Kish design effect from unequal weighting

Computes Kish's design effect due to unequal weighting, \(deff = 1 + CV^2(w) = m \sum w^2 / (\sum w)^2\), and the effective sample size \(n_\mathrm{eff} = m / deff\) it implies. It is the standard one-number summary of what a weighting cascade cost in precision, and it is what the summary() and plot() methods of a prepped recipe report step by step.

Usage

design_effect(w)

Arguments

w

vector of weights (zeros are dropped; negative weights are kept active, but see the note above on the design effect).

Value

list with deff, n_eff, cv and n.

Details

Zero weights are dropped (they are the "dropped-unit" marker); negative weights – a valid but unusual output of unbounded linear/GREG calibration – are kept active, so the count n matches collect_weights(). Be aware, however, that the Kish formula assumes non-negative weights: a negative weight shrinks \(\sum w\) and enlarges \(\sum w^2\) at once, so with negatives present deff is inflated and no longer interpretable as an effective-sample summary. prep() raises an alert when a calibration produces negative weights; prefer bounds to keep the factor positive if you need the design effect to be meaningful.

Examples

design_effect(sample_survey$pw)
#> $deff
#> [1] 1.055596
#> 
#> $n_eff
#> [1] 442.4043
#> 
#> $cv
#> [1] 0.2357872
#> 
#> $n
#> [1] 467
#>