n primary units (segments, grid cells) are visited and m secondary
units (points, tracts, establishments) are observed in each, at a cost
c1 * n + c2 * n * m. With s2_between the variance among primary-unit
means and s2_within the variance within primary units, the variance of
the estimated mean per secondary unit is
(1 - n/N) s2_between / n + (1 - m/M) s2_within / (n m) and the
cost-optimal m is sqrt(c1 s2_within / (c2 (s2_between - s2_within / M)))
(Cochran 1977, Section 10.6). The cost c1 of visiting a primary unit is
where the routing enters: it is not a constant but depends on how spread
the sample is; take it from cost_variance_frontier() at the candidate
n and iterate.
Usage
two_stage_allocation(
n_primary,
m_secondary,
s2_between,
s2_within,
c1,
c2,
target_variance = NULL,
target_cv = NULL,
budget = NULL,
mean = NULL
)Arguments
- n_primary, m_secondary
Primary units in the population and secondary units per primary unit (an average when they differ; need not be a whole number). The allocation tries every whole
mup tom_secondaryand keeps the cheapest pair meeting the target (or the least variance within the budget), so small populations are handled exactly.- s2_between, s2_within
Variance among primary-unit means and within primary units.
- c1, c2
Cost per primary unit visited and per secondary unit observed.
- target_variance, target_cv, budget
Exactly one: target variance of the estimated total, target coefficient of variation (needs
mean), or budget.- mean
Population mean per secondary unit, for
target_cv.
Value
A list of class fieldopt_two_stage: n, m, m_optimal,
cost, variance_mean, variance_total, se_total, cv (when mean
is given), bounded.
Examples
two_stage_allocation(
n_primary = 5000, m_secondary = 20, s2_between = 4, s2_within = 25,
c1 = 300, c2 = 20, budget = 1e5
)
#>
#> ── Two-stage allocation ────────────────────────────────────────────────────────
#> n = 185 primary units with m = 12 secondary units each (optimal m 11.68): cost
#> 99900, variance of the total 253300000 (SE 15910).
two_stage_allocation(5000, 20, 4, 25, c1 = 300, c2 = 20, target_cv = 0.03, mean = 12)
#>
#> ── Two-stage allocation ────────────────────────────────────────────────────────
#> n = 36 primary units with m = 13 secondary units each (optimal m 11.68): cost
#> 20160, variance of the total 1.29e+09 (SE 35920, CV 2.99%).