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On the implementation of stratified two-stage simple random sampling without replacement, with possible collapsed

Philippe Aubry1

  • 1OFB - Office français de la biodiversité - Direction surveillance, évaluation, données - Unité données et appui méthodologique, Saint Benoist, BP 20, F-78612 Le Perray-en-Yvelines, France.

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Summary

This study introduces methods for collapsing strata in complex two-stage stratified sampling to improve variance estimation. It provides algorithms and highlights that high correlation doesn't guarantee low bias in collapsed strata variance estimation.

Keywords:
Collapsed strataCombinatorial optimizationComputation in stratified two-stage simple random sampling without replacement, with possible collapsed strataExpansion estimatorGrouping strataSize variableStratified two-stage samplingWoodruff’s method

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Area of Science:

  • Statistics
  • Survey Methodology
  • Data Analysis

Background:

  • Two-stage stratified sampling designs are complex.
  • Insufficient sampled units within strata hinder variance estimation.
  • Collapsing strata is a technique to address small sample sizes within strata.

Purpose of the Study:

  • To present computer-implementable formulas for estimators and their variances in stratified two-stage sampling.
  • To introduce novel methods for grouping strata when collapsing.
  • To analyze the bias properties of variance estimators in collapsed strata.

Main Methods:

  • Developed formulas for total, mean, and ratio estimators and their sampling variances.
  • Implemented algorithms for stratified two-stage simple random sampling without replacement.
  • Introduced deterministic and stochastic methods for grouping strata based on codes or minimizing within-group inertia.

Main Results:

  • Provided ready-to-use algorithms for complex sampling designs.
  • Demonstrated that the bias of the sampling variance estimator for collapsed strata is not invariant to linear transformations.
  • Showed that a high correlation between a size variable and the variable of interest does not ensure a low-bias estimator.

Conclusions:

  • The proposed methods and formulas facilitate accurate estimation in complex surveys.
  • Care must be taken when collapsing strata, as statistical properties can be affected by transformations.
  • The relationship between variable correlation and bias in collapsed strata requires careful consideration.