Frames, strata and cost-aware allocation

Agricultural and establishment surveys often combine a list frame (known farms, cheap to reach by phone or on a planned visit) and an area frame (segments of territory, expensive to reach, complete in coverage). Even within one frame, strata differ in the cost of a unit. frame_allocation() is Cochran’s (1977, Section 5.5) allocation with a cost per unit in each stratum, in its two forms.

library(fieldopt)
library(dplyr, warn.conflicts = FALSE)
strata <- tibble(
  stratum = c("list: large", "list: small", "area"),
  size = c(300, 2500, 900),
  sd = c(80, 12, 35),
  cost = c(60, 35, 190)
)
strata
#> # A tibble: 3 × 4
#>   stratum      size    sd  cost
#>   <chr>       <dbl> <dbl> <dbl>
#> 1 list: large   300    80    60
#> 2 list: small  2500    12    35
#> 3 area          900    35   190

Minimum cost for a target variance

Fix the precision wanted for the estimated total and find the cheapest allocation that reaches it. The optimum has n_h proportional to N_h S_h / sqrt(c_h), rounded up and kept between 2 and N_h.

a <- frame_allocation(strata, target_variance = 2.5e7)
a
#> 
#> ── Allocation ──────────────────────────────────────────────────────────────────
#> Minimum cost for target variance 2.5e+07: cost 22640, variance 24720000.
#> # A tibble: 3 × 5
#>   stratum      size    sd  cost     n
#>   <chr>       <dbl> <dbl> <dbl> <int>
#> 1 list: large   300    80    60    88
#> 2 list: small  2500    12    35   143
#> 3 area          900    35   190    65

Minimum variance for a budget

Fix the budget instead and find the allocation with the smallest variance within it. Rounding is repaired so that the cost never exceeds the budget.

b <- frame_allocation(strata, budget = 30000)
b
#> 
#> ── Allocation ──────────────────────────────────────────────────────────────────
#> Minimum variance for budget 30000: cost 29980, variance 17830000.
#> # A tibble: 3 × 5
#>   stratum      size    sd  cost     n
#>   <chr>       <dbl> <dbl> <dbl> <int>
#> 1 list: large   300    80    60   116
#> 2 list: small  2500    12    35   191
#> 3 area          900    35   190    86

Where the field cost comes in

The cost column above is a planning value per unit. For the area frame it should include the travel, and that is what the routing part of the package estimates: run cost_variance_frontier() on the area frame with a cost model, read the mean cost per unit at the sample size under consideration, and use it as the stratum cost. The allocation then reflects how expensive the segments really are to visit, and the loop can be iterated once or twice until the stratum costs and sample sizes agree.

f <- cost_variance_frontier(
  area_frame,
  depot = "depot", cost_model = model,
  n_grid = c(40, 60, 80, 100), y = "crop"
)
area_cost <- f |>
  slice_min(abs(n - a$n[3]), n = 1) |>
  pull(cost_per_unit)
strata |>
  mutate(cost = if_else(stratum == "area", area_cost, cost)) |>
  frame_allocation(target_variance = 2.5e7)

Overlap between the frames (farms present in both) is not modelled here; the allocation treats the frames as strata of a single design.

Several study variables

A survey rarely has one target: crop area and cattle, or production and the number of holdings, each with its own precision. multivariate_allocation() finds the cheapest allocation that meets every target at once (Bethel 1989); the multipliers say which targets bind, and glance() shows the variance attained for each.

multi <- tibble(
  stratum = c("list", "area-high", "area-low"),
  size = c(2000, 300, 500), cost = c(40, 200, 160),
  sd_crop = c(12, 35, 18), sd_cattle = c(30, 10, 25)
)
ma <- multivariate_allocation(
  multi,
  sd = c("sd_crop", "sd_cattle"),
  target_cv = c(0.04, 0.06), totals = c(120000, 90000)
)
ma
#> 
#> ── Multivariate allocation ─────────────────────────────────────────────────────
#> Minimum cost 11200 for 2 targets.
#> sd_crop: target 23040000, attained 22600000 (binding).
#> sd_cattle: target 29160000, attained 28670000 (binding).
#> # A tibble: 3 × 4
#>   stratum    size  cost     n
#>   <chr>     <dbl> <dbl> <int>
#> 1 list       2000    40   169
#> 2 area-high   300   200     7
#> 3 area-low    500   160    19
glance(ma)
#> # A tibble: 2 × 7
#>   variable    target  attained multiplier     n  cost bounded
#>   <chr>        <dbl>     <dbl>      <dbl> <int> <dbl> <lgl>  
#> 1 sd_crop   23040000 22603942.  0.0000616   195 11200 FALSE  
#> 2 sd_cattle 29160000 28668674.  0.000305    195 11200 FALSE

References

Bethel, J. (1989). Sample allocation in multivariate surveys. Survey Methodology, 15, 47–57.

Cochran, W. G. (1977). Sampling Techniques, third edition. Wiley.