inclusion_prob: Calculate inclusion probabilities

Description

Calculate stratified (first-order) inclusion probabilities.

Usage

inclusion_prob(x, n, strata = NULL, alpha = 0.001, cutoff = Inf)
becomes_ta(x, alpha = 0.001, cutoff = Inf)

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.

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.

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. See vignette("take-all") for details. 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.

Examples

Run this code

# Make inclusion probabilities for a population with units
# of different size
x <- c(1:10, 100)
(pi <- inclusion_prob(x, 5))

# The last unit is sufficiently large to be included in all
# samples with two or more units
becomes_ta(x)

# Determine the number of take-all units before drawing a sample
n_ta <- function(x, n, ...) {
  sum(becomes_ta(x, ...) <= n, na.rm = TRUE)
}

n_ta(x, 7)

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