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Sample allocation

Source: R/allocate.R
allocate.Rd

Compute the proportional, power, Neyman, and optimal sample allocations.

Usage

allocate(
  allocation,
  N.h,
  n.samp = NULL,
  S.h = NULL,
  c.h = NULL,
  cost = NULL,
  variance = NULL,
  power = NULL,
  lbound = 2
)

Arguments

allocation

type of allocation, must be one of "proportional", "power", "neyman", or "optimal".

N.h

vector of population stratum sizes (\(N_h\), all positive values), for example c(150, 600, 250).

required for all allocation types.

n.samp

total sample size to be allocated (positive integer of length 1).

required for the following allocation types: proportional, power, and Neyman, and NULL otherwise.

S.h

vector of stratum unit standard deviations (positive values same length as N.h) (\(S_h\)).

required for the following allocation types: Neyman, and optimal, and NULL otherwise.

c.h

vector of cost per unit in stratum h (positive values same length as N.h) (\(c_h\)).

required for the optimal allocation only, and NULL otherwise.

cost

total variable cost (positive value) \((C – c_0)\).

required for the cost-constrained optimal allocation only, and NULL otherwise.

variance

fixed variance target for estimated mean (positive value) (\(V_0\)).

required for the precision-constrained optimal allocation only, and NULL otherwise.

power

power value for power allocation (\(0 \le \alpha \le 1\)).

required for the power allocation only, and NULL otherwise.

lbound

minimum stratum-level (positive integer of length 1). Default value is 2.

Value

Integer vector same length of N.h final allocation

Method

The allocate function allocates a sample size n on H strata using one of the following allocation methods:

  1. Proportional allocation [n.samp, N.h, allocation = "proportional"] $$n_h = n \times \frac{N_h}{\sum\limits_{h=1}^H N_h}$$ where
    \(n\): total sample size to be allocated (function input is n.samp), and
    \(N_h\): population size of stratum h (function input is N.h).

  2. Power allocation [n.samp, N.h, power, allocation = "power"] $$n_h = n \times \frac{N_h^\alpha}{\sum\limits_{h=1}^H N_h^\alpha}$$ where
    \(\alpha\): a power value to control over-under-sampling with \(0 \le \alpha \le 1\) (function input is power).

  3. Neyman allocation [n.samp, N.h, S.h, allocation = "neyman"] $$n_h = n \times \frac{N_h S_h}{\sum\limits_{h=1}^H N_h S_h}$$ where
    \(S_h\): standard deviation of stratum h (function input is S.h).

  4. Optimal allocation

    • cost-constrained [N.h, S.h, c.h, cost, allocation = "optimal"] $$n_h = (C−c_0) \times \frac{N_h S_h / \sqrt{c_h}}{\sum\limits_{h=1}^H N_h S_h \sqrt{c_h}}$$ where
      \(c_h\): cost per unit in stratum h (function input is c.h), and
      \((C – c_0)\): total variable cost (function input is cost)

    • precision-constrained [N.h, S.h, c.h, variance, allocation = "optimal"] $$n_h = N_h S_h / \sqrt{c_h} \times \frac{\sum\limits_{h=1}^H N_h S_h \sqrt{c_h}}{V_0 \left(\sum\limits_{h=1}^H N_h \right)^2 + \sum\limits_{h=1}^H N_h S_h^2}$$ where
      \(V_0\): fixed variance target for estimated mean (function input is variance)

The table below presents the relevant inputs for each type; when irrelevant inputs are entered, an error message will be displayed.

allocationN.hn.sampS.hc.hcostvariancelboundpower
proportional✓✓✓
power✓✓✓✓
neyman✓✓✓✓✓
optimal: cost-constrained✓✓✓✓✓
optimal: precision-constrained✓✓✓✓✓

Examples

# The first step is getting a frame summary
#  Summarize the IPEDS dataset by OBEREG
# - N: number of universities per region
# - SD_ENRTOT: standard deviation of total enrollment per region
# - Filter out rows with missing ENRTOT to ensure accurate variance estimates

ipeds_summary <- ipeds |>
  tidytable::filter(!is.na(ENRTOT)) |>
  tidytable::group_by(OBEREG) |>
  tidytable::summarize(
    N = tidytable::n(),
    SD_ENRTOT = stats::sd(ENRTOT)
  ) |>
  tidytable::ungroup()

# Example of proportional allocation
ipeds_summary |>
  tidytable::mutate(
    n = allocate("proportional", N.h = N, n.samp = 500)
  )
#> Sample allocation of 500 using proportional with the relevant inputs:
#>   N.h = 7, 299, 971, 851, 468, 1467, 633, 216, 870, 132
#> 
#> Output:
#> 2, 25, 82, 72, 40, 124, 53, 18, 73, 11
#> # A tidytable: 10 × 4
#>    OBEREG                                                      N SD_ENRTOT     n
#>    <fct>                                                   <int>     <dbl> <int>
#>  1 U.S. Service schools                                        7     1680.     2
#>  2 New England (CT, ME, MA, NH, RI, VT)                      299    11800.    25
#>  3 Mid East (DE, DC, MD, NJ, NY, PA)                         971     5956.    82
#>  4 Great Lakes (IL, IN, MI, OH, WI)                          851     7537.    72
#>  5 Plains (IA, KS, MN, MO, NE, ND, SD)                       468     5830.    40
#>  6 Southeast (AL, AR, FL, GA, KY, LA, MS, NC, SC, TN, VA,…  1467     7293.   124
#>  7 Southwest (AZ, NM, OK, TX)                                633    11149.    53
#>  8 Rocky Mountains (CO, ID, MT, UT, WY)                      216    14784.    18
#>  9 Far West (AK, CA, HI, NV, OR, WA)                         870     7641.    73
#> 10 Other U.S. jurisdictions (AS, FM, GU, MH, MP, PR, PW, …   132     2981.    11

# Example of power allocation
ipeds_summary |>
  tidytable::mutate(
    n = allocate("power", N.h = N, power = 0.5, n.samp = 500)
  )
#> Sample allocation of 500 using power with the relevant inputs:
#>   N.h = 7, 299, 971, 851, 468, 1467, 633, 216, 870, 132
#>   power = 0.5
#> 
#> Output:
#> 6, 39, 70, 66, 49, 87, 57, 33, 67, 26
#> # A tidytable: 10 × 4
#>    OBEREG                                                      N SD_ENRTOT     n
#>    <fct>                                                   <int>     <dbl> <int>
#>  1 U.S. Service schools                                        7     1680.     6
#>  2 New England (CT, ME, MA, NH, RI, VT)                      299    11800.    39
#>  3 Mid East (DE, DC, MD, NJ, NY, PA)                         971     5956.    70
#>  4 Great Lakes (IL, IN, MI, OH, WI)                          851     7537.    66
#>  5 Plains (IA, KS, MN, MO, NE, ND, SD)                       468     5830.    49
#>  6 Southeast (AL, AR, FL, GA, KY, LA, MS, NC, SC, TN, VA,…  1467     7293.    87
#>  7 Southwest (AZ, NM, OK, TX)                                633    11149.    57
#>  8 Rocky Mountains (CO, ID, MT, UT, WY)                      216    14784.    33
#>  9 Far West (AK, CA, HI, NV, OR, WA)                         870     7641.    67
#> 10 Other U.S. jurisdictions (AS, FM, GU, MH, MP, PR, PW, …   132     2981.    26

# Example of Neyman allocation
ipeds_summary |>
  tidytable::mutate(
    n = allocate("neyman", N.h = N, n.samp = 500, S.h = SD_ENRTOT)
  )
#> Sample allocation of 500 using neyman with the relevant inputs:
#>   N.h = 7, 299, 971, 851, 468, 1467, 633, 216, 870, 132
#>   S.h = 1680.11385668608, 11800.2993217881, 5956.31998862919, 7536.59885863143, 5830.00927558341, 7293.44943657165, 11149.0070418081, 14783.8369003426, 7641.36426053871, 2981.44889034106
#> 
#> Output:
#> 2, 38, 62, 69, 29, 115, 76, 34, 71, 4
#> # A tidytable: 10 × 4
#>    OBEREG                                                      N SD_ENRTOT     n
#>    <fct>                                                   <int>     <dbl> <int>
#>  1 U.S. Service schools                                        7     1680.     2
#>  2 New England (CT, ME, MA, NH, RI, VT)                      299    11800.    38
#>  3 Mid East (DE, DC, MD, NJ, NY, PA)                         971     5956.    62
#>  4 Great Lakes (IL, IN, MI, OH, WI)                          851     7537.    69
#>  5 Plains (IA, KS, MN, MO, NE, ND, SD)                       468     5830.    29
#>  6 Southeast (AL, AR, FL, GA, KY, LA, MS, NC, SC, TN, VA,…  1467     7293.   115
#>  7 Southwest (AZ, NM, OK, TX)                                633    11149.    76
#>  8 Rocky Mountains (CO, ID, MT, UT, WY)                      216    14784.    34
#>  9 Far West (AK, CA, HI, NV, OR, WA)                         870     7641.    71
#> 10 Other U.S. jurisdictions (AS, FM, GU, MH, MP, PR, PW, …   132     2981.     4

# Example of Neyman allocation with a lower bound of 5
ipeds_summary |>
  tidytable::mutate(
    n = allocate("neyman", N.h = N, n.samp = 500, S.h = SD_ENRTOT, lbound = 5)
  )
#> Sample allocation of 500 using neyman with the relevant inputs:
#>   N.h = 7, 299, 971, 851, 468, 1467, 633, 216, 870, 132
#>   S.h = 1680.11385668608, 11800.2993217881, 5956.31998862919, 7536.59885863143, 5830.00927558341, 7293.44943657165, 11149.0070418081, 14783.8369003426, 7641.36426053871, 2981.44889034106
#> 
#> Output:
#> 5, 38, 62, 68, 29, 113, 75, 34, 71, 5
#> # A tidytable: 10 × 4
#>    OBEREG                                                      N SD_ENRTOT     n
#>    <fct>                                                   <int>     <dbl> <int>
#>  1 U.S. Service schools                                        7     1680.     5
#>  2 New England (CT, ME, MA, NH, RI, VT)                      299    11800.    38
#>  3 Mid East (DE, DC, MD, NJ, NY, PA)                         971     5956.    62
#>  4 Great Lakes (IL, IN, MI, OH, WI)                          851     7537.    68
#>  5 Plains (IA, KS, MN, MO, NE, ND, SD)                       468     5830.    29
#>  6 Southeast (AL, AR, FL, GA, KY, LA, MS, NC, SC, TN, VA,…  1467     7293.   113
#>  7 Southwest (AZ, NM, OK, TX)                                633    11149.    75
#>  8 Rocky Mountains (CO, ID, MT, UT, WY)                      216    14784.    34
#>  9 Far West (AK, CA, HI, NV, OR, WA)                         870     7641.    71
#> 10 Other U.S. jurisdictions (AS, FM, GU, MH, MP, PR, PW, …   132     2981.     5