Weighted sampling

Draws rows with probability governed by a weights column. The weights are used as supplied: prob= normalises by their sum, so rescaling them changes nothing and the design does not offer it.

Usage

design_weighted(
  weights,
  n,
  replace = FALSE,
  method = c("successive", "systematic", "poisson"),
  na_rm = FALSE
)

Arguments

weights: Column of positive, finite weights.
n: Number of rows to draw. Under method = "poisson" this is the expected number, not a guarantee.
replace: Sample with replacement? Only available for method = "successive".
method: One of "successive", "systematic" or "poisson". See "Choosing a method".
na_rm: Drop rows whose weight is NA instead of raising an error.

Value

A design object, for use with draw().

Choosing a method

The three methods differ in what the weights actually control, which decides whether the sample can be used for estimation.

"successive": The default, and what base::sample() does. The weights govern each sequential draw, not the probability that a row ends up in the sample, and the realised inclusion probabilities are not proportional to the weights. How far off depends on the weights: with a single dominant unit the two nearly coincide (about 1% apart), but with a moderate spread the gap is large — weights 1:10 at n = 5 give inclusion probabilities up to 35% away from proportional. Treating them as if they were proportional biases a Horvitz-Thompson total by around 1% at n = 15 out of 30 in simulation, small in size but unmistakable in sign. There is no closed form for pi, so inclusion_prob() refuses to give one. Fine when you want a weighted selection; wrong as the basis for an estimate.
"systematic": Randomised systematic probability-proportional-to- size (Madow 1949; Hartley and Rao 1962). Shuffles the rows, then walks the cumulative weights with a fixed step from a random start, giving pi_i = n * p_i exactly. Rows heavy enough that n * p_i > 1 are taken with certainty and the rest rescaled, repeatedly, until every probability is valid. Fixed sample size. The shuffle means every pair of rows can appear together, but their joint probabilities have no closed form, so ht_total() uses Deville's approximation for the variance, which needs only the first-order probabilities.
"poisson": Each row is included independently with probability pi_i = n * p_i, capped at 1. Inclusion probabilities are exactly proportional to size and every pair can co-occur, at the cost of a random sample size averaging n.

replace = TRUE is with-replacement PPS and applies only to method = "successive".

References

Madow, W. G. (1949). On the theory of systematic sampling, II. Annals of Mathematical Statistics, 20, 333–354.

Hartley, H. O. and Rao, J. N. K. (1962). Sampling with unequal probabilities and without replacement. Annals of Mathematical Statistics, 33, 350–374.

Brewer, K. R. W. and Hanif, M. (1983). Sampling with Unequal Probabilities. Springer.

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

Examples

df <- data.frame(id = 1:20, w = 1:20)
draw(df, design_weighted("w", n = 5), seed = 1)
#>   id  w
#> 1 18 18
#> 2 16 16
#> 3 12 12
#> 4  6  6
#> 5 19 19

# Inclusion probabilities proportional to size, and checkable
d <- design_weighted("w", n = 5, method = "systematic")
round(inclusion_prob(df, d), 3)
#>  [1] 0.024 0.048 0.071 0.095 0.119 0.143 0.167 0.190 0.214 0.238 0.262 0.286
#> [13] 0.310 0.333 0.357 0.381 0.405 0.429 0.452 0.476

# Which is what makes an unbiased total possible
s <- draw(df, d, seed = 1, weights = TRUE)
sum(s$w * s$.weight)   # estimates sum(df$w) = 210
#> [1] 210