make_ppswor_approx_matrix function

Create a quadratic form's matrix to represent a variance estimator for PPSWOR designs, based on commonly-used approximations

Create a quadratic form's matrix to represent a variance estimator for PPSWOR designs, based on commonly-used approximations

Several variance estimators for designs that use unequal probability sampling without replacement (i.e., PPSWOR), variance estimation tends to be more accurate when using an approximation estimator that uses the first-order inclusion probabilities (i.e., the basic sampling weights) and ignores the joint inclusion probabilities. This function returns the matrix of the quadratic form used to represent such variance estimators.

make_ppswor_approx_matrix(probs, method = "Deville-1")

Arguments

  • probs: A vector of first-order inclusion probabilities

  • method: A string specifying the approximation method to use. See the "Details" section below. Options include:

    • "Deville-1"
    • "Deville-2"
    • "Beaumont-Emond"

Returns

A symmetric matrix whose dimension matches the length of probs.

Deville's Estimators

The "Deville-1" and "Deville-2" approximations have been shown to be effective for designs that use a fixed sample size with a high-entropy sampling method. This includes most PPSWOR sampling methods, but unequal-probability systematic sampling is an important exception.

Deville's variance estimators generally take the following form:

v^(Y^)=∑i=1nci(y˘i−1∑i=knck∑k=1ncky˘k)2 \hat{v}(\hat{Y}) = \sum_{i=1}^{n} c_i (\breve{y}_i - \frac{1}{\sum_{i=k}^{n}c_k}\sum_{k=1}^{n}c_k \breve{y}_k)^2

where y˘i=yi/πi\breve{y}_i = y_i/\pi_i is the weighted value of the the variable of interest, and cic_i are constants that depend on the approximation method used.

The matrix of the quadratic form, denoted Σ\Sigma, has its ijij-th entry defined as follows:

σii=ci(1−ci∑k=1nck) when i=jσij=−cicj∑k=1nck when i≠j \sigma_{ii} = c_i (1 - \frac{c_i}{\sum_{k=1}^{n}c_k}) \textit{ when } i = j \\\sigma_{ij}=\frac{-c_i c_j}{\sum_{k=1}^{n}c_k} \textit{ when } i \neq j \\

When πi=1\pi_{i} = 1 for every unit, then σij=0\sigma_{ij}=0 for all i,ji,j. If there is only one sampling unit, then σ11=0\sigma_{11}=0; that is, the unit is treated as if it was sampled with certainty.

The constants cic_i are defined for each approximation method as follows, with the names taken directly from Matei and Tillé (2005).

  • "Deville-1" :
ci=(1−πi)nn−1 c_i=\left(1-\pi_i\right) \frac{n}{n-1}
  • "Deville-2" :
ci=(1−πi)[1−∑k=1n(1−πk∑k=1n(1−πk))2]−1 c_i = (1-\pi_i) \left[1 - \sum_{k=1}^{n} \left(\frac{1-\pi_k}{\sum_{k=1}^{n}(1-\pi_k)}\right)^2 \right]^{-1}

Both of the approximations "Deville-1" and "Deville-2" were shown in the simulation studies of Matei and Tillé (2005) to perform much better in terms of MSE compared to the strictly-unbiased Horvitz-Thompson and Yates-Grundy variance estimators. In the case of simple random sampling without replacement (SRSWOR), these estimators are identical to the usual Horvitz-Thompson variance estimator.

Beaumont-Emond Estimator

Beaumont and Emond (2022) proposed a variance estimator for unequal probability sampling without replacement. This estimator is simply the Horvitz-Thompson variance estimator with the following approximation for the joint inclusion probabilities.

πkl≈πkπln−1(n−1)+(1−πk)(1−πl) \pi_{kl} \approx \pi_k \pi_l \frac{n - 1}{(n-1) + \sqrt{(1-\pi_k)(1-\pi_l)}}

In the case of cluster sampling, this approximation should be applied to the clusters rather than the units within clusters.

References

Matei, Alina, and Yves Tillé. 2005. “Evaluation of Variance Approximations and Estimators in Maximum Entropy Sampling with Unequal Probability and Fixed Sample Size.” Journal of Official Statistics 21(4):543–70.

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  • Maintainer: Ben Schneider
  • License: GPL (>= 3)
  • Last published: 2025-02-09

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