---
title: "Interviews to Catch: Statistical Pipeline"
output:
  rmarkdown::html_vignette:
    highlight: null
vignette: >
  %\VignetteIndexEntry{Interviews to Catch: Statistical Pipeline}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

## Introduction

This vignette explains how creel interview data becomes a catch rate estimate,
and how that catch rate is combined with an effort estimate to produce a total
catch estimate with a meaningful measure of uncertainty.

The audience is a creel biologist who knows what catch rate and total catch mean
in practice, but may not be familiar with ratio estimators or the delta method.
Both topics are developed from first principles with plain-language explanations
alongside the formulas.

The pipeline has three stages:

1. Interview data is collected: for each intercepted angler you record trip
   duration (hours fished) and catch.
2. A catch rate (CPUE, catch per angler-hour) is estimated from the sample of
   interviews using a ratio estimator.
3. Total catch is estimated as the product of estimated effort and estimated
   CPUE, with uncertainty propagated through the delta method.

## From Interviews to Catch Rate

Each creel interview captures two key numbers for a completed trip: the
angler's total catch $c_i$ and the total time they spent fishing $h_i$
(their trip duration in hours). Catch per unit effort (CPUE) is catch per
angler-hour.

The challenge is that we observe only a sample of anglers — those who happened
to be at count locations during sampling periods — not the full population.
We need to estimate the population-level CPUE from this sample.

Two estimators are commonly used for this purpose.

### Ratio-of-Means (ROM)

The ratio-of-means estimator divides total sample catch by total sample hours:

$$\hat{R}_{\text{ROM}} = \frac{\sum_{i=1}^{n} c_i}{\sum_{i=1}^{n} h_i}$$

where:

- $c_i$ is the catch of angler $i$
- $h_i$ is the trip duration (hours fished) of angler $i$
- $n$ is the number of interviews

Plain-language interpretation: **total catch in the sample divided by total
hours in the sample.**

ROM naturally down-weights anglers with very short trips. An angler who fishes
for 6 hours contributes six times as much to the denominator as an angler who
fishes for 1 hour, so longer-trip anglers appropriately dominate the estimate
when trip durations are variable.

### Mean-of-Ratios (MOR)

The mean-of-ratios estimator computes each angler's individual catch rate first,
then averages across anglers:

$$\hat{R}_{\text{MOR}} = \frac{1}{n} \sum_{i=1}^{n} \frac{c_i}{h_i}$$

where:

- $c_i / h_i$ is the catch rate for angler $i$
- $n$ is the number of interviews

Plain-language interpretation: **the average of each angler's individual catch
rate.**

MOR treats each angler as an equally weighted observation regardless of how
long they fished. Because each angler's rate is computed before averaging, MOR
can also be applied to incomplete trips (anglers still on the water at
interview time), where you know catch so far and elapsed time even though the
trip has not finished.

### Side-by-Side Comparison

Consider ten anglers interviewed on a single survey day:

| Angler | Catch ($c_i$) | Hours ($h_i$) | Individual CPUE ($c_i / h_i$) |
|--------|---------------|----------------|-------------------------------|
| 1      | 2             | 4              | 0.50                          |
| 2      | 0             | 1              | 0.00                          |
| 3      | 6             | 3              | 2.00                          |
| 4      | 1             | 2              | 0.50                          |
| 5      | 3             | 2              | 1.50                          |
| 6      | 4             | 4              | 1.00                          |
| 7      | 0             | 2              | 0.00                          |
| 8      | 2             | 2              | 1.00                          |
| 9      | 1             | 1              | 1.00                          |
| 10     | 5             | 3              | 1.67                          |

**ROM by hand:**

$$\hat{R}_{\text{ROM}} = \frac{2+0+6+1+3+4+0+2+1+5}{4+1+3+2+2+4+2+2+1+3} = \frac{24}{24} = 1.00 \text{ fish/hr}$$

**MOR by hand:**

$$\hat{R}_{\text{MOR}} = \frac{0.50+0.00+2.00+0.50+1.50+1.00+0.00+1.00+1.00+1.67}{10} = \frac{9.17}{10} = 0.917 \text{ fish/hr}$$

The estimates differ because Anglers 2 and 7 each fished only 1–2 hours with
zero catch. Under ROM, their unproductive hours are diluted across the 24 total
hours in the sample (small influence). Under MOR, each angler's zero rate
receives equal weight to every other angler (larger influence on the average).

We can verify these arithmetic steps in R directly:

```{r comparison}
# Ten-angler example
catch <- c(2, 0, 6, 1, 3, 4, 0, 2, 1, 5)
hours <- c(4, 1, 3, 2, 2, 4, 2, 2, 1, 3)

rom <- sum(catch) / sum(hours)
mor <- mean(catch / hours)

cat("ROM:", round(rom, 3), "fish/hr\n")
cat("MOR:", round(mor, 3), "fish/hr\n")
```

These match the by-hand calculations above.

Now confirm the ROM result with `estimate_catch_rate()` from tidycreel:

```{r rom_package}
library(tidycreel)

mini_calendar <- data.frame(
  date       = as.Date("2024-06-01"),
  day_type   = "weekday",
  open_time  = 6,
  close_time = 20
)

mini_counts <- data.frame(
  date       = as.Date("2024-06-01"),
  day_type   = "weekday",
  count_time = as.POSIXct("2024-06-01 10:00", tz = "UTC"),
  n_anglers  = 10
)

mini_interviews <- data.frame(
  date          = as.Date("2024-06-01"),
  catch_total   = c(2, 0, 6, 1, 3, 4, 0, 2, 1, 5),
  hours_fished  = c(4, 1, 3, 2, 2, 4, 2, 2, 1, 3),
  trip_status   = "complete",
  trip_duration = c(4, 1, 3, 2, 2, 4, 2, 2, 1, 3)
)

mini_design <- creel_design(mini_calendar, date = date, strata = day_type) |>
  add_counts(mini_counts) |>
  add_interviews(mini_interviews,
    catch         = catch_total,
    effort        = hours_fished,
    n_anglers     = 1, # every interview is a single angler
    trip_status   = trip_status,
    trip_duration = trip_duration
  )

# ROM estimate (default estimator for complete trips)
rom_est <- estimate_catch_rate(mini_design)
print(rom_est)
```

The `estimate` value matches the by-hand ROM result of 1.00 fish/hr.

**When to use each estimator:**

- **ROM** (`estimator = "ratio-of-means"`, the default) is appropriate for
  access-point creel surveys using complete trip interviews. Longer trips
  contribute proportionally more to the denominator, which correctly reflects
  the actual fishing-time distribution in the population. ROM is the default in
  tidycreel for instantaneous-count surveys following Pollock et al. (1994).

- **MOR** (`estimator = "mor"`) is appropriate when working with incomplete
  trips — anglers still fishing who are interviewed before their trip ends. Each
  angler's elapsed catch rate is an independent observation of unknown final
  duration, so equal weighting is more defensible. In tidycreel, `estimator =
  "mor"` requires `trip_status` to include incomplete trips and will not run on
  a complete-only dataset.

In practice, ROM and MOR agree when trip durations are homogeneous. They
diverge most when a few anglers have very short or very long trips.

## From Catch Rate to Total Catch

Total catch is the product of estimated effort and estimated catch rate:

$$\hat{TC} = \hat{E} \times \hat{R}$$

where $\hat{E}$ is the estimated total angler-hours for the survey period and
$\hat{R}$ is the estimated catch rate (fish per angler-hour). Both quantities
are estimated with uncertainty, so the uncertainty in their product requires
special treatment.

### Delta Method Variance

The delta method is a general technique for approximating the variance of a
function of random variables by linearizing the function around the estimates.
For a product of two estimated quantities, the idea is: treat each factor as
fixed and ask how much the product would change if that factor varied by its
standard error. The variances from each factor, scaled by the other factor,
add up to give the total variance.

For a product of two independent estimates, the delta method gives:

$$\text{Var}(\hat{TC}) \approx \hat{E}^2 \cdot \text{Var}(\hat{R}) + \hat{R}^2 \cdot \text{Var}(\hat{E}) + 2\hat{E}\hat{R} \cdot \text{Cov}(\hat{E}, \hat{R})$$

where:

- $\hat{E}$ is the estimated effort
- $\hat{R}$ is the estimated catch rate (CPUE)
- $\text{Var}(\hat{R})$ is the variance of the catch rate estimate
- $\text{Var}(\hat{E})$ is the variance of the effort estimate
- $\text{Cov}(\hat{E}, \hat{R})$ is the covariance between effort and catch rate

**Interpreting the three terms:**

1. $\hat{E}^2 \cdot \text{Var}(\hat{R})$ — How much of the total-catch
   uncertainty comes from imprecise catch-rate estimation, treating effort as
   if it were known exactly.

2. $\hat{R}^2 \cdot \text{Var}(\hat{E})$ — How much of the total-catch
   uncertainty comes from imprecise effort estimation, treating catch rate as
   if it were known exactly.

3. $2\hat{E}\hat{R} \cdot \text{Cov}(\hat{E}, \hat{R})$ — A covariance
   adjustment. This term is usually small. tidycreel assumes zero covariance
   between the effort and catch-rate samples because counts and interviews are
   collected through independent sampling processes, so this term vanishes and
   the formula simplifies to the first two terms only.

### Worked Numeric Example

Suppose effort estimation on this survey day produced an estimate of
$\hat{E} = 40$ angler-hours with $\text{SE}(\hat{E}) = 8$ (so
$\text{Var}(\hat{E}) = 64$), and the ROM catch rate estimate from the
ten-angler sample is $\hat{R} = 1.00$ fish/hr with
$\text{SE}(\hat{R}) = 0.30$ (so $\text{Var}(\hat{R}) = 0.09$).

**Term 1** (catch-rate uncertainty):

$$\hat{E}^2 \cdot \text{Var}(\hat{R}) = 40^2 \times 0.09 = 1600 \times 0.09 = 144$$

**Term 2** (effort uncertainty):

$$\hat{R}^2 \cdot \text{Var}(\hat{E}) = 1.00^2 \times 64 = 1 \times 64 = 64$$

**Total variance** (covariance term is zero):

$$\text{Var}(\hat{TC}) \approx 144 + 64 = 208$$

$$\text{SE}(\hat{TC}) = \sqrt{208} \approx 14.4 \text{ fish}$$

**Total catch estimate:** $\hat{TC} = 40 \times 1.00 = 40$ fish, with
SE $\approx$ 14.4 fish (CV $\approx$ 36%).

In this example, Term 1 contributes 144 / 208 = 69% of the total variance, so
catch-rate imprecision (from the small interview sample of ten anglers) is the
dominant source of uncertainty. Collecting more interviews would reduce the CV
more than increasing count precision would.

We can confirm this structure using the package's example datasets, which span
a full survey season and provide enough sampling days for variance estimation:

```{r total_catch}
data(example_calendar)
data(example_counts)
data(example_interviews)

season_design <- creel_design(example_calendar, date = date, strata = day_type) |>
  add_counts(example_counts) |>
  add_interviews(example_interviews,
    catch         = catch_total,
    effort        = hours_fished,
    n_anglers     = n_anglers,
    trip_status   = trip_status,
    trip_duration = trip_duration
  )

total <- estimate_total_catch(season_design)
print(total)
```

The `estimate` column is $\hat{E} \times \hat{R}$. The `se` column is the
delta method standard error combining uncertainty from both the effort estimate
(counts variance) and the catch rate estimate (interviews variance). Dividing
`se` by `estimate` gives the CV, which you can compare to your target precision.

## Summary

Creel interview data flows through the following pipeline to produce a total
catch estimate with a quantified uncertainty:

1. **Interviews to catch rate:** The ratio-of-means (ROM) estimator divides
   total sample catch by total sample hours. It is the default for
   instantaneous-count surveys because it correctly weights anglers by trip
   length. The mean-of-ratios (MOR) estimator treats each angler equally and is
   used in tidycreel when analyzing incomplete trips via `estimator = "mor"`.

2. **Catch rate to total catch:** Total catch is effort multiplied by catch
   rate. The delta method propagates uncertainty from both the effort estimate
   and the catch-rate estimate into the total catch SE. Examining which term
   dominates tells you whether collecting more counts or more interviews would
   most efficiently reduce total catch uncertainty.

By default, tidycreel uses ROM following Pollock et al. (1994) for
instantaneous-count surveys with complete trip interviews.

## See Also

For the full API walkthrough using real survey data, see
[Interview-Based Estimation](interview-estimation.html).

## References

Pollock, K.H., Jones, C.M., and Brown, T.L. (1994). *Angler Survey Methods and
Their Applications in Fisheries Management*. American Fisheries Society Special
Publication 25. Bethesda, Maryland.

Bence, J.R., and Smith, K.D. (1999). An age-structured model with
density-dependent recruitment applied to the Lake Michigan alewife stock.
*Transactions of the American Fisheries Society*, 128(2), 265–284.