class: center, middle, inverse, title-slide # Lecture 13 ## Estimating survival: Open capture-mark-recapture models ### <br/><br/><br/>WILD6900 (Spring 2020) --- ## Readings > ### Kéry & Schaub 171-239 > ### [Powell & Gale chp. 10](https://docs.wixstatic.com/ugd/95e73b_3a230cc8eaa64148ad17a8a36c98240b.pdf) --- ## From closed-population models to open-population models #### All CMR studies have a similar basic design: .pull-left[ During each sampling occasion - individuals are captured - marked or identified - released alive ] .pull-right[ <img src="https://upload.wikimedia.org/wikipedia/commons/6/6f/Band_on_pelican_leg_%285282438747%29.jpg" width="50%" style="display: block; margin: auto;" /> ] -- #### But whats happens next? -- - In the `\(\large M_0\)` model, we assumed the population was closed to any change in `\(\large N\)` during our study + No births, deaths, emigration, or immigration -- + Any `\(\large 0\)` in the capture histories was due to detection error ??? Image courtesy of USFWS Mountain-Prairie, via Wikimedia Commons --- ## From closed-population models to open-population models #### **Open population** models relax this assumption - Individuals can enter (births or immigration) or leave the population (deaths or emigration) between sampling occasions - `\(\large 0\)`'s in the capture histories could be because individuals are there but not detected or because they are not there -- #### Many different forms of open population models - Allow estimation of: + survival + recruitment + movement --- ## Open-population models #### In this lecture, we will focus on estimating survival/emigration -- <img src="figs/cjs1.png" width="75%" style="display: block; margin: auto;" /> --- ## Open-population models #### In this lecture, we will focus on estimating survival/emigration <img src="figs/cjs2.png" width="75%" style="display: block; margin: auto;" /> -- - *Condition on first capture* --- ## Open-population models #### On the occassion after release, 4 possible scenarios: - 1) Individual survives and is re-captured (capture history = `11`) - 2) Individual survives but is not recaptured (capture history = `10`) - 3) Individual dies and is **not available** for recapture (capture history = `10`) - 4) Individual survives but leaves the study area and is **not available** for recapture (capture history = `10`) --- ## Open-population models #### On the occassion after release, 4 possible scenarios: - 1) Individual survives and is re-captured (capture history = `11`) - 2) Individual survives but is not recaptured (capture history = `10`) - 3) Individual dies and is **not available** for recapture (capture history = `10`) - 4) Individual survives but leaves the study area and is **not available** for recapture (capture history = `10`) #### Not possible to distuingish between scenarios 3 & 4 without additional data - `\(\LARGE \phi_t = s_t \times (1 - \epsilon_t)\)` - `\(\LARGE \phi_t\)`: **Apparent survival** (prob. individual survives *and* remains within study area) ??? `\(\LARGE \epsilon_t\)`: probability an individual leaves the study area --- ## Cormack-Jolly-Seber model #### How do we distinguish between scenarios 2 & 3/4? -- #### **CJS model** - Parameters + `\(\LARGE \phi\)`: Apparent survival probability + `\(\Large p\)`: Recapture probability ??? Note that for simplicity, we have removed the `\(t\)` subscript. This implies that apparent survival probability and recapture probability are constant across all occasions. However, the CJS model can estimate different apparent survival and recapture probabilities for each occasion, that is `\(\phi_t\)` and `\(p_t\)` --- ## Cormack-Jolly-Seber model #### How do we distinguish between scenarios 2 & 3/4? <img src="figs/cjs_flowchart.png" width="75%" style="display: block; margin: auto;" /> --- ## Cormack-Jolly-Seber model <table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 10px; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:center;"> Individual </th> <th style="text-align:center;"> Capture history </th> <th style="text-align:center;"> Probability </th> </tr> </thead> <tbody> <tr> <td style="text-align:center;"> Indv 1 </td> <td style="text-align:center;"> 111 </td> <td style="text-align:center;"> \(\phi_1 p_2 \phi_2 p_3\) </td> </tr> <tr> <td style="text-align:center;"> Indv 2 </td> <td style="text-align:center;"> 101 </td> <td style="text-align:center;"> </td> </tr> <tr> <td style="text-align:center;"> Indv 3 </td> <td style="text-align:center;"> 110 </td> <td style="text-align:center;"> </td> </tr> <tr> <td style="text-align:center;"> Indv 4 </td> <td style="text-align:center;"> 100 </td> <td style="text-align:center;"> </td> </tr> </tbody> </table> #### Individual 1 - survived interval 1 `\(\large (\phi_1)\)`, recaptured on occasion 2 `\(\large (p_2)\)`, survived occasion 2 `\(\large (\phi_2)\)`, recapture on occasion 3 `\(\large (p_3)\)` --- ## Cormack-Jolly-Seber model <table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 10px; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:center;"> Individual </th> <th style="text-align:center;"> Capture history </th> <th style="text-align:center;"> Probability </th> </tr> </thead> <tbody> <tr> <td style="text-align:center;"> Indv 1 </td> <td style="text-align:center;"> 111 </td> <td style="text-align:center;"> \(\phi_1 p_2 \phi_2 p_3\) </td> </tr> <tr> <td style="text-align:center;"> Indv 2 </td> <td style="text-align:center;"> 101 </td> <td style="text-align:center;"> \(\phi_1 (1-p_2) \phi_2 p_3\) </td> </tr> <tr> <td style="text-align:center;"> Indv 3 </td> <td style="text-align:center;"> 110 </td> <td style="text-align:center;"> </td> </tr> <tr> <td style="text-align:center;"> Indv 4 </td> <td style="text-align:center;"> 100 </td> <td style="text-align:center;"> </td> </tr> </tbody> </table> #### Individual 2 - survived interval 1 `\(\large (\phi_1)\)`, not recaptured on occasion 2 `\(\large (1-p_2)\)`, survived occasion 2 `\(\large (\phi_2)\)`, recapture on occasion 3 `\(\large (p_3)\)` --- ## Cormack-Jolly-Seber model <table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 10px; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:center;"> Individual </th> <th style="text-align:center;"> Capture history </th> <th style="text-align:center;"> Probability </th> </tr> </thead> <tbody> <tr> <td style="text-align:center;"> Indv 1 </td> <td style="text-align:center;"> 111 </td> <td style="text-align:center;"> \(\phi_1 p_2 \phi_2 p_3\) </td> </tr> <tr> <td style="text-align:center;"> Indv 2 </td> <td style="text-align:center;"> 101 </td> <td style="text-align:center;"> \(\phi_1 (1-p_2) \phi_2 p_3\) </td> </tr> <tr> <td style="text-align:center;"> Indv 3 </td> <td style="text-align:center;"> 110 </td> <td style="text-align:center;"> \(\phi_1 p_2 \phi_2 (1-p_3)+(1-\phi_2)\) </td> </tr> <tr> <td style="text-align:center;"> Indv 4 </td> <td style="text-align:center;"> 100 </td> <td style="text-align:center;"> </td> </tr> </tbody> </table> #### Individual 3 - survived interval 1 `\(\large (\phi_1)\)`, recaptured on occasion 2 `\(\large (p_2)\)` -- + survived occasion 2 `\(\large (\phi_2)\)`, not recaptured on occasion 3 `\(\large (1 - p_3)\)`; **or** -- + died during occasion 2 `\(\large (1-\phi_2)\)` --- ## Cormack-Jolly-Seber model <table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 10px; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:center;"> Individual </th> <th style="text-align:center;"> Capture history </th> <th style="text-align:center;"> Probability </th> </tr> </thead> <tbody> <tr> <td style="text-align:center;"> Indv 1 </td> <td style="text-align:center;"> 111 </td> <td style="text-align:center;"> \(\phi_1 p_2 \phi_2 p_3\) </td> </tr> <tr> <td style="text-align:center;"> Indv 2 </td> <td style="text-align:center;"> 101 </td> <td style="text-align:center;"> \(\phi_1 (1-p_2) \phi_2 p_3\) </td> </tr> <tr> <td style="text-align:center;"> Indv 3 </td> <td style="text-align:center;"> 110 </td> <td style="text-align:center;"> \(\phi_1 p_2 \phi_2 (1-p_3)+(1-\phi_2)\) </td> </tr> <tr> <td style="text-align:center;"> Indv 4 </td> <td style="text-align:center;"> 100 </td> <td style="text-align:center;"> \((1-\phi_1) + \phi_1 (1-p_2) (1-\phi_2 p_3)\) </td> </tr> </tbody> </table> #### Individual 4 - died during interval 1 `\(\large (1-\phi_1)\)`; **or** -- - survived occasion 1 `\(\large (\phi_1)\)`, not recaptured on occasion 2 `\(\large (1 - p_2)\)`, died during occasion 2 `\((1-\phi_2)\)`; **or** -- - survived occasion 1 `\(\large (\phi_1)\)`, not recaptured on occasion 2 `\(\large (1 - p_2)\)`, survived occasion 2 `\(\large (\phi_2)\)`, not recaptured on occasion 3 `\(\large (1 - p_3)\)` --- ## CJS model as a state-space model #### Using the tools we've learned this semester, it's relativley straightfoward to write the CJS model as a state-space model: -- #### Process model, capture occassion 1 `$$\large z_{i,f_1} = 1$$` -- #### Process model, capture occasion 2+ -- `$$\large z_{i,t} \sim Bernoulli(z_{i,t-1}\phi)$$` -- #### Observation model `$$\large y_{i,t} \sim Bernoulli(z_{i,t}p)$$` --- ## CJS model with time-variation #### As for the other models we've seen this semester, it's possible to add temporal variation to the CJS model -- `$$\Large logit(\phi_t) = \mu + \epsilon_t$$` `$$\Large \epsilon_t \sim normal(0, \tau_{\phi})$$` -- `$$\Large logit(p_t) = \mu + \xi_t$$` `$$\Large \xi_t \sim normal(0, \tau_{p})$$` --- ## Identifiability of the CJS model with time-variation #### In the fully time-dependent model, `\(\large \phi_T\)` and `\(\large p_T\)` are not identifiable - the model will return posteriors for both parameters (because each has a prior) but the model will not be able to separately estimate both parameters + posteriors will actually be for `\(\phi_T \times p_T\)` --- ## Identifiability of the CJS model with time-variation #### Why is this? -- - In a CMR study with two occassions (**note** - never do this!): + 100 individuals captured on first occasion + 60 of those individuals recaptured on the second occasion -- - Expected number of recaptures `\(= N \times \phi \times p\)` + `\(60 = 100 \times 0.8 \times 0.75\)` + `\(60 = 100 \times 0.9 \times 0.667\)` + `\(60 = 100 \times 0.6 \times 1.00\)` -- - No unique solution + Separating `\(p\)` from `\(\phi\)` requires *internal* zeros --- ## Identifiability of the CJS model with time-variation #### What can you do: 1) Constant `\(\large p\)` 2) Covariates on `\(\large \phi\)` and `\(\large p\)` 3) Informative priors --- ## Assumptions of the CJS model #### 1) Every animal has the same chance of capture, p #### 2) Every animal has same probability of surviving `\(\large \phi\)` #### 3) Marks are not lost #### 4) Samples are instantaneous (short periods) #### 5) All emigration is permanent (`101` must indicate `\(1-p\)`) #### 6) Fates of animals are independent of other animals