Calculate inclusion probabilities

Calculate stratified (first-order) inclusion probabilities.

Usage

inclusion_prob(x, n, strata = gl(1, length(x)), alpha = 0.001, cutoff = Inf)

becomes_ta(x, alpha = 0.001, cutoff = Inf)

Arguments

x

A positive and finite numeric vector of sizes for units in the population (e.g., revenue for drawing a sample of businesses).

n

A positive integer vector giving the sample size for each stratum, ordered according to the levels of strata. A single value is recycled for all strata. Non-integers are truncated towards 0.

strata

A factor, or something that can be coerced into one, giving the strata associated with units in the population. The default is to place all units into a single stratum.

alpha

A numeric vector with values between 0 and 1 for each stratum, ordered according to the levels of strata. Units with inclusion probabilities greater than or equal to 1 - alpha are set to 1 for each stratum. A single value is recycled for all strata. The default is slightly larger than 0.

cutoff

A positive numeric vector of cutoffs for each stratum, ordered according to the levels of strata. Units with x >= cutoff get an inclusion probability of 1 for each stratum. A single value is recycled for all strata. The default does not apply a cutoff.

Value

inclusion_prob() returns a numeric vector of inclusion probabilities for each unit in the population.

becomes_ta() returns an integer vector giving the sample size at which a unit enters the take-all stratum.

Details

Within a stratum, the inclusion probability for a unit is given by \(\pi = nx / \sum x\). These values can be greater than 1 in practice, and so they are constructed iteratively by taking units with \(\pi \geq 1 - \alpha\) (from largest to smallest) and assigning these units an inclusion probability of 1, with the remaining inclusion probabilities recalculated at each step. If \(\alpha > 0\), then any ties among units with the same size are broken by their position.

The becomes_ta() function reverses this operations and finds the critical sample size at which a unit enters the take-all stratum. This value is undefined for units that are always included in the sample (because their size exceeds cutoff) or never included.

See also

sps() for drawing a sequential Poisson sample.

Examples

# Make inclusion probabilities for a population with units
# of different size
x <- c(1:10, 100)
(pi <- inclusion_prob(x, 5))
#>  [1] 0.07272727 0.14545455 0.21818182 0.29090909 0.36363636 0.43636364
#>  [7] 0.50909091 0.58181818 0.65454545 0.72727273 1.00000000

# The last unit is sufficiently large to be included in all
# samples with two or more units
becomes_ta(x)
#>  [1] 11 11 10 10  9  9  8  8  7  7  2

# Use the inclusion probabilities to calculate the variance of the
# sample size for Poisson sampling
sum(pi * (1 - pi))
#> [1] 1.963636