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

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.