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`	Also takes
`srswor`	Without	Fixed	-	-
`srswr`	With	Fixed	-	-
`systematic`	Without	Fixed	-	-
`bernoulli`	Without	Random	-	`prn`
`pps_systematic`	Without	Fixed	Required	-
`pps_brewer`	Without	Fixed	Required	-
`pps_cps`	Without	Fixed	Required	-
`pps_sampford`	Without	Fixed	Required	-
`pps_poisson`	Without	Random	Required	`prn`
`pps_sps`	Without	Fixed	Required	`prn`
`pps_pareto`	Without	Fixed	Required	`prn`
`pps_multinomial`	With	Fixed	Required	-
`pps_chromy`	Minimum	Fixed	Required	-
`cube`	Without	Fixed	Optional	`aux`
`lpm2`	Without	Fixed	Optional	`spread`
`scps`	Without	Fixed	Optional	`spread`, `prn`

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. prn in the last column marks the methods that accept permanent random numbers for coordination, and it is always optional. cube takes aux optionally, whereas lpm2 and scps require spread.

What each method does

srswor, the default, is simple random sampling.
srswr allows a unit to be drawn more than once.
systematic selects at a fixed interval.
bernoulli makes an independent trial for each unit.
pps_systematic is order-sensitive, with exact first-order probabilities. Sorted with control, it spreads the sample over the sorted variables.
pps_brewer is Brewer's general method for any n, with exact first-order and approximate joint probabilities. PPS_BREWER in SAS and SPSS selects two units per stratum.
pps_cps has the highest entropy and exact joint probabilities.
pps_sampford has exact Sampford joint probabilities. It is the PPS_SAMPFORD method of SAS and SPSS.
pps_poisson is the PPS analog of Bernoulli sampling.
pps_sps (sequential Poisson) and pps_pareto meet their probability targets only approximately.
pps_multinomial allows any number of hits, with the Hansen-Hurwitz estimator.
pps_chromy is sequential and order-sensitive, the method of SAS PPS_SEQ.
cube is the cube method of Deville and Tillé (2004).
lpm2 and scps spread the sample in space.

"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.

Choosing a PPS method

When a protocol names the method, use that method.

Otherwise a general fixed-size method is the usual choice. pps_brewer, pps_cps and pps_sampford select any number of units, ignore the frame order and give every pair of units a positive joint probability, so the variance can be estimated from the design. The joint probabilities are exact for CPS and Sampford and approximate for Brewer.

pps_systematic and pps_chromy depend on the frame order. Sorted with control, they spread the sample over the sorted variables, which is why many household survey protocols use them. Systematic PPS gives some pairs of units joint probability zero, so no design-unbiased variance estimator exists for it, and a frame whose order resonates with the sampling interval can make its standard errors far too small. as_svydesign() uses Brewer's approximation for a pps_systematic stage and warns, and a pps_chromy stage is treated as drawn with replacement (variance_estimation).

pps_poisson, pps_sps and pps_pareto take permanent random numbers, for samples coordinated across waves or surveys.

Spreading the sample

A protocol that sorts the frame by geography wants the sample spread over the areas it sorts by. samplyr reaches that spread in four ways, and each pays for it differently.

control with pps_systematic or pps_chromy gives implicit stratification, with the variance limits given above.
Finer strata in stratify_by(), down to two units per stratum, with a general method. Every joint probability stays positive, at the cost of more strata and smaller takes in each.
method = "cube" with aux = c(bound(province)) fixes the count in each province to an integer adjacent to its expected count.
method = "lpm2" or "scps" with spread spreads the sample in space.

The last two keep exact first-order probabilities, but their joint probabilities are refused and their variance comes from subbootstrap or mrbbootstrap replicates only (variance_estimation). No method dominates the others, and the choice depends on the variance the analysis needs.

Selection and inference

Every method above draws a sample when its inputs are met, but selection support and variance support are separate contracts. variance_estimation gives how each method's variance is estimated and how exact it is.

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.

The two order-sampling methods part ways in small pools. Drawing 3 of 12 units with sizes from 1 to 20, over 20,000 draws, pps_sps realized inclusion probabilities up to 0.047 away from its targets, and a Horvitz-Thompson count built on the targets came out 6.6% low. pps_pareto stayed within 0.009 of its targets, with a bias under 1%. For coordinated samples from small pools, prefer pps_pareto.

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.

lpm2: Grafström, A., Lundström, N.L.P. and Schelin, L. (2012). Spatially balanced sampling through the pivotal method. Biometrics, 68(2), 514-520. doi:10.1111/j.1541-0420.2011.01699.x

scps: Grafström, A. (2012). Spatially correlated Poisson sampling. Journal of Statistical Planning and Inference, 142(1), 139-147. doi:10.1016/j.jspi.2011.07.003

Grafström, A. and Matei, A. (2018). Coordination of spatially balanced samples. Survey Methodology, 44(2), 215-238.

Matei, A., Smith, P.A., Smeets, M.J.E. and Klingwort, J. (2023). Targetted double control of burden in multiple surveys. Survey Methodology, 49(2), 363-384.