Two-phase sample allocation

Allocate a two-phase (double) sample: a large cheap phase 1, then a subsample measured on the expensive variable. Minimizes cost for a target precision, or precision for a fixed budget.

Usage

n_twophase(frame, ...)

# Default S3 method
n_twophase(
  frame,
  ...,
  phase1_cost,
  cv = NULL,
  budget = NULL,
  between = NULL,
  mu = NULL,
  N = Inf,
  n_phase1 = NULL,
  assurance = NULL,
  phase1_deff = 1,
  resp_rate = 1,
  single_deff = 1,
  single_resp_rate = 1,
  single_cost = NULL,
  fixed_cost = 0,
  plan = NULL
)

# S3 method for class 'svyplan_prec'
n_twophase(frame, ..., cv = NULL, budget = NULL)

Arguments

frame

For the default method: a data frame with one row per phase-2 stratum, in the n_alloc() column vocabulary (N, sd, mean, unit_cost, take_all) but on a narrower contract: sd is required where n_alloc() also accepts var, and the cost and p/mean rules differ, so a table built for one may need adjusting before it is passed to the other. Columns:

N: Stratum size (required). Only relative size matters, since the stratum weights are N / sum(N).
sd: Within-stratum standard deviation of the expensive variable (required).
mean: Stratum mean. Supply it to let the between-stratum component be derived, which is what makes stratifying worthwhile. Omit it, or pass between = 0, when the strata are not expected to differ in level, as in nonresponse follow-up.
unit_cost: Incremental phase-2 cost per unit. Default 1. Zero means the stratum is already measured, which with take_all = TRUE is exactly a phase-1 respondent group.
deff: Design effect of the phase-2 subsample in this stratum, default 1. This one applies to the residual variation that phase 2 has to measure, under the phase-2 field design. See Details for why it is not the same design effect as phase1_deff.
resp_rate: Expected phase-2 completion rate in this stratum, in (0, 1], default 1. It divides the stratum's contribution to the variance, so a stratum that responds poorly is treated as carrying less information per issued unit.
take_all: Logical. TRUE carries every unit phase 1 successfully classified into phase 2. On the issued-unit scale nu is measured on, that is nu = resp_rate, the phase-1 classification rate, and not 1. Among classified units the fraction is one. The two coincide only when phase 1 classifies everyone. Default FALSE.
stratum: Optional label.

For svyplan_prec objects: a precision result from prec_twophase().

...

Additional arguments passed to methods. Unused arguments are rejected.

phase1_cost

Cost per phase-1 unit (> 0). This is paid on every unit screened, whether or not it reaches phase 2.

cv

Target coefficient of variation. Specify exactly one of cv or budget.

budget

Total budget. Specify exactly one of cv or budget.

between

Between-stratum variance component. NULL (default) derives it from the mean column as sum(W * (mean - ybar)^2), or uses 0 when no mean is given. Supply it directly when you know the stratification gain but not the stratum means.

mu

Population mean, used to turn the variance into a coefficient of variation. NULL (default) derives it from mean when present. Required in cv mode when no mean column is supplied.

N

Population size for the phase-1 finite population correction. Inf (default) applies none.

n_phase1

Fixed phase-1 sample size, or NULL (default) to optimize it. Supply it when the screener has already run, when phase 1 is an existing survey or panel, or when its size is set by field capacity rather than by this design. The relative allocation across strata is unchanged, since it is S_h sqrt(d_2h / c_h) in every mode. What changes is the overall scale, which is then pinned by the budget left after paying for phase 1, or by what it takes to reach cv at that size. Because the optimizing choice of n_phase1 is the best member of this family, fixing it can only match or lose to leaving it free, and the gap is what a phase-1 size you did not choose is costing. Two ways it can fail. The budget may not reach phase 2 at all, once the screening and any take_all strata are paid for. And a target cv may sit below the floor that remains when phase 2 carries every classified unit through, which is the between-stratum component d_1 A / r_1 plus the phase-2 residual at nu_h = resp_rate. No amount of subsampling can beat it, because both parts are already paid for.

assurance

Probability in (0, 1), or NULL (default). Planning at the expected respondent count leaves roughly half of all designs short. Supplying a level reports, alongside the expected design, the issued sizes for which the required respondents arrive with at least that probability, from the binomial distribution of respondents.

The level is marginal, holding stratum by stratum. The chance that every stratum clears its target at once is the product over strata and so is lower, materially so with many strata: three strata at 0.95 give about 0.86 together. Raise the level if you need a familywise guarantee. It is also conditional on the phase-1 pool: the assured phase-2 issue is compared against what phase 1 supplies, and a stratum that needs more than its pool is reported in a warning, because the answer there is to enlarge phase 1 rather than to over-issue. Phase-1 composition is itself random unless phase 1 is a census, so the expected stratum shares behind that pool are not simultaneous lower bounds either.

phase1_deff

Design effect of the phase-1 sample, default 1. It multiplies the between-stratum component, which is the part of the variance phase 1 is responsible for. Set it above 1 when phase 1 is clustered, as an area screener is.

resp_rate

Expected phase-1 response, screening or successful classification rate, in (0, 1], default 1. It divides the between-stratum component and, because phase 2 can only draw from the units phase 1 actually classified, it also caps every subsampling fraction at resp_rate. In a nonresponse follow-up frame leave it at 1: there the strata are response status, so classification succeeds for every unit and setting it again would count the same loss twice.

single_deff

Design effect of the single-phase comparator, default 1. It is a separate number from the stratum deff column because the comparator need not be fielded the same way as phase 2.

single_resp_rate

Expected response rate of the single-phase comparator, in (0, 1], default 1.

single_cost

Cost per unit of the single-phase design that skips phase 1 and measures the expensive variable directly. NULL (default) uses sum(share * unit_cost), which is the right baseline when unit_cost is what measuring one unit costs. In a nonresponse follow-up design it is not: a unit_cost of 0 there means "already measured", not "free", and the right baseline is the cost of one completed interview without any follow-up, phase1_cost / resp_rate. Supply it in that case.

fixed_cost

Fixed overhead, not depending on sample size. Default 0. In budget mode only budget - fixed_cost is allocatable.

plan

Optional svyplan() object providing design defaults.

Value

A svyplan_twophase object with components:

n: Named numeric vector c(n_phase1 = , n_phase2 = ), both continuous, counting units issued. n_phase2 is the expected issue, sum(W_h * nu_h) * n_phase1.
responding: The same two quantities in expected respondents, r_1 n_a and sum(r_{2h} W_h nu_h) n_a. Issued and responding are different quantities and neither is an effective sample size. There is no single effective size for a two-phase design, because the two variance components carry different design effects.
cv: Coefficient of variation achieved.
cost: Total cost, including fixed_cost.
detail: Per-stratum table: stratum, N, share, sd, unit_cost, deff, resp_rate, nu (the subsampling fraction, as a share of the phase-1 units issued, and with resp_rate below 1 it is capped there, since phase 2 can only draw from the units phase 1 classified, and the fraction taken among those classified units is nu / resp_rate), n_issued, n_resp (its expected responding part), and take_all, which is TRUE for strata that were supplied pinned and for those the allocator truncated at the classification rate.
single_phase: The design that skips phase 1 and measures the expensive variable directly, as c(n = , cv = , cost = ), plus reaches_target, and better, TRUE when that plainer design wins. Two-phase sampling is not always an improvement and this comparison is the check that says so. In cv mode a comparator that cannot reach the target from a frame of size N is capped there, reports the coefficient of variation a census of it would achieve, and never wins on cost alone.
operational: The whole-unit field design: n (both phases), the per-stratum n_int, and the cost and cv it actually achieves. Budget mode floors and then buys back whole units in order of variance reduction per unit cost, so the integer design stays inside the budget, while cv mode rounds up. Both are bounded by the phase-2 pool, so a stratum whose expected phase-1 yield rounds below one unit is left unsampled with a warning, and the operational cv is then Inf. With assurance set it also carries assured, assured_phase1, pool and assured_cost.
params: Validated inputs, read back by prec_twophase(). They include the design decisions a stratum table cannot express, so that n_twophase(prec_twophase(fit)) rebuilds this problem rather than a new unconstrained one: the take_all set as supplied, a fixed n_phase1, assurance, and the comparator settings. There is no predict() method for this class.

Details

One allocator covers the two designs that usually get separate treatments. Double sampling for stratification subsamples every phase-2 stratum. Nonresponse follow-up carries the phase-1 respondents through untouched and subsamples only the nonrespondents. Both are the same problem with different strata marked take_all.

Write $W_h$ for the stratum weight, $S_h$ for the within-stratum standard deviation, $c_h$ for the incremental phase-2 unit cost, $c_a$ for the phase-1 unit cost, and $\nu_h$ for the fraction of the phase-1 stratum carried into phase 2. Ignoring the finite population correction,

$$V = \frac{1}{n_a}\left[A + \sum_h \frac{W_h S_h^2}{\nu_h}\right], \qquad C = n_a\left[c_a + \sum_h c_h W_h \nu_h\right],$$

where $A$ is the between-stratum component. Minimizing $VC$, which is free of $n_a$, gives

$$\nu_h = \frac{S_h}{\sqrt{c_h}} \sqrt{\frac{\tilde c_a}{\tilde A}}, \qquad \tilde A = A + \sum_{h \in P} W_h S_h^2, \qquad \tilde c_a = c_a + \sum_{h \in P} c_h W_h,$$

where $P$ is the set of pinned strata. The shape is Neyman-like, $\nu_h \propto S_h/\sqrt{c_h}$, but the overall scale is set by the between-stratum variance: weak stratification pushes every $\nu_h$ up, toward keeping everything phase 1 found.

Any $\nu_h$ above the cap is truncated there and the stratum joins $P$, which changes $\tilde A$ and $\tilde c_a$ and so the remaining strata are re-solved. The cap is resp_rate, not 1: phase 2 can only subsample the units phase 1 succeeded in classifying, so a stratum is "take-all" once it keeps all of those, not all of $N_h$. Strata are pinned one at a time in decreasing $S_h/\sqrt{c_h}$, because pinning lowers the multiplier and can bring others back below the cap.

The finite population correction

With N set, the correction applies component by component and not to the variance as a whole:

$$V = A\left(\frac{1}{n_a}-\frac{1}{N}\right) + \sum_h W_h S_h^2 \left\{\frac{1}{\nu_h n_a}-\frac{1}{N}\right\}.$$

Phase 1 estimates the stratum weights, so a phase-1 census makes the between-stratum term vanish. The phase-2 terms vanish only when phase 2 also measures every unit it found. Scaling the whole variance by $1 - n_a/N$ would subtract $1/(\nu_h N)$ in place of $1/N$ and so report a phase-1 census as a zero-variance design. Note that with resp_rate below 1 a phase-1 census is not a complete classification of the frame, and the residual it leaves is real rather than an artifact.

Two design effects, not one

The variance has two components and each carries its own design effect,

$$V \approx \frac{1}{n_a}\left[d_1 A + \sum_h \frac{d_{2h} W_h S_h^2}{\nu_h}\right],$$

so the optimum becomes $\nu_h \propto S_h\sqrt{d_{2h}/c_h}$ and $\tilde A$ is built from $d_1 A$ and the pinned strata's $d_{2h} W_h S_h^2$. Inflating the combined variance by a single design effect instead is a different and wrong model.

The two are design effects for different variables. phase1_deff applies to what phase 1 explains, the between-stratum contrast. The deff column applies to what is left for phase 2 to measure, the within-stratum residual. A clustered phase 1 raises the first. A stratifier that absorbs geographic variation lowers the second.

They are not independent, and the trap runs in the direction that looks attractive. A stratifier built purely from between-cluster structure shrinks the residual and so lowers deff, but it drives the fitted values toward constancy within clusters and so raises phase1_deff toward the cluster size. Under a clustered phase 1 the first term can then dominate and the design loses to a plain single-phase sample even though the residual design effect looks favorable. When the stratification explains little, the phase-2 residual is close to the whole variance and the stratum deff should be close to single_deff. Setting it well below with a weak stratifier assumes away the cost of the design.

Response rates

All costs and sample sizes count units issued, so a cost quoted per completed interview has to be converted before it is passed in: with a contact cost and a completion cost, the issued-basis figure is c_contact + resp_rate * c_complete.

Response enters each component separately, d_1 becoming d_1/r_1 and d_{2h} becoming d_{2h}/r_{2h}, so a phase-2 stratum that responds poorly is subsampled differently rather than the whole design being inflated by one factor. A response rate common to everything scales the sample needed for a precision target but cancels from the relative allocation. Phase- or stratum-specific rates change it.

This is an expected-information calculation and not a bias adjustment. Dividing by a response rate assumes response is ignorable within the strata you supplied, which is a substantive assumption about the strata, not a property of the arithmetic. No sample size removes nonresponse bias. Where that assumption is uncomfortable, modeling the nonresponse as a follow-up phase, as below, is the alternative the design itself offers.

Nonresponse follow-up is the two-stratum case: respondents with unit_cost = 0 and take_all = TRUE, nonrespondents free, and no between-stratum component. The optimum reduces to $\nu = \sqrt{c_1/(c_2\theta)}$ for a phase-1 response rate $\theta$, which is the standard result.

References

Fuller, W. A. (2009). Sampling Statistics, Sect. 3.3. Wiley.

Neyman, J. (1938). Contribution to the theory of sampling human populations. Journal of the American Statistical Association, 33(201), 101–116.

Saerndal, C.-E., Swensson, B., and Wretman, J. (1992). Model Assisted Survey Sampling, Sect. 15.4. Springer.

Valliant, R., Dever, J. A., and Kreuter, F. (2018). Practical Tools for Designing and Weighting Survey Samples, 2nd edition, Sect. 17.5.2. Springer.

Examples

# Double sampling for stratification: a cheap screener splits the frame
# into four groups, the expensive measurement goes to a subsample.
frame <- data.frame(
  stratum   = c("A", "B", "C", "D"),
  N         = c(3500, 2500, 2500, 1500),
  sd        = c(12, 25, 8, 40),
  mean      = c(40, 70, 35, 90),
  unit_cost = c(2, 5, 1, 9)
)
n_twophase(frame, phase1_cost = 1, budget = 50000)
#> Two-phase allocation (4 phase-2 strata)
#> field design: n_phase1 = 16924 | n_phase2 = 8094
#> cv = 0.0050, cost = 50000
#> 
#>  stratum share    sd unit_cost     nu n_int
#>        A 0.350 12.00      2.00 0.4154  2462
#>        B 0.250 25.00      5.00 0.5474  2316
#>        C 0.250  8.00      1.00 0.3917  1659
#>        D 0.150 40.00      9.00 0.6528  1657
#> 
#> single-phase is better here: n = 14085 at cv 0.0046, so skip phase 1
#> # summary() for the continuous optimum and the comparator

# Weak stratification pushes the fractions up, and strata that reach 1
# are reported as take-all.
flat <- transform(frame, mean = c(55, 56, 55, 56))
n_twophase(flat, phase1_cost = 1, budget = 50000)
#> Two-phase allocation (4 phase-2 strata)
#> field design: n_phase1 = 11476 | n_phase2 = 10030
#> cv = 0.0037, cost = 50000
#> 
#>  stratum share    sd unit_cost     nu n_int take_all
#>        A 0.350 12.00      2.00 0.8085  3250         
#>        B 0.250 25.00      5.00 1.0000  2869        *
#>        C 0.250  8.00      1.00 0.7623  2190         
#>        D 0.150 40.00      9.00 1.0000  1721        *
#> 
#> single-phase is better here: n = 14085 at cv 0.0033, so skip phase 1
#> # summary() for the continuous optimum and the comparator

# Nonresponse follow-up: respondents are already measured, so they cost
# nothing more and are all kept. Only the nonrespondents are subsampled.
theta <- 0.5
nrfu <- data.frame(
  stratum   = c("respondents", "nonrespondents"),
  N         = c(theta, 1 - theta),
  sd        = c(1, 1),
  unit_cost = c(0, 200),
  take_all  = c(TRUE, FALSE)
)
# a unit_cost of 0 means "already measured", not "free", so the
# single-phase baseline has to be named: one completed interview
n_twophase(nrfu, phase1_cost = 50, budget = 100000, mu = 1,
           single_cost = 50 / theta)
#> Two-phase allocation (2 phase-2 strata)
#> field design: n_phase1 = 828 | n_phase2 = 707
#> cv = 0.0382, cost = 1e+05
#> 
#>         stratum share   sd unit_cost     nu n_int take_all
#>     respondents 0.500 1.00      0.00 1.0000   414        *
#>  nonrespondents 0.500 1.00    200.00 0.7071   293         
#> 
#> single-phase is better here: n = 1000 at cv 0.0316, so skip phase 1
#> # summary() for the continuous optimum and the comparator