Unifying design-based and model-based sampling theory -- some suggestions to clear the cobwebs

TL;DR

This paper unifies the design-based and model-based sampling theories into a single framework, clarifying their differences, definitions, and common misconceptions to improve understanding and application.

Contribution

It provides a unified framework with consistent notation, clarifies the concept of population variance, and critiques standard presentations to enhance clarity in sampling theory.

Findings

Unified framework for design-based and model-based sampling

Clarification of population variance definitions

Advocacy for explicit conditioning and Bessel's correction

Abstract

This paper gives a holistic overview of both the design-based and model-based paradigms for sampling theory. Both methods are presented within a unified framework with a simple consistent notation, and the differences in the two paradigms are explained within this common framework. We examine the different definitions of the "population variance" within the two paradigms and examine the use of Bessel's correction for a population variance. We critique some messy aspects of the presentation of the design-based paradigm and implore readers to avoid the standard presentation of this framework in favour of a more explicit presentation that includes explicit conditioning in probability statements. We also discuss a number of confusions that arise from the standard presentation of the design-based paradigm and argue that Bessel's correction should be applied to the population variance.