\alpha_{m}) = a_m \] and

\[\log(\beta_{m}) = b_m\]

, respectively, where $a_m$ and $b_m$ are the values by method.

The expected number of deer removed using the $m$th method on the $i$th island during the $d$th day ($\mu_{m,i,d}$) is given by

\[\lambda_{m,i,d} = \text{Efficiency}_{m,i,d} \cdot \text{Effort}_{m,i,d}\]

where $\text{Effort}_{m,i,d}$ is the effort (heli hours) of the $m$th method on the $i$th island during the $d$th day.

The actual number of deer removed using the $m$th method on the $i$th island during the $d$th day ($\mu_{m,i,d}$) is given by the relationship

\[\text{Removed}_{m,i,d} \sim \text{NBinomial}(\lambda_{m,i,d}, \phi)\]

The total population on each island at the start of the study was given by

\[\text{Population}_{i} = \text{TotalRemoved}_i + \Psi_i\]

where $\Psi_i$, which is the total number of remaining deer on the $i$th island, was based on the number of scent trails detected by dogs at the end of the surveys and the effective coverage of those dogs according to the relationship

\[\text{ScentTrails}_i \sim \text{Binomial}(\Psi_{i},\text{Coverage}_i)\]

The priors were as follows:

\[a_m \sim \text{Normal}(1, 2)\]

\[b_m \sim \text{Normal}(0, 2)\] \[\phi \sim \text{Normal}(0, 1)\ \text{T}(0,)\]

\[\Psi_i \sim \text{Normal}(0, \text{Area}_i \cdot 5) \text{T}(0,)\]

Model Templates

Removal

model{
  for(i in 1:nIsland) {
    bRemaining[i] ~ dnorm(0, (5 * Area[i])^-2) T(0,)
    Detections[i] ~ dbin(ObsEff[i], round(bRemaining[i]))
    bPopn[i,1] <- round(TotalRemoved[i] + bRemaining[i])
    bDensity[i,1] <- bPopn[i,1] / Area[i]
    for(j in 2:nDay) { 
      bPopn[i,j] <- bPopn[i,j-1] - DeerTotal[i,j-1]
      bDensity[i,j] <- bPopn[i,j] / Area[i]
    }
  }
  alpha0 <- 0

  for(i in 1:nMethod) {
    alpha_method[i] ~ dnorm(1, 2^-2)
  }

  beta0 <- 0

  for(i in 1:nMethod) {
    beta_method[i] ~ dnorm(0, 2^-2)
  }

  phi ~ dnorm(0, 1^-2) T(0,)
  for(i in 1:nObs) {
    eDensity[i] <- bDensity[Island[i],Day[i]]
    eEffort[i] <- Hours[i] * HourlyRate[i]
    log(eAlpha[i]) <- alpha0 + alpha_method[Method[i]]
    log(eBeta[i]) <- beta0 + beta_method[Method[i]]
    log(eEfficiency[i]) <- eAlpha[i] +  log((eDensity[i] / 100)^eBeta[i])
    eDeer[i] <- eEffort[i] * eEfficiency[i] 
    eR[i] <- 1/phi
    eP[i] <- eR[i] / (eR[i] + eDeer[i])
    Deer[i] ~ dnegbin(eP[i], eR[i])
  }

Block 1. Model description.

Results

Tables

Costs

Table 1. The estimated hourly costs (in $) by crew member and equipment.

Type	HourlyRate
BaitHunter	180
BaitHunterLead	280
BaitHunterAvg	230
BoatPlusOperator	90
HunterAvg	135
Dog	40
DogHunter	90
HeliHunter	405
HeliPlusOperator	1560

Removal

Table 2. The number of scent trail detections by dogs and the effective coverage by island.

Island	ScentTrials	EffectiveCoverage
Ramsay Island	10	0.30
Murchison Island	2	0.75
House Island	0	0.90

Table 3. Model comparison using Pareto Smoothed Importance-Sampling Leave-One-Out Cross-Validation (PSIS) criterion. ‘ic’ is the information criterion value (IC) on the deviance scale; ‘se’ is the standard error of the IC; ‘npars’ is the number of effective parameters, ‘delta ic’ is the difference between the model’s IC and the minimum IC; ‘delta se’ is the standard error of the difference in IC; ‘weight’ summarizes the relative support for each model; and ‘k outliers’ is the proportion of data points with Pareto $\hat{k}$ values exceeding 0.7.

model	ic	se	npars	delta ic	delta se	weight
ambm	1019.026	41.54826	9.472834	0.00000	0.00000	9.995015e-01
amb0	1034.285	43.11210	7.304104	15.25862	8.83647	4.857535e-04
a0bm	1041.568	43.08811	7.276514	22.54128	10.85570	1.273524e-05
a0b0	1065.989	43.28138	3.383383	46.96279	15.62084	6.337856e-11

Table 4. Parameter descriptions.

Parameter	Description
`Area[i]`	Surface area of `i`^th island (km2)
`Day[i]`	Days since April 20th 2017 of `i`^th outing
`DeerTotal[i,j]`	Number of deer removed from `i`^th Island on `j`^th `Day` since April 20th 2017
`Deer[i]`	Number of deer removed during `i`^th outing
`Detections[i]`	Number of scent trail detections on `i`^th island at end of operations
`HourlyRate[i]`	Relative cost of `i`^th outing (helicopter crew hourly rate)
`Hours[i]`	Duration of `i`^th outing (hours)
`Island[i]`	Island of `i`^th outing
`ObsEff[i]`	Effective coverage of scent trail surveys on `i`^th island at end of operations
`TotalRemoved[i]`	Total number of deer removed from `i`^th island during operations
`alpha0`	Intercept for `log(eAlpha)`
`alpha_method[i]`	Effect of `i`^th method on `alpha0`
`bDensity[i,j]`	Expected density of deer per km2 on `i`^th island at start of `j`^th `Day` since April 20th 2017
`bPopn[i,j]`	Expected number of deer on `i`^th island at start of `j`^th `Day` since April 20th 2017
`bRemaining[i]`	Expected number of deer remaining on `i`^th island at end of operations
`beta0`	Intercept for `log(eBeta)`
`beta_method[i]`	Effect of `i`^th method on `beta0`
`eAlpha[i]`	`log(Efficiency)` at a density of 100 deer per km2
`eBeta[i]`	Effect of `log(bDensity * 100)` on `log(Efficiency)`
`phi`	Extra Poisson variation

Table 5. Model terms (with 95% CIs).

term	estimate	lower	upper	svalue
alpha_method[1]	1.6200	0.98100	2.030	12.000
alpha_method[2]	1.2700	1.01000	1.480	12.000
alpha_method[3]	1.1300	0.83300	1.370	12.000
alpha_method[4]	-0.9830	-3.28000	0.310	2.500
alpha_method[5]	-0.4640	-3.44000	0.985	0.624
bRemaining[1]	0.2730	0.01290	1.170	12.000
bRemaining[2]	2.8100	1.57000	5.870	12.000
bRemaining[3]	31.9000	19.20000	50.400	12.000
beta_method[1]	0.9550	0.00441	1.470	4.330
beta_method[2]	0.0176	-0.41600	0.334	0.124
beta_method[3]	0.1220	-0.25000	0.429	1.050
beta_method[4]	-2.0700	-4.64000	-0.662	12.000
beta_method[5]	-0.6440	-1.28000	0.155	3.040
phi	0.1330	0.02300	0.310	12.000

Table 6. Model convergence.

n	K	logLik	IC	nchains	niters	nthin	ess	rhat	converged
485	14	NA	NA	4	1000	200	1672	1.002	TRUE

Table 7. Model sensitivity.

all	analysis	sensitivity	bound
all	1.002	1.001	1.205

Table 8. Model sensitivity by parameter.

parameter	analysis	sensitivity	bound
alpha_method	1.000	1.000	1.193
bDensity	1.002	1.000	1.001
beta_method	1.000	1.001	1.205
bPopn	1.002	1.000	1.001
bRemaining	1.002	1.001	1.001

Table 9. Model sensitivity by fixed terms.

term	analysis	sensitivity	bound
alpha_method[1]	1.000	1.000	1.001
alpha_method[4]	1.000	1.000	1.193
alpha_method[5]	1.000	1.000	1.096
beta_method[1]	1.000	1.001	1.001
beta_method[4]	1.000	1.000	1.205
beta_method[5]	1.000	1.000	1.049
bRemaining[1]	1.001	1.001	1.001
bRemaining[2]	1.002	1.000	1.001

Table 10. Model posterior predictive checks.

moment	observed	median	lower	upper	svalue
zeros	0.5443299	0.5434298	0.4922049	0.5947105	0.0123122
mean	-0.2247702	-0.2483574	-0.3332791	-0.1558049	0.7757029
variance	0.9341431	0.8680227	0.7537828	0.9898375	1.8459070
skewness	0.7119498	0.7010032	0.4946737	0.9341840	0.1108871
kurtosis	0.1692013	-0.1078410	-0.5601321	0.6157612	1.4793099

Table 11. The total number of deer removed, the area (km2) and the removal density (deer/km2) by island.

Island	Removed	Area	RemovalDensity
House	10	0.3292	30.376671
Murchison	26	3.9987	6.502113
Ramsay	412	16.2276	25.388844

Table 12. The total number of deer removed and the estimated number of remaining deer by island (with CIs).

Island	Removed	lower98	lower95	estimate	upper95	upper98
House	10	0	0	0	1	1
Murchison	26	2	2	3	6	7
Ramsay	412	17	19	32	50	54

Table 13. The total number of deer removed and the estimated percent eradication completion by island (with CIs).

Island	Removed	lower98	lower95	estimate	upper95	upper98
House	10	0.9090909	0.9090909	1.0000000	1.0000000	1.0000000
Murchison	26	0.7878788	0.8125000	0.8965517	0.9285714	0.9285714
Ramsay	412	0.8841202	0.8917749	0.9279279	0.9559165	0.9603730

Table 14. The total costs spent in heli hours by island.

Island	Cost	Area	Cost/km2
House	10.44707	0.3292	31.734732
Murchison	33.69046	3.9987	8.425353
Ramsay	282.61980	16.2276	17.415995

Table 15. The estimated total completion cost by island and method (with CIs).

Island	Method	lower98	lower95	estimate	upper95	upper98
House	Indicator Dog	0.0000000	0.0000000	0.000000	1.376753	2.478155
House	Boat	0.0000000	0.0000000	0.000000	1.648855	2.748092
Murchison	Indicator Dog	0.2753505	0.5507011	3.854908	18.173136	24.781550
Murchison	Boat	1.0992366	2.1984733	18.137405	122.564886	169.837557
Ramsay	Indicator Dog	16.2429289	18.7169536	42.954686	129.696993	163.304904
Ramsay	Boat	35.7251908	46.7175573	194.015267	1124.093130	1777.075420

Table 16. The estimated proportional efficiency of indicator dog hunting by method and density.

Method	Density	estimate	lower	upper
Combined	2	2.5248852	0.6743702	6.582835
Boat	2	0.2672479	-0.4415746	1.954678

Table 17. The total number of deer removed and the proportion of the encounters removed by method.

Method	Removed	EncountersRemoved
Bait Station	90	0.90
Boat	101	0.57
Helicopter	138	0.49
Indicator Dog	75	0.71
Miscellaneous	15	0.75
NA	29	0.74

Table 18. The total number of deer removed on Ramsay Island and the proportion of the encounters removed by method.

Method	Removed	EncountersRemoved
Bait Station	90	0.90
Boat	97	0.56
Helicopter	136	0.49
Indicator Dog	64	0.68
Miscellaneous	5	0.71
NA	20	0.69

Figures

Removal

Figure 1. The removal efficiency by date, method and island with the estimated removal efficiency.

Figure 2. The estimated removal efficiency by density and method.

Figure 3. The estimated removal efficiency by method and density.

Figure 4. The posterior probability of the percent eradication completion by island.

Figure 5. The estimated eradication completion by cumulative effort by method for Ramsay with a starting population of 466 deer.

Acknowledgements

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

Parks Canada
- Nadine Wilson
- Peter Dyment
- Charlotte Houston
- Patrick Bartier
- Christine Bentley
- Emily Gonzales
- Kent Prior
Coastal Conservation
- Chris Gill
- Greg Howald
Poisson Consulting
- Seb Dalgarno
Simon Fraser University
- Carl Schwarz
David Will
Pete McLelland
Norm MacDonald
Gerry Morigeau

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Sitka Black-­Tailed Deer | Poisson Consulting

Llgaay Gwii sdiihlda (Restoring Balance project) 2014-2019

Background

Data Preparation

Statistical Analysis

Model Descriptions

Model

Model Templates

Removal

Results

Tables

Costs

Removal

Figures

Removal

Acknowledgements

References

Sitka Black-Tailed Deer | Poisson Consulting