Calculate sampling days required under Neyman-optimal allocation

Determines how many sampling days are needed to achieve a target coefficient of variation on the effort estimate, then allocates those days across strata using Neyman (optimal) allocation. Unlike creel_n_effort(), which distributes days proportionally to stratum size, this function concentrates days in strata with higher between-day variance, minimising total days for a given precision.

Usage

optimal_n(cv_target, N_h, ybar_h, s2_h, cost_ratio = 1)

Arguments

cv_target: Numeric scalar. Target coefficient of variation for the effort estimate (e.g., 0.20 for 20 percent). Must be in (0, 1].
N_h: Named numeric vector. Total available days per stratum (e.g., c(weekday = 65, weekend = 28)). Values must be >= 1.
ybar_h: Numeric vector of same length as N_h. Pilot mean effort per day per stratum (e.g., angler-hours per day). Values must be >= 0.
s2_h: Numeric vector of same length as N_h. Pilot variance of effort per day per stratum. Values must be >= 0.
cost_ratio: Numeric scalar or named numeric vector of same length as N_h. Relative cost of sampling one day in each stratum. Default 1 (equal costs). When costs differ across strata, days are preferentially allocated to cheaper, high-variance strata. Values must be > 0.

Value

A named integer vector. Elements named after strata in N_h give the optimal sampling days per stratum; element "total" gives Cochran's overall sample size before allocation, and "allocated" the sum of the per-stratum values actually returned. Budget against "allocated"; see creel_n_effort() for why the two differ.

Details

Total sample size uses the cost-generalised Cochran (1977) formula (eq. 5.25 / 5.34 with finite-population correction):

$$n = \left\lceil \frac{A \cdot C}{V_0 + \sum_h N_h s_h^2} \right\rceil$$

where $A = \sum_h N_h s_h / \sqrt{c_h}$, $C = \sum_h N_h s_h \sqrt{c_h}$, $V_0 = (CV_{target} \cdot \hat{E})^2$, $\hat{E} = \sum_h N_h \bar{y}_h$, and $s_h = \sqrt{s_h^2}$. When all $c_h = 1$ (equal costs) this reduces to $(\sum_h N_h s_h)^2 / (V_0 + \sum_h N_h s_h^2)$, which gives the same n_total as creel_n_effort() (per-stratum allocation differs: Neyman uses $n_h \propto N_h s_h$ vs proportional $n_h \propto N_h$).

Per-stratum allocation uses the cost-adjusted Neyman formula (Cochran 1977 eq. 5.30):

$$n_h = \left\lceil n \cdot \frac{N_h s_h / \sqrt{c_h}}{\sum_h N_h s_h / \sqrt{c_h}} \right\rceil$$

where $c_h$ is the relative sampling cost for stratum $h$. With equal costs (cost_ratio = 1), this reduces to the standard Neyman formula $n_h \propto N_h s_h$.

Because each stratum is ceiling-ed independently, sum(n_h) may slightly exceed n_total.

References

Cochran, W.G. 1977. Sampling Techniques, 3rd ed. Wiley, New York.

McCormick, J.L. and Quist, M.C. 2017. Sample size estimation for on-site creel surveys. North American Journal of Fisheries Management 37:970-983. doi:10.1080/02755947.2017.1342723

Examples

# Two-stratum weekday/weekend example
optimal_n(
  cv_target = 0.20,
  N_h   = c(weekday = 65, weekend = 28),
  ybar_h = c(50, 60),
  s2_h   = c(400, 500)
)
#>   weekday   weekend     total allocated 
#>         3         2         4         5 

# Weekend sampling costs twice as much -- shift days toward weekdays
optimal_n(
  cv_target  = 0.20,
  N_h        = c(weekday = 65, weekend = 28),
  ybar_h     = c(50, 60),
  s2_h       = c(400, 500),
  cost_ratio = c(weekday = 1, weekend = 2)
)
#>   weekday   weekend     total allocated 
#>         3         2         4         5