Lecture 10

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# Lecture 10
## Estimating abundance: N-mixture models
### <br/><br/><br/>WILD6900 (Spring 2020)

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## Readings

> Kéry & Schaub 383-411

> [Powell & Gale chp. 18](https://docs.wixstatic.com/ugd/95e73b_233898561a20413687b6b7236b77e371.pdf)

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## Systematic bias in state-space models

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## State-space models

#### Produce unbiased estimates of `$\Large N$` **only** when false-positives and false-negatives cancel each other out on average  
<br/>

#### Produce unbiased estimates of population **indices** `$\Large (Np)$` if detection probability has no pattern ove time  
<br/>

#### Do **not** produce unbiased estimates of `$\Large N$` or `$\Large Np$` if their are temporal patterns in detection probability or false-positive rates

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## Estimating abundance

#### Unbiased estimates of `$\large N$` require estimating `$\large p$`

#### *Many* methods available:

- Mark-recapture

- Removal sampling

- Distance sampling

- Double observer

- N-mixture models

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## Estimating abundance

#### Unbiased estimates of `$\large N$` require estimating `$\large p$`

#### *Many* methods available:

- Mark-recapture

- Removal sampling

- Distance sampling

- Double observer

- **N-mixture models ([Royle 2004](https://www.jstor.org/stable/pdf/3695558.pdf?casa_token=VmzoVcHywNkAAAAA:L2gjCIWETinhTyRUYVrexii5S6sD3oHYFIhb7ZNMdXQuTGeSckUtF6Yg360fsPkILD2-KW0cosorvn-BJ4t07629KUaH1ekcm6l5ZSoI0raU_1Vfje8))**

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# N-mixture models

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## N-mixture models

#### The basic idea

- `$\large \mathbf J$` sites surveyed

+ Each site has an expected abundance `$\large \lambda$`

--
    + State model:  
    
`$$\Large N_j \sim Poisson(\lambda)$$`

--
- Each site is surveyed `$\large \mathbf K$` times

+ During each visit, probability `$\large p$` of detecting each individual

--
    + Observation model:  
    
`$$\Large y_{j,k} \sim binomial(N_j, p)$$`

---
## N-mixture models

#### The data

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> Site </th>
   <th style="text-align:right;"> N </th>
   <th style="text-align:right;"> Visit1 </th>
   <th style="text-align:right;"> Visit2 </th>
   <th style="text-align:right;"> Visit3 </th>
   <th style="text-align:right;"> Visit4 </th>
   <th style="text-align:right;"> Visit5 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;font-weight: bold;border-right:1px solid;"> 3 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0 </td>
   <td style="text-align:right;background-color: yellow !important;"> 1 </td>
   <td style="text-align:right;background-color: yellow !important;"> 1 </td>
   <td style="text-align:right;background-color: yellow !important;"> 2 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;font-weight: bold;border-right:1px solid;"> 5 </td>
   <td style="text-align:right;background-color: yellow !important;"> 4 </td>
   <td style="text-align:right;background-color: yellow !important;"> 2 </td>
   <td style="text-align:right;background-color: yellow !important;"> 2 </td>
   <td style="text-align:right;background-color: yellow !important;"> 2 </td>
   <td style="text-align:right;background-color: yellow !important;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;font-weight: bold;border-right:1px solid;"> 1 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0 </td>
   <td style="text-align:right;background-color: yellow !important;"> 1 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;font-weight: bold;border-right:1px solid;"> 1 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0 </td>
   <td style="text-align:right;background-color: yellow !important;"> 1 </td>
   <td style="text-align:right;background-color: yellow !important;"> 1 </td>
  </tr>
</tbody>
</table>

---
## N-mixture models

#### JAGS code

```
model{
  # Priors
  lambda ~ dgamma(0.01, 0.01)
  p ~ dbeta(1, 1)
  
  # Likelihood
  for(j in 1:J){
    ## State model
    N[j] ~ dpois(lambda)
    
    ## Observation model
    for(k in 1:K){
      y[j, k] ~ dbinom(p, N[j])
    } # end k loop
  } # end j loop
  
} # end model
```

---
## N-mixture models

#### Extensions

1) Covariates and random effects

`$$\large log(\lambda_j) = \alpha_0 + \alpha_1x_j + \epsilon_j$$`

`$$\large \epsilon_j \sim normal(0, \tau)$$`

`$$\large logit(p_{j,k}) = \beta_0 + \beta_1x_j + \beta_2x_{j,k}$$`

---
## N-mixture models

#### Extensions

1) Covariates and random effects

2) Open N-mixture model ([Dail & Madsen 2011](https://www.jstor.org/stable/pdf/41242495.pdf?casa_token=Y3RTUwUZKs8AAAAA:QFHRWtxjqlOQfRoAaw_3Z5qkbbXZ3WMcjnwFW1pf04FDmhgAPQBn44eGWi4mLASdYpgVUMmKt3_oUX7MKTHZKUfJReftyPUNwQL6oATonf7uTGc3rZ4))

`$$\large N_{j,1} \sim Poisson(\lambda)$$`
`$$\large N_{j,t} = S_{j,t} + G_{j,t}$$`

`$$\large S_{j,t} \sim binomial(N_{j,t-1}, \phi)$$`
`$$\large G_{j,t} \sim Poisson(\gamma N_{j,t-1})$$`

---
## N-mixture models

#### Extensions

1) Covariates and random effects

3) Other distributions

- negative binomial, zero-inflated Poisson, multinomial

---
## N-mixture models

#### Extensions

1) Covariates and random effects

3) Other distributions

4) Integrated N-mixture models

- Integrated N-mixture/known fate model ([Schmidt et al. 2015](https://esajournals.onlinelibrary.wiley.com/doi/full/10.1890/15-0385.1))  
    
- Integrated N-mixture/distance sampling model ([Hostetter et al. 2019](https://esajournals.onlinelibrary.wiley.com/doi/full/10.1002/ecs2.2586))

---
## Assumptions of the N-mixture model

1) Poisson and binomial distributions are true descriptions of state/observation processes

2) Abundance at each site is random and independent of abundance at all other sites

3) Population is closed between surveys

4) Observers do not double count individuals

5) All `$\large N$` individuals have the same detection probability `$\large p$`

---
## Advantages and disadvantages of the N-mixture model

#### Advantages

- Count data is "cheap" to collect (relative to mark-recapture)

- Does not require auxillary information (distance, double observer, etc.)

- Analysis is straightforward

#### Disadvantages

- Count data has less information about `$\large p$` than mark-recapture data

- Requires lots of replication

- Trade-off between temporal replication and spatial replication

- Inference can be sensitive to violating assumptions

---
## Controversy

#### [Barker et al. (2017)](https://onlinelibrary.wiley.com/doi/full/10.1111/biom.12734)

- Mark-recapture data provides auxillary information about `$\large p$` without reference to `$\large N$`

- Without auxillary information about `$\large p$`, count data cannot distinguish between N-mixture model or other possible models of `$\large N$`

---
## Controversy

#### [Barker et al. (2017)](https://onlinelibrary.wiley.com/doi/full/10.1111/biom.12734)

#### [Kery (2017)](https://esajournals.onlinelibrary.wiley.com/doi/pdf/10.1002/ecy.2093)

- No issues with identifiability of Poisson N-mixture model based on 137 bird data sets from 2,037 sites

- Some parameters not identifiability with negative binomial model (especially with small sample sizes)

+ problematic because NB often selected based on AIC

---
## Controversy

#### [Barker et al. (2017)](https://onlinelibrary.wiley.com/doi/full/10.1111/biom.12734)

#### [Kery (2017)](https://esajournals.onlinelibrary.wiley.com/doi/pdf/10.1002/ecy.2093)

#### [Link et al. (2018)](https://esajournals.onlinelibrary.wiley.com/doi/abs/10.1002/ecy.2362)

- Estimates from N-mixture models sensitive to violation of double counting and constant `$\large \lambda/p$`

- Small violations unlikely to be detected using goodness‐of‐fit tests but can influence inference about `$\large N$`