Lecture 12

class: center, middle, inverse, title-slide

# Lecture 12
## Estimating abundance: Closed-population capture-mark-recapture
### <br/><br/><br/>WILD6900 (Spring 2020)

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## Readings

> ### Kéry & Schaub 134-170

> ### [Powell & Gale chp. 8](https://docs.wixstatic.com/ugd/95e73b_6d832c61405d4b3cbd9d99bbb4530f7b.pdf)

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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

---
## 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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# Capture-mark-recapture

---
## Capture-mark-recapture (CMR)

#### CMR includes a **wide** variety of related techniques and models

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#### Traditionally, CMR referred to methods of capturing, marking, and then recapturing individuals at some point in the future

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#### This results in a **capture-history** for each individual:

##### Individual 1: `\(101101\)`  
##### Individual 2: `\(011001\)`

???

Individual 1 was captured on occasions 1, 3, 4, and 6

Individual 2 was captured on occasions 2,3, and 6.

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#### From the capture-histories, possible to estimate `\(\large p\)`

--
#### With `\(\large p\)`, possible to estimate `\(\large N\)`

`$$\large N = \frac{C}{p}$$`

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## Capture-mark-recapture

#### From capture-histories, we can estimate:

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- abundance

--
- detection probability

--
- survival

--
- movement

--
- recruitment

--
- individual growth

--
- populations trends

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## Capture-mark-recapture

#### CMR methods are not restricted to physical captures and recaptures

<img src="figs/pelican.jpg" width="600" style="display: block; margin: auto;" />

???

Capture histories can be created from resighting marks that can be seen from a distance

Photo Credit: Rick Kimble/USFWS

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## Capture-mark-recapture

#### CMR methods are not restricted to physical captures and recaptures

<img src="figs/blrf.jpg" width="600" style="display: block; margin: auto;" />

???

Capture histories can be created from automated systems that record ID's, like PIT tags

Image courtesy of: USFWS Mountain-Prairie via Wikimedia Commons

---
## Capture-mark-recapture

#### CMR methods are not restricted to physical captures and recaptures

<img src="https://upload.wikimedia.org/wikipedia/commons/3/37/Wild_Sumatran_tiger.jpg" width="600" style="display: block; margin: auto;" />

???

Capture histories can be created from "natural marks", e.g. camera trapping

Image courtesy of: Arddu, via Wikimedia Commons

---
## Capture-mark-recapture

#### CMR methods are not restricted to physical captures and recaptures

<img src="https://upload.wikimedia.org/wikipedia/commons/2/2a/Grizzly_hair_snare_%28Northern_Divide_Grizzly_Bear_Project%29_%284428177124%29.jpg" width="600" style="display: block; margin: auto;" />

???

Capture histories can be created from "natural marks", e.g. genetic markers

Image courtesy of: GlacierNPS, via Wikimedia Commons

---
## Capture-mark-recapture

#### CMR methods are not restricted to physical captures and recaptures

#### *What ties all of these methods together is that we have individual-level capture-histories*

---
## Capture-mark-recapture

#### Key to closed CMR is that because individuals do not enter or leave the population, we assume any 0 in the capture history is non-detection `\(\large (1-p)\)`

- in simplest model, easy to translate capture history into probabilistic statements based only on `\(p\)`:

.pull-left[
##### Individual 1: `\(\large 101101\)`  
##### Individual 2: `\(\large 011001\)`  
]

.pull-right[
#### `\(\large p(1-p)pp(1-p)p\)`  
#### `\(\large (1-p)pp(1-p)(1-p)p\)`
]

<br/>
--

#### In the CMR literature, the constant `\(\large p\)` model is known at the `\(M_0\)` model ([Otis et al. (1978)](https://www.jstor.org/stable/pdf/3830650.pdf?casa_token=Dr4Y3XD5DJAAAAAA:IQA7BkqTFg7_0h2kqbrn20CGZLBbTBBbuAsCWsvdRWD-muQDenn-UVYfpdL3q09ZTmiGGxQBm5vnFV9RomNxOdtBq2c98vOSELdt6-hRyoY52ipEYj4))

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## The `\(\large M_0\)` model in JAGS

#### In all closed CMR models, `\(\large N\)` is unknown

- we know `\(n\)` but how many individuals were not detected at all?

--
- data augmentation!

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class: inverse, middle, center

# Data augmentation

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## Data augmentation

### Imagine an occupancy study:

- `\(\large M\)` sites are surveyed  
    + `\(\large N\)` sites are occupied `\(\large (z = 1)\)`
    + `\(\large M - N\)` sites are unoccupied `\(\large (z = 0)\)`

--
- species is detected `\((y_i=1)\)` at `\(n\)` sites
    + species is not detected `\((y_i=0)\)` at `\(M-n\)` sites  
    
--
`$$\Large y_i \sim Bernoulli(z_i p)$$`

---
## Data augmentation

### Imagine an occupancy study:

#### How many sites are actually occupied `\(\large (N)\)`?

- if `\(\large \psi\)` is the probability of occupancy

`$$\Large z_i \sim Bernoulli(\psi)$$`

--
- and

`$$\Large N = \sum_{i=1}^M z_i$$`

---
## Data augmentation

#### These ideas can be applied to CMR studies

#### Imagine a CMR study:

- `\(\large n\)` individuals were detected during the study

--
- `\(\large N-n\)` individuals were not detected
    + how do we know how many individuals were not detected?

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- Add `\(\large M-n\)` individuals to the data
    + Choose `\(M \gt \gt N\)`
    + All of these "augmented" individuals have `\(\large y=0\)`

--
`$$\large z_i \sim Bernoulli(\psi)$$`

`$$\large N = \sum_{i = 1}^M z_i$$`
---
## The `\(\large M_0\)` model in JAGS

#### In all closed CMR models, `\(\large N\)` is unknown

- we know `\(n\)` but how many individuals were not detected at all?

- data augmentation!

--
`$$\Large z_i \sim Bernoulli(\psi)$$`

--
`$$\Large y_{ik} \sim Bernoulli(z_ip)$$`

--
`$$\Large \psi \sim beta(1,1)$$`

`$$\Large p \sim beta(1,1)$$`

--
`$$\Large N = \sum_{i=1}^M z_i$$`
---
## The `\(\large M_0\)` model in JAGS

```r
model{
  psi ~ dbeta(1, 1)
  p ~ dbeta(1, 1)
  
  for(i in 1:M){
    z[i] ~ dbern(psi)
    
    for(t in 1:T){
      y[i, t] ~ dbern(p * z[i])
    } # end t
  } # end i
  
  N <- sum(z[1:M])
}
```

---
## Capture-mark-recapture

####  In addition to the `\(\large M_0\)` model, Otis et al. 1978 outlined 3 other basic *closed* CMR "models":

--
- `\(\large M_t\)`: variation in `\(p\)` among occasions

`$$\large p_1(1-p_2)p_3p_4(1-p_5)p_6$$`

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- `\(\large M_h\)`: variation in `\(p\)` among individuals

`$$\large p_i(1-p_i)p_ip_i(1-p_i)p_i$$`

--
- `\(\large M_b\)`: behavioral responses (trap happiness/shyness)

`$$\large p(1-c)cc(1-c)c$$`

.footnote[
[1] David Otis was a post-doc at USU in the Coop Unit when he wrote this seminal paper
]

---
## Capture-mark-recapture

#### Otis et al. 1978 was seminal because it clarified ways that `\(\large p\)` might vary

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However, `\(\large M_0\)`, `\(\large M_t\)`, `\(\large M_h\)`, `\(\large M_b\)` are not single models but instead families of models that allow `\(p\)` to vary for different reasons

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#### In modern Bayesian analysis of closed CMR models, we can use the tools you learned so far in this course to model complex variation in `\(\large p\)`

- group effects

- hierarchical structure

- continuous covariates (via GLM)

---
## The `\(\large M_t\)` model in JAGS

#### How do we model `\(\large p\)` as a function of occassion?

```r
model{
  psi ~ dbeta(1, 1)
  for(t in 1:T){
    p[t] ~ dbeta(1, 1)
  }
  
  for(i in 1:M){
    z[i] ~ dbern(psi)
    
    for(t in 1:T){
      y[i, t] ~ dbern(p[t] * z[i])
    } # end t
  } # end i
  
  N <- sum(z[1:M])
}
```

---
## The `\(\large M_t\)` model in JAGS

```r
model{
  psi ~ dbeta(1, 1)
  for(t in 1:T){
    lp[t] ~ dnorm(mu.p, tau.p)
    logit(p[t]) <- lp[t]
  }
  mu.p ~ dnorm(0, 0.1)
  tau.p ~ dgamma(0.25, 0.25)
  for(i in 1:M){
    z[i] ~ dbern(psi)
    
    for(t in 1:T){
      y[i, t] ~ dbern(p[t] * z[i])
    } # end t
  } # end i
}
```

- What is the difference between this model and the previous? How will this change influence estimates of `\(p_t\)`?

---
## Model `\(\large M_b\)`

#### Behavioral responses are common in studies that require phyisical captures

- probability of capture `\(\large \neq\)` probability of recapture `\((\large c)\)`

- individuals learn to avoid traps (trap shy; `\(\large p > c\)`)

- individuals learn to seek out traps (trap happy; `\(\large c > p\)`)

--
#### In the `\(\large M_b\)` models, `\(\large p\)` depends on whether an individual has previously been captured

- response can be permanent or ephemeral

---
## Model `\(\large M_b\)` in JAGS

```r
model{
  psi ~ dbeta(1, 1)
  p ~ dbeta(1, 1)
  c ~ dbeta(1, 1)
  
  for(i in 1:M){
    z[i] ~ dbern(psi)
    
    y[i, 1] ~ dbern(p * z[i])
    
    for(t in 2:T){
      y[i, t] ~ dbern(z[i] * ((1 - y[i, t-1]) * p + y[i, t-1] * c))
    } # end t
  } # end i
  
  N <- sum(z[1:M])
  trap.response <- c - p
}
```

---
## Model `\(\large M_h\)`

#### Individual heterogeniety not captured by covariates

- treat individuals as random effect

`$$\large logit(p_i) \sim normal(logit(\mu_p), \tau_p)$$`

--
- models with heterogeneity are not always identifiable 
    + different assumptions about `\(p_i\)` will give different estimates of `\(N\)` but model selection does not differentiate between models 
    
--
- however, *not* modeling individual heterogeneity leads to known bias
    + assuming constant `\(p\)` underestimates `\(N\)`

---
## Model `\(\large M_h\)` in JAGS

```r
model{
  psi ~ dbeta(1, 1)
  mean.p ~ dbeta(1, 1)
  mean.lp <- log(mean.p) - log(1 - mean.p)
  tau.p ~ dgamma(0.25, 0.25)
  
  for(i in 1:M){
    z[i] ~ dbern(psi)
    logit(p[i]) ~ dnorm(mean.lp, tau.p)T(-5, 5)
    
    y[i] ~ dbin(p[i] * z[i], T)
  } # end i
  
  N <- sum(z[1:M])
}
```

---
## Extensions of the Otis et al. models

#### Many extensions possible

- `\(\large M_{tb}\)`

- `\(\large M_{th}\)`

- `\(\large M_{tbh}\)`

- and many more limited only by your data and your modeling skills

--
#### As additional variation is added, more data is needed to estimate parameters

- all else equal, each new parameter results in lower precision for `\(\large N\)` (no free lunch!)