Estimate total harvest from a mark-recapture population estimate

Computes a total harvest estimate and its uncertainty using the delta method, given a closed-population angler population estimate from estimate_angler_n and a known harvest rate.

The point estimate is \(\hat{H} = \hat{N} \times r\) where \(r\) is the harvest rate in fish per angler. The delta-method standard error is \(SE(\hat{H}) = r \times SE(\hat{N})\), propagating only the uncertainty in \(\hat{N}\) (harvest-rate uncertainty is not propagated in this release).

Usage

estimate_mr_harvest(
  angler_n,
  harvest_rate,
  harvest_rate_se = NULL,
  conf_level = 0.95,
  ci_method = c("logit", "delta", "bootstrap")
)

Arguments

angler_n

A creel_estimates object returned by estimate_angler_n.

harvest_rate

numeric scalar. Harvest per angler, in fish per angler, over the same period angler_n counts anglers for. Must be \(> 0\). This is a rate, not a proportion, and is not bounded above: a fishery averaging 1.4 fish per angler is a legal value. In the notation of Hansen & Van Kirk (2018) eq. (1), \(H = N \cdot D \cdot V\), this argument is the product \(D \times V\) — mean days fished per angler times mean daily harvest per angler — not \(V\) alone. Supply harvest_rate_se to propagate its uncertainty.

harvest_rate_se

numeric scalar or NULL. The standard error of harvest_rate, on the same fish-per-angler scale. When supplied, the total variance is Goodman's (1960) product form, \(\hat{N}^2 \sigma_r^2 + r^2 \sigma_{\hat{N}}^2 - \sigma_{\hat{N}}^2 \sigma_r^2\), via the same helper the three estimate_total_*() functions already use.

When NULL (default), the reported SE reflects uncertainty in \(\hat{N}\) alone and is a lower bound; the function says so at runtime. The component is absent rather than zero, because a zero would be indistinguishable from having propagated the rate's error and found none. Rasmussen et al. (1998) draw the distinction explicitly: the subtractive product formula is the one for terms "estimated from a sample", and differs from the population formula "used when the terms in the product are known, not estimated".

Supplying it also changes the confidence interval. The default logit interval scales the endpoints of the \(\hat{N}\) interval by harvest_rate, which is exact only while that rate is a known positive constant. Once the rate is estimated those endpoints are themselves random, so the function falls back to a symmetric interval built from the full product SE.

conf_level

numeric. Confidence level for the CI. Default 0.95.

ci_method

character(1). CI construction method: "delta" (default) uses the analytic delta-method formula; "bootstrap" propagates the bootstrap samples stored in attr(angler_n, "boot_samples") (produced by calling estimate_angler_n(..., ci_method = "bootstrap") first).

Value

A creel_estimates S3 object with method = "mark-recapture-harvest" and an estimates tibble with columns: parameter, estimate, se, ci_lower, ci_upper.

Details

The harvest rate is treated as a known constant. This is a simplification made by this implementation, not by the cited method: Hansen & Van Kirk (2018) estimate both factors of the rate, give each a log-normal sampling distribution, and resample them alongside \(\hat{N}\) in the bootstrap that produces their harvest CIs. Holding the rate fixed therefore makes the reported se a lower bound on the true uncertainty. Propagation of harvest-rate uncertainty via a two-source delta method is a planned future extension.

The delta-method interval is also symmetric, which for a mark-recapture estimate is optimistic at the lower end and can place ci_lower below zero when recaptures are few; see the same note under estimate_angler_n.

References

Hansen, J. M., & Van Kirk, R. W. (2018). A mark-recapture-based approach for estimating angler harvest. North American Journal of Fisheries Management, 38(2), 400–410. doi:10.1002/nafm.10038

Examples

# Step 1: estimate angler population
result <- estimate_angler_n(M = 200L, n = 50L, m = 10L)

# Step 2: compute total harvest
harvest <- estimate_mr_harvest(angler_n = result, harvest_rate = 0.35)
#> ℹ `harvest_rate_se` was not supplied, so the reported SE excludes harvest-rate
#>   uncertainty.
#>   It reflects uncertainty in the abundance estimate alone and is a lower bound.
#>   Supply `harvest_rate_se` to propagate the rate's own error (Goodman 1960).
print(harvest)
#> 
#> ── Creel Survey Estimates ──────────────────────────────────────────────────────
#> Method: mark-recapture-harvest
#> Variance: delta
#> Confidence level: 95%
#> 
#> # A tibble: 1 × 5
#>   parameter     estimate    se ci_lower ci_upper
#>   <chr>            <dbl> <dbl>    <dbl>    <dbl>
#> 1 total_harvest     326.  81.1     212.     600.