Selection methods

The sixteen selection methods samplyr ships, what each one requires, and how registered methods extend the set. draw() chooses among them with its method argument.

Details

Sixteen methods are built in, in three families: equal probability, PPS (probability proportional to size, all requiring mos), and balanced.

Method	Replacement	Size	mos	Other input	Notes
srswor	Without	Fixed	-	-	The default. Standard SRS
srswr	With	Fixed	-	-	Allows duplicates
systematic	Without	Fixed	-	-	Periodic selection
bernoulli	Without	Random	-	prn	Independent trial per unit
pps_systematic	Without	Fixed	Required	-	Order-sensitive, exact first-order probabilities
pps_brewer	Without	Fixed	Required	-	Exact first-order, approximate joint probabilities
pps_cps	Without	Fixed	Required	-	Highest entropy, exact joint probabilities
pps_sampford	Without	Fixed	Required	-	Exact Sampford joint probabilities
pps_poisson	Without	Random	Required	prn	PPS analog of Bernoulli
pps_sps	Without	Fixed	Required	prn	Sequential Poisson with approximate probability targets
pps_pareto	Without	Fixed	Required	prn	Pareto with approximate probability targets
pps_multinomial	With	Fixed	Required	-	Any hit count, Hansen-Hurwitz
pps_chromy	Min. repl.	Fixed	Required	-	As SAS PPS_SEQ
cube	Without	Fixed	Optional	aux optional	Deville & Tillé 2004
lpm2	Without	Fixed	Optional	spread required	Spatial spread
scps	Without	Fixed	Optional	spread required	Spatial spread

Every method takes either n or frac, except pps_cps, which requires n. For fixed-size methods, frac follows the round parameter (ceiling by default). Bernoulli and PPS Poisson use an unrounded expected target. The prn column marks the methods that accept permanent random numbers for coordination. It is always optional.

"Min. repl." is probability minimum replacement: pps_chromy draws a unit either \(\lfloor E \rfloor\) or \(\lceil E \rceil\) times, where \(E\) is its expected number of hits, so a unit is never hit more often than its size warrants.

Selection and inference

All methods above can draw samples when their input requirements are met. Selection support and variance support are separate contracts. This table is a quick comparison. as_svydesign() and as_svrepdesign() describe the supported stage/phase compositions and sample-specific checks.

Method or family	First-order quantity	Joint information	Analysis route
srswor	Exact inclusion probabilities	SRS formulas (not exposed by the joint helper)	SRS with FPC, standard replicates or RWYB
srswr, pps_multinomial	Exact expected hit counts	Exact joint hits for PPS	WR draw occurrences with standard replicates or RWYB
systematic, pps_systematic	Exact inclusion probabilities	Exact order-specific matrix for PPS (zero pairs may occur)	SRS/Brewer or generic replicate approximation under the systematic policy
bernoulli, pps_poisson	Exact independent probabilities	Exact Poisson matrix for PPS	Analytic single-stage Poisson variance, or RWYB for supported clustered/multistage designs
pps_brewer, pps_cps, pps_sampford	Exact inclusion probabilities	Approximate for Brewer, exact for CPS/Sampford	Brewer by default, an explicit joint matrix, or PPS-compatible replicates (including approximate RWYB)
pps_sps, pps_pareto	Approximate targets	High-entropy approximation using targets	Brewer or generic PPS-compatible replicates (no built-in RWYB mapping)
pps_chromy	Exact expected hit counts	Monte Carlo joint hits	WR or generic replicate approximation to PMR (no built-in RWYB mapping)
Unconstrained cube	Exact inclusion probabilities under the method contract	High-entropy approximation	Approximate linearization or generic replicates (no built-in RWYB mapping)
Bounded cube, lpm2, scps	Exact inclusion probabilities under the method contract	Refused	Generic subbootstrap or mrbbootstrap only (linearization refused)
Custom methods	Declared exact or approximate quality	Registered joint support	Depends on the variance-family declaration and adapter checks

Joint information here refers to joint_expectation(), which exposes PPS/balanced-family quantities. Ordinary SRS, WR and independent Bernoulli formulas still apply. A stage's joint matrix is not automatically a matrix for the final units of a multistage design. Exact first-order probabilities do not establish exact variance or confidence-interval coverage.

RWYB means the explicit type = "rwyb" option with svrep. type = "auto" does not select it. PPS WOR remains approximate. Generic replicates do not recreate ordering, balancing, spatial spreading or hard constraints. Poisson sampling is refused by generic replicate methods. See as_svrepdesign() for missing-parent and singleton restrictions, and as_svydesign() for Poisson and two-phase export limits.

SPS, Pareto and custom approximate targets require allow_approximate = TRUE in exante_probabilities() and exante_overlaps(). Estimates using these targets need not be design-unbiased. Unknown probabilities are refused. Probability and variance declarations by a custom-method author are contracts, not proofs.

Fixed vs random sample size

Where the table says Fixed, n is the realized sample size. Where it says Random, n is the expected size: it is converted to frac = n / N (with N the stratum or frame size) and the realized count varies around it.

For pps_poisson, the raw inclusion probabilities are computed as \(\pi_i = f \cdot x_i / \bar{x}\) where \(f\) is frac and \(x_i\) is the MOS value. Any \(\pi_i > 1\) is clipped to 1, so the expected sample size \(E[n] = \sum \min(\pi_i, 1)\) can be less than \(f \cdot N\) when large units dominate the MOS distribution. Use certainty_size or certainty_prop to handle these dominant units explicitly.

Declaring certainty units does more than remove them. The remainder is re-resolved over the reduced target and the reduced MOS total, so the surviving chances can rise. The remaining expected take equals the reduced target only if no remaining probability needs clipping. See draw() for the certainty-adjusted n and frac contract.

This is not silent. execute() warns with class samplyr_warning_poisson_shortfall once per stage when a pool's resolved expectation falls more than 5% below what that pool could reach, naming the pools affected and how many chances were clipped. The comparison is against the reachable target rather than the request: a pool asked for more units than it holds has already had its target reduced by the population, which samplyr_warning_nominal_cap reports, and only the further reduction that saturation caused is charged here. A design reduced both ways gets both warnings.

The shortfall is a gap between the nominal and realized design, not a bias. Horvitz-Thompson estimates from a saturated Poisson design remain unbiased, because the weights are the reciprocals of the resolved probabilities.

When an allocation method is set in stratify_by() (equal, proportional, neyman, optimal, power), specify total sample size via n. Combining alloc with frac is not supported.

References

srswor, srswr, systematic, bernoulli, pps_systematic, pps_multinomial: Cochran, W.G. (1977). Sampling Techniques, 3rd ed. Wiley.

pps_brewer: Brewer, K.R.W. (1975). A simple procedure for sampling PPS WOR. Australian Journal of Statistics, 17(3), 166-172.

pps_cps: Hájek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a finite population. Annals of Mathematical Statistics, 35(4), 1491-1523.

Chen, X.-H., Dempster, A.P. and Liu, J.S. (1994). Weighted finite population sampling to maximize entropy. Biometrika, 81(3), 457-469.

pps_sampford: Sampford, M.R. (1967). On sampling without replacement with unequal probabilities of selection. Biometrika, 54(3/4), 499-513.

pps_poisson: Tillé, Y. (2006). Sampling Algorithms. Springer.

pps_sps: Ohlsson, E. (1998). Sequential Poisson sampling. Journal of Official Statistics, 14(2), 149-162.

pps_pareto: Rosén, B. (1997). Asymptotic theory for order sampling. Journal of Statistical Planning and Inference, 62(2), 135-158.

pps_chromy: Chromy, J.R. (1979). Sequential sample selection methods. Proceedings of the Survey Research Methods Section, ASA, 401-406.

balanced: Deville, J.-C. and Tillé, Y. (2004). Efficient balanced sampling: the cube method. Biometrika, 91(4), 893-912.

Chauvet, G. (2009). Stratified balanced sampling. Survey Methodology, 35(1), 115-119.

See also

draw() to set a method on a stage, joint_expectation() for which methods yield exact second-order quantities, as_svydesign() for how each family is exported to survey

Other design specification: add_stage(), cluster_by(), draw(), sampling_design(), stratify_by()