---
title: "Stat 140 Capture/Recapture Lab"
author: "Evan Ray"
date: "October 4, 2017"
output:
  html_document:
    fig_height: 3
    fig_width: 5
  pdf_document:
    fig_height: 2.8
    fig_width: 6
  word_document:
    fig_height: 3
    fig_width: 5
---


```{r include=FALSE}
# Don't delete this chunk if you are using the mosaic package
# This loads the mosaic and dplyr packages
require(mosaic)
require(readr)
require(ggplot2)
require(dplyr)
```

```{r include=FALSE}
# Some customization.  You can alter or delete as desired (if you know what you are doing).

# knitr settings to control how R chunks work.
require(knitr)
opts_chunk$set(
  tidy=FALSE,     # display code as typed
  size="small"    # slightly smaller font for code
)
```

# Introduction

Naturalists often want to estimate the sizes of populations that are difficult to measure directly.  The capture/recapture method (also referred to as mark/recapture) allows one to estimate, for example, the number of fish in a lake.  The idea is also the basis of methods that the Census Bureau has developed that can be used in adjusting population figures obtained in the decennial census (although this is controversial).

## Objectives

There are a few things I'd like you to learn from this lab:

 1. What is the capture/recapture method?

 2. What are some potential limitations of the method, and how do they relate to sampling?

 3. How do methodological limitations of a sampling method affect estimates of quantities in the population we sampled from?

 4. Simulation studies are a valuable tool for studying statistical methods.

**Note**: In this lab, we will use a few R commands you haven't seen before.  I'd like you to try to work through and understand the R code used in this lab, but you won't be specifically responsible for knowing the new commands used in this lab for future work.  The main purpose is to use R as a tool for learning about the statistical ideas having to do with sampling in objective 3 above.

## Outline

We will explore the capture/recapture method in the context of estimating the total number of goldfish in a pond.  Our exploration will have four phases:

1. Simulation of the method using goldfish crackers to make the ideas concrete.

2. Simulation of the method using R.

3. Modified simulations in R to explore how failures in the sampling process could affect the estimates of population size.

4. (If time) Simulation in R to explore how the sample sizes drawn affect the estimates of population size.


# Phase 1: A Physical Simulation

#### 1. Set up

Pour 100 Cheddar goldfish onto a plate.  This represents the population of fish in the pond.  For the rest of this exercise, we're going to pretend that we don't know the total number of fish in the pond and we're going to try to estimate that number.

#### 2. Capture (or Mark)

Draw a random sample of 30 fish from the lake.  These are the ``captured'' fish.  We're going to mark each of these fish with an identifying tag so that we will know we've captured this fish before if we see it again in the next step.  If these were real fish, we would use physical tags, but in this simulation we'll mark the fish by making them change color.  Replace each of the 30 fish you "captured" with a fish of the other flavor.  (The new goldfish replace the old ones -- don't put the old ones back into the pond!)

#### 3. Recapture

Stir the fish around in the pond so that the marked and unmarked fish are well mixed together.  Then draw a new random sample of 30 fish from the lake.  When you do this, close your eyes so that you don't introduce any bias into your sampling -- each fish should have an equal chance of being selected.  Count the number of fish in the new sample that are marked (call this number $m$).

#### 4. Estimate the proportion of fish in the pond that were captured in the Capture step

If the fish were well mixed in between the Capture and Recapture steps, then it's reasonable to estimate the overall proportion of fish in the lake that have been marked by the proportion of fish sampled in the Recapture step that were marked.  Let's call the overall proportion of fish in the pond that were marked $p$.  Since we sampled 30 fish in the Recapture step and $m$ of those were marked, our estimate of that proportion is

$\hat{p} = \frac{m}{30}$

#### 5. Estimate the total number of fish in the pond

Denote the total number of fish by $N$; this is what we want to estimate.  We know that we marked 30 fish, because we captured 30 fish in the Capture step.  Therefore, the true proportion of fish that were marked is $p = \frac{30}{N}$.  Setting this equal to our estimate of $p$ from step 4 and rearranging, we can estimate $N$ by

$\widehat{N} = \frac{30}{\widehat{p}}$

#### 6. Put your estimates of $p$ and $N$ on the board



#### 7. Brainstorm some possible limitations of the Capture/Recapture method outlined here.  Think about the different sampling schemes and common sampling mistakes discussed in Chapter 11.

\vspace{5cm}

\newpage

# Phase 2: Simulation in R

## 2 (a): A single simulation in R

The following R chunks contain code that does steps 1 through 5 in the physical simulation above.  After running the code in each chunk, look at the objects that have been created in the "Global Environment" pane and make sure you understand what happened (even if you wouldn't have been able to write the code to do it).

#### 1. Set up

We first define three variables that will determine the total number of fish in the pond, how many fish we catch and mark in the Capture step, and how many we catch in the Recapture step.  Then we create a data frame called pond_population with a unique id for each fish in the pond (the numbers 1 through N) and a variable called marked, which will be 0 if the fish has not been marked and 1 if the fish has been marked.  To start, all fish are unmarked.

Run the chunk below, and then take a look at the pond_population data frame in your Global Environment if you're not sure what it looks like.

```{r}
N <- 100 # number of fish in the pond
C <- 30 # number of fish to sample in the Capture step
R <- 30 # number of fish to sample in the Recapture step

# Make a data frame to represent the population of all fish in the pond.
pond_population <- data.frame(
  fish_id = seq(from = 1, to = N),
  marked = rep(0, N)
)
```

#### 2. Capture (or Mark)

In the capture step, we capture C fish and mark them.

In R, we simulate capturing the fish by randomly sampling $C$ of the fish ids.
Then, we update the "marked" variable for those fish, setting it equal to 1.

Run the R chunk below, then look at the captured_fish_ids variable (for example, by typing captured_fish_ids into the console) and the updated pond_population data frame in your Global Environment.

```{r}
# Capture Step: capture C fish and mark them
captured_fish_ids <- sample(pond_population$fish_id, size = C)
pond_population$marked[captured_fish_ids] <- 1
```

#### 3. Recapture

In the recapture step, we capture R fish.  Among those $R$ fish, we count the number that were marked.

In R, we again simulate capturing the fish by randomly sampling $R$ of the fish ids.
Then, we count the number of those fish that were captured by adding up the values of the "marked" variable for those captured fish, and save that count in a variable called m.  Remember that in steps 1 and 2 above we set up the marked variable so that it is 0 for unmarked fish and 1 for marked fish.  That means that if I add up these values for the fish in my sample, I get a count of how many fish in the sample were marked.

Run the R chunk below, then look at the recaptured_fish_ids and m.

```{r}
## Recapture step: capture R fish.  Of those, count how many were marked
recaptured_fish_ids <- sample(pond_population$fish_id, size = R)
m <- sum(pond_population$marked[recaptured_fish_ids])
```

#### 4. Estimate the proportion of fish in the pond that were captured in the Capture step

We can use R to calculate the estimated proportion $\hat{p} = \frac{m}{R}$ as follows.  Go ahead and run this chunk.

```{r}
## Estimate proportion of fish in the pond that were marked in the Capture step:
## The number of fish sampled in the Recapture step that were marked divided by
## the total number of fish sampled in the Recapture step.
p_hat <- m / R
```

#### 5. Estimate the total number of fish in the pond

We can use R to calculate the estimated total number of fish in the pond, $\widehat{N} = \frac{C}{\widehat{p}}$, as follows.  Go ahead and run this chunk.

```{r}
## Estimate the total number of fish in the pond:
## The number of fish that were marked in the Capture step divided by the
## estimated proportion of fish that were 
N_hat <- C / p_hat
```

We've now done one simulation!

## 2 (b): Repeating the simulation

It's great that we can do the simulation once, but we really want to do it hundreds or thousands of times and save all of the results.  Running the above chunks thousands of times by hand would be tedious; let's automate this process.

That's what the R chunk below does.  This R chunk puts together all of the code in the previous 5 chunks, with two additions:
 1. The line `simulation_results <- do(1000) * {` says to "do everything between the following curly braces 1000 times", and save the results in a new data frame called simulation_results.  The code between the curly braces does one simulation, so effectively we're repeating the simulation 1000 times.
 2. Near the end of the R chunk, after doing the computations for one simulation, we make a data frame that stores the p_hat and N_hat values obtained in that simulation run.  R will automatically assemble these data frames from the different simulation runs, so the final result (what gets stored in the simulation_results) is a data frame with 1000 rows (one for each simulation run) and two variables (the p_hat and N_hat from each simulation run).

Go ahead and run this R chunk, and then take a look at the simulation_results data frame that gets created.  (If you get an error -- did you run the R chunk on lines 19-24 of this Rmd document?)

```{r}
N <- 100 # number of fish in the pond
C <- 30 # number of fish to sample in the Capture step
R <- 30 # number of fish to sample in the Recapture step

simulation_results <- do(1000) * {
  ## Setup step: make a data frame representing all of the fish in the pond
  pond_population <- data.frame(
    fish_id = seq(from = 1, to = N),
    marked = rep(0, N)
  )

  ## Capture step: capture C fish and add a variable to the data frame
  ## with a 1 if the fish was marked and a 0 if it was not
  captured_fish_ids <- sample(pond_population$fish_id, size = C)
  pond_population$marked[captured_fish_ids] <- 1

  ## Recapture step: capture R fish.  Of those, count how many were marked
  recaptured_fish_ids <- sample(pond_population$fish_id, size = R)
  m <- sum(pond_population$marked[recaptured_fish_ids])
  
  ## Estimate proportion of fish in the pond that were marked in the Capture step:
  ## The number of fish sampled in the Recapture step that were marked divided by
  ## the total number of fish sampled in the Recapture step.
  p_hat <- m / R
  
  ## Estimate the total number of fish in the pond:
  ## The number of fish that were marked in the Capture step divided by the
  ## estimated proportion of fish that were 
  N_hat <- C / p_hat
  
  data.frame(p_hat = p_hat, N_hat = N_hat)
}
```

#### Make a histogram or density plot of p_hat, and a second histogram or density plot of N_hat.  The data frame with these variables is called simulation_results.  Also, calculate the mean and median of each of these values.

SOLUTION:

```{r}
# Your code goes here
ggplot() +
  geom_histogram(aes(x = p_hat), data = simulation_results)

ggplot() +
  geom_histogram(aes(x = N_hat), data = simulation_results)

mean(simulation_results$p_hat)
median(simulation_results$p_hat)

mean(simulation_results$N_hat)
median(simulation_results$N_hat)
```

#### Because we set up the simulation study, we know that the true proportion of fish that were marked (the true p) was $p = 30/100 = 0.3$, and that the true population size was $N = 100$.  What do you observe about the distribution of estimates of the population size?

SOLUTION:

The sampling distribution of population proportion estimates is symmetric and centered at the true value of 0.3.

The sampling distribution of population size estimates is skewed right.  The median value is at the true population size that we were trying to estimate, 100 -- but the mean is too large.

# Phase 3: Exploration of Limitations

So far, everything we've done has assumed that we were equally likely to capture any of the fish in the Capture and Recapture steps, and that we could definitely tell whether or not each fish we caught in the Recapture step had been previously caught and marked in the Capture step.  The estimates of the proportion of fish in the pond that were marked in the Capture step and the total population size that we calculated depend on these assumptions.  What happens if these assumptions are wrong, and some fish are more or less likely to be captured or we can't tell if a fish we see in the Recapture step was previously marked?  Here, we will modify our simulation study in order to explore a couple of ways this could happen and what the effects on our estimates of population size are.

## 3 (a) Trap-happy and Trap-shy (sample volunteers/sample conveniently)

In the wild, it happens that some animals are more likely to be captured ("trap-happy") and other may be less likely to be captured ("trap-shy").  We can represent this in our R code above by adding a `prob` argument when we call the `sample` function.  This argument should be a vector of length $N$ giving the relative weights for how likely each fish is to be captured.  For example, suppose the first 20 fish are ten times as likely to be captured as the remaining 80 fish.  We can use the following lines to sample fish with these weights (after running this chunk, if you're not sure what the sampling_weights vector looks like, take a look at it by typing sampling_weights into the console):

```{r}
sampling_weights <- c(rep(10, 20), rep(1, 80))
sample(pond_population$fish_id, size = C, prob = sampling_weights)
```

The R chunk below contains the original code from section 2 (b) above, to do the simulation study assuming all of the fish are equally likely to be captured.  Modify the calls to the `sample()` function in both the Capture and Recapture steps to use unequal sampling weights for the different fish by setting a `prob = ` argument.

```{r}
N <- 100 # number of fish in the pond
C <- 30 # number of fish to sample in the Capture step
R <- 30 # number of fish to sample in the Recapture step

simulation_results <- do(1000) * {
  ## Setup step: make a data frame representing all of the fish in the pond
  pond_population <- data.frame(
    fish_id = seq(from = 1, to = N),
    marked = rep(0, N)
  )

  ## Capture step: capture C fish and add a variable to the data frame
  ## with a 1 if the fish was marked and a 0 if it was not
  sampling_weights <- c(rep(10, 20), rep(1, 80))
  captured_fish_ids <- sample(pond_population$fish_id, size = C, prob = sampling_weights)
  pond_population$marked[captured_fish_ids] <- 1

  ## Recapture step: capture R fish.  Of those, count how many were marked
  recaptured_fish_ids <- sample(pond_population$fish_id, size = R, prob = sampling_weights)
  m <- sum(pond_population$marked[recaptured_fish_ids])
  
  ## Estimate proportion of fish in the pond that were marked in the Capture step:
  ## The number of fish sampled in the Recapture step that were marked divided by
  ## the total number of fish sampled in the Recapture step.
  p_hat <- m / R
  
  ## Estimate the total number of fish in the pond:
  ## The number of fish that were marked in the Capture step divided by the
  ## estimated proportion of fish that were 
  N_hat <- C / p_hat
  
  data.frame(p_hat = p_hat, N_hat = N_hat)
}
```

#### Make a histogram or density plot of p_hat, and a second histogram or density plot of N_hat.  The data frame with these variables is called simulation_results.  Also, calculate the mean and median of each of these values.

SOLUTION:

```{r}
# Your code goes here
ggplot() +
  geom_histogram(aes(x = p_hat), data = simulation_results)

ggplot() +
  geom_histogram(aes(x = N_hat), data = simulation_results)

mean(simulation_results$p_hat)
median(simulation_results$p_hat)

mean(simulation_results$N_hat)
median(simulation_results$N_hat)
```

#### You will notice that the distribution of population size estimates has changed.  Explain how the changes we made to the simulation study code affect the estimates of the proportion of the population that was marked in the Capture step and why this means that the population size estimates are larger or smaller.

SOLUTION:

Because we are likely to sample the same fish in both the Capture and Recapture phases, a high proportion of the fish we sample in the Recapture step will have been marked.  That means that our estimate of the overall proportion of the population that has been marked is too high.  In turn, that means that our estimate of the population size is too small.

## 3 (b) Marked fish less likely to be Recaptured (sample volunteers/sample conveniently)

It might also be the case that fish that were marked in the Capture step are more or less likely to be caught again in the Recapture step.  For example, maybe the experience of being tagged was traumatic and the fish will strenuously avoid being caught again.  Or maybe you treated the fish you caught really well, and the ones who were captured the first time will try to be captured again.  We can again represent this in the simulation study by modifying the sampling weights.  Now, we'll only need to modify the weights in the Recapture step, for just those fish who were captured in the Capture step.

If the fish that were tagged are one tenth as likely to be recaptured than the fish that were not tagged, we can use the following code to calculate the weights to use in the Recapture step:
```{r}
sampling_weights <- rep(1, 100)
sampling_weights[pond_population$marked == 1] <- 0.1
```

The R chunk below contains the original code from section 2 (b) above, to do the simulation study assuming all of the fish are equally likely to be captured.  Modify the call to the `sample()` function Recapture step to reflect the scenario where the fish that were marked in the Capture step are less likely to be caught in the Recapture step.  You will need to put the calculation of the sampling weights in just before the Recapture step, and modify the call to the `sample()` function in the Recapture step (but not the Capture step).

```{r}
N <- 100 # number of fish in the pond
C <- 30 # number of fish to sample in the Capture step
R <- 30 # number of fish to sample in the Recapture step

simulation_results <- do(1000) * {
  ## Setup step: make a data frame representing all of the fish in the pond
  pond_population <- data.frame(
    fish_id = seq(from = 1, to = N),
    marked = rep(0, N)
  )

  ## Capture step: capture C fish and add a variable to the data frame
  ## with a 1 if the fish was marked and a 0 if it was not
  captured_fish_ids <- sample(pond_population$fish_id, size = C)
  pond_population$marked[captured_fish_ids] <- 1

  ## Recapture step: capture R fish.  Of those, count how many were marked
  sampling_weights <- rep(1, 100)
  sampling_weights[pond_population$marked == 1] <- 0.1
  recaptured_fish_ids <- sample(pond_population$fish_id, size = R, prob = sampling_weights)
  m <- sum(pond_population$marked[recaptured_fish_ids])
  
  ## Estimate proportion of fish in the pond that were marked in the Capture step:
  ## The number of fish sampled in the Recapture step that were marked divided by
  ## the total number of fish sampled in the Recapture step.
  p_hat <- m / R
  
  ## Estimate the total number of fish in the pond:
  ## The number of fish that were marked in the Capture step divided by the
  ## estimated proportion of fish that were 
  N_hat <- C / p_hat
  
  data.frame(p_hat = p_hat, N_hat = N_hat)
}
```

#### Make a histogram or density plot of p_hat, and a second histogram or density plot of N_hat.  The data frame with these variables is called simulation_results.  Also, calculate the mean and median of each of these values.

SOLUTION:

```{r}
# Your code goes here
ggplot() +
  geom_histogram(aes(x = p_hat), data = simulation_results)

ggplot() +
  geom_histogram(aes(x = N_hat), data = simulation_results)

mean(simulation_results$p_hat)
median(simulation_results$p_hat)

mean(simulation_results$N_hat)
median(simulation_results$N_hat)
```


#### You will notice that the distribution of population size estimates has changed.  Explain how the changes we made to the simulation study code affect the estimates of the proportion of the population that was marked in the Capture step and why this means that the population size estimates are larger or smaller.

SOLUTION:

Our population size estimates are now much larger than the true population size.  This is because we are not likely to sample the same fish in the Capture and Recapture phases: the fish we got in the Capture step try to avoid being caught again in the Recapture step.  As a result, a low proportion of the fish we sample in the Recapture step will have been marked.  That means that our estimate of the overall proportion of the population that has been marked is too small.  In turn, that means that our estimate of the population size is too large.

#### Optional: If you'd like, you can also repeat this but where marked fish are 10 times as likely to be caught in the Recapture step as unmarked fish.

```{r}
N <- 100 # number of fish in the pond
C <- 30 # number of fish to sample in the Capture step
R <- 30 # number of fish to sample in the Recapture step

simulation_results <- do(1000) * {
  ## Setup step: make a data frame representing all of the fish in the pond
  pond_population <- data.frame(
    fish_id = seq(from = 1, to = N),
    marked = rep(0, N)
  )

  ## Capture step: capture C fish and add a variable to the data frame
  ## with a 1 if the fish was marked and a 0 if it was not
  captured_fish_ids <- sample(pond_population$fish_id, size = C)
  pond_population$marked[captured_fish_ids] <- 1

  ## Recapture step: capture R fish.  Of those, count how many were marked
  sampling_weights <- rep(1, 100)
  sampling_weights[pond_population$marked == 1] <- 10
  recaptured_fish_ids <- sample(pond_population$fish_id, size = R, prob = sampling_weights)
  m <- sum(pond_population$marked[recaptured_fish_ids])
  
  ## Estimate proportion of fish in the pond that were marked in the Capture step:
  ## The number of fish sampled in the Recapture step that were marked divided by
  ## the total number of fish sampled in the Recapture step.
  p_hat <- m / R
  
  ## Estimate the total number of fish in the pond:
  ## The number of fish that were marked in the Capture step divided by the
  ## estimated proportion of fish that were 
  N_hat <- C / p_hat
  
  data.frame(p_hat = p_hat, N_hat = N_hat)
}
```

```{r}
# Your code goes here
ggplot() +
  geom_histogram(aes(x = p_hat), data = simulation_results)

ggplot() +
  geom_histogram(aes(x = N_hat), data = simulation_results)

mean(simulation_results$p_hat)
median(simulation_results$p_hat)

mean(simulation_results$N_hat)
median(simulation_results$N_hat)
```

Now our population size estimates are much smaller than the true population size.  This is because we are likely to sample the same fish in the Capture and Recapture phases: the fish we got in the Capture step try to be caught again in the Recapture step.  As a result, a high proportion of the fish we sample in the Recapture step will have been marked.  That means that our estimate of the overall proportion of the population that has been marked is high.  In turn, that means that our estimate of the population size is small.

## 3 (c) Tags not securely attached (response bias)

If the tags are not securely attached to the fish, they could fall off between the Capture and Recapture steps.  This would mean that in the Recapture step, we wouldn't know for sure whether or not any fish we catch that don't have tags were caught previously and marked.

In our simulation study, we can represent this by marking fewer than the intended number of fish.  For example, if we intended to mark 30 fish but 1/3 of the tags fell off, we could represent this in the R code by sampling only 20 fish to mark in the Capture step (but still use the number 30 when calculating the estimated population size `N_hat`):
```{r}
captured_fish_ids <- sample(pond_population$fish_id, size = 20)
```

The R chunk below contains the original code from section 2 (b) above, to do the simulation study assuming all of the fish are equally likely to be captured and that the tags don't fall off between the Capture step and the Recapture step.  Modify the calls to the `sample()` function Recapture step to reflect the scenario where the fish that were marked in the Capture step are not all tagged successfully.  As above, let's assume that 1/3 of the tags fall off between the Capture and Recapture steps.  You will need to modify the call to the `sample()` function in the Capture step (but not the Recapture step).

```{r}
N <- 100 # number of fish in the pond
C <- 30 # number of fish to sample in the Capture step
R <- 30 # number of fish to sample in the Recapture step

simulation_results <- do(1000) * {
  ## Setup step: make a data frame representing all of the fish in the pond
  pond_population <- data.frame(
    fish_id = seq(from = 1, to = N),
    marked = rep(0, N)
  )

  ## Capture step: capture C fish and add a variable to the data frame
  ## with a 1 if the fish was marked and a 0 if it was not
  captured_fish_ids <- sample(pond_population$fish_id, size = 20)
  pond_population$marked[captured_fish_ids] <- 1

  ## Recapture step: capture R fish.  Of those, count how many were marked
  recaptured_fish_ids <- sample(pond_population$fish_id, size = R)
  m <- sum(pond_population$marked[recaptured_fish_ids])
  
  ## Estimate proportion of fish in the pond that were marked in the Capture step:
  ## The number of fish sampled in the Recapture step that were marked divided by
  ## the total number of fish sampled in the Recapture step.
  p_hat <- m / R
  
  ## Estimate the total number of fish in the pond:
  ## The number of fish that were marked in the Capture step divided by the
  ## estimated proportion of fish that were 
  N_hat <- C / p_hat
  
  data.frame(p_hat = p_hat, N_hat = N_hat)
}
```

#### Make a histogram or density plot of p_hat, and a second histogram or density plot of N_hat.  The data frame with these variables is called simulation_results.  Also, calculate the mean and median of each of these values.

SOLUTION:

```{r}
# Your code goes here
ggplot() +
  geom_histogram(aes(x = p_hat), data = simulation_results)

ggplot() +
  geom_histogram(aes(x = N_hat), data = simulation_results)

mean(simulation_results$p_hat)
median(simulation_results$p_hat)

mean(simulation_results$N_hat)
median(simulation_results$N_hat)
```


#### You will notice that the distribution of population size estimates has changed.  Explain how the changes we made to the simulation study code affect the estimates of the proportion of the population that was marked in the Capture step and why this means that the population size estimates are larger or smaller.

SOLUTION:

Our population size estimates are now much larger than the true population size.  This is because as far as we can tell, a low proportion of the fish we sample in the Recapture step have been marked (as some of their tags fell off).  That means that our estimate of the overall proportion of the population that has been marked is small.  In turn, that means that our estimate of the population size is large.

# Phase 4: The effect of sample size on estimates of population size.

So far, we have used a sample size of 30 fish in each of the Capture and Recapture steps.  What if the boss tells us that we now only have enough money to catch 10 or 20 fish in each phase.  How much will that impact our estimates of population size?  Can we make an argument that our sample sizes should be even larger, maybe 40 fish?  Repeat the above simulation study, but using different numbers of fish caught to the Capture and Recapture steps:

 * Mark $C = 10$ fish in the Capture step, and catch $R = 10$ fish in the Recapture step
 * Mark $C = 20$ fish in the Capture step, and catch $R = 20$ fish in the Recapture step
 * Mark $C = 40$ fish in the Capture step, and catch $R = 40$ fish in the Recapture step

## 4. (a) $C = 10$, $R = 10$

Modify the code in the R chunk below to mark $C = 10$ fish in the Capture step, and catch $R = 10$ fish in the Recapture step

```{r}
N <- 100 # number of fish in the pond
C <- 10 # number of fish to sample in the Capture step
R <- 10 # number of fish to sample in the Recapture step

simulation_results <- do(1000) * {
  ## Setup step: make a data frame representing all of the fish in the pond
  pond_population <- data.frame(
    fish_id = seq(from = 1, to = N),
    marked = rep(0, N)
  )

  ## Capture step: capture C fish and add a variable to the data frame
  ## with a 1 if the fish was marked and a 0 if it was not
  captured_fish_ids <- sample(pond_population$fish_id, size = C)
  pond_population$marked[captured_fish_ids] <- 1

  ## Recapture step: capture R fish.  Of those, count how many were marked
  recaptured_fish_ids <- sample(pond_population$fish_id, size = R)
  m <- sum(pond_population$marked[recaptured_fish_ids])
  
  ## Estimate proportion of fish in the pond that were marked in the Capture step:
  ## The number of fish sampled in the Recapture step that were marked divided by
  ## the total number of fish sampled in the Recapture step.
  p_hat <- m / R
  
  ## Estimate the total number of fish in the pond:
  ## The number of fish that were marked in the Capture step divided by the
  ## estimated proportion of fish that were 
  N_hat <- C / p_hat
  
  data.frame(p_hat = p_hat, N_hat = N_hat)
}
```

#### Make a histogram or density plot of p_hat, and a second histogram or density plot of N_hat.  The data frame with these variables is called simulation_results.  Also, calculate the mean and median of each of these values.

SOLUTION:

```{r}
# Your code goes here
ggplot() +
  geom_histogram(aes(x = p_hat), data = simulation_results)

ggplot() +
  geom_histogram(aes(x = N_hat), data = simulation_results)

mean(simulation_results$p_hat)
median(simulation_results$p_hat)

mean(simulation_results$N_hat)
median(simulation_results$N_hat)
```

## 4. (b) $C = 20$, $R = 20$

Modify the code in the R chunk below to mark $C = 20$ fish in the Capture step, and catch $R = 20$ fish in the Recapture step

```{r}
N <- 100 # number of fish in the pond
C <- 20 # number of fish to sample in the Capture step
R <- 20 # number of fish to sample in the Recapture step

simulation_results <- do(1000) * {
  ## Setup step: make a data frame representing all of the fish in the pond
  pond_population <- data.frame(
    fish_id = seq(from = 1, to = N),
    marked = rep(0, N)
  )

  ## Capture step: capture C fish and add a variable to the data frame
  ## with a 1 if the fish was marked and a 0 if it was not
  captured_fish_ids <- sample(pond_population$fish_id, size = C)
  pond_population$marked[captured_fish_ids] <- 1

  ## Recapture step: capture R fish.  Of those, count how many were marked
  recaptured_fish_ids <- sample(pond_population$fish_id, size = R)
  m <- sum(pond_population$marked[recaptured_fish_ids])
  
  ## Estimate proportion of fish in the pond that were marked in the Capture step:
  ## The number of fish sampled in the Recapture step that were marked divided by
  ## the total number of fish sampled in the Recapture step.
  p_hat <- m / R
  
  ## Estimate the total number of fish in the pond:
  ## The number of fish that were marked in the Capture step divided by the
  ## estimated proportion of fish that were 
  N_hat <- C / p_hat
  
  data.frame(p_hat = p_hat, N_hat = N_hat)
}
```

#### Make a histogram or density plot of p_hat, and a second histogram or density plot of N_hat.  The data frame with these variables is called simulation_results.  Also, calculate the mean and median of each of these values.

SOLUTION:

```{r}
# Your code goes here
ggplot() +
  geom_histogram(aes(x = p_hat), data = simulation_results)

ggplot() +
  geom_histogram(aes(x = N_hat), data = simulation_results)

mean(simulation_results$p_hat)
median(simulation_results$p_hat)

mean(simulation_results$N_hat)
median(simulation_results$N_hat)
```

## 4. (c) $C = 40$, $R = 40$

Modify the code in the R chunk below to mark $C = 40$ fish in the Capture step, and catch $R = 40$ fish in the Recapture step

```{r}
N <- 100 # number of fish in the pond
C <- 40 # number of fish to sample in the Capture step
R <- 40 # number of fish to sample in the Recapture step

simulation_results <- do(1000) * {
  ## Setup step: make a data frame representing all of the fish in the pond
  pond_population <- data.frame(
    fish_id = seq(from = 1, to = N),
    marked = rep(0, N)
  )

  ## Capture step: capture C fish and add a variable to the data frame
  ## with a 1 if the fish was marked and a 0 if it was not
  captured_fish_ids <- sample(pond_population$fish_id, size = C)
  pond_population$marked[captured_fish_ids] <- 1

  ## Recapture step: capture R fish.  Of those, count how many were marked
  recaptured_fish_ids <- sample(pond_population$fish_id, size = R)
  m <- sum(pond_population$marked[recaptured_fish_ids])
  
  ## Estimate proportion of fish in the pond that were marked in the Capture step:
  ## The number of fish sampled in the Recapture step that were marked divided by
  ## the total number of fish sampled in the Recapture step.
  p_hat <- m / R
  
  ## Estimate the total number of fish in the pond:
  ## The number of fish that were marked in the Capture step divided by the
  ## estimated proportion of fish that were 
  N_hat <- C / p_hat
  
  data.frame(p_hat = p_hat, N_hat = N_hat)
}
```

#### Make a histogram or density plot of p_hat, and a second histogram or density plot of N_hat.  The data frame with these variables is called simulation_results.  Also, calculate the mean and median of each of these values.

SOLUTION:

```{r}
# Your code goes here
ggplot() +
  geom_histogram(aes(x = p_hat), data = simulation_results)

ggplot() +
  geom_histogram(aes(x = N_hat), data = simulation_results)

mean(simulation_results$p_hat)
median(simulation_results$p_hat)

mean(simulation_results$N_hat)
median(simulation_results$N_hat)
```



## 4. (d) Wrapup

Compare the distributions of population size estimates in each scenario and with the scenario of $C = 30$ and $R = 30$ that we started off with.  How much does the sample size affect the sampling distribution of population size estimates?

SOLUTION:

Moving to a sample size of only 10 fish in each phase would be a disaster: about 1/3 of the time, we did not catch any of the fish in the recatch phase that we had initially captured in the capture phase.  In those cases, we get a $\hat{p}$ of 0, with corresponding population size estimates of infinity.  That's not very useful.  Using a sample size of 20 fish in each phase is not quite as bad, but does result in a few simulation runs where the estimated population size is infinity.  Using a larger sample size of 40 is helpful, since it reduces the minimum population size estimate we obtained to about 70, and the maximum population size estimate to about 180.
