Superpopulations and Superpopulation Models-An Introduction

We describe a population as a set of identifiable units. We require the units in the superpopulation to be potentially identifiable. Parameters are defined for units in the population, possibly with response error. We then define a superpopulation as a vector of sums of indicator random variables arising from a random permutation of the population units. The superpopulation defined here is a collapsed version of the superpopulation used in the rest of this research. This development serves as a link between early descriptions of mixed models and the expanded superpopulation defined later.

Examples of Superpopulations arising from a Random Permutation Superpopulation Model

Four simple examples of populations and superpopulations under a random permutation superpopulation model are described.

Superpopulations and Superpopulation Models

This document is a review of definitions given to 'superpopulations' and 'superpopulation models' in the literature. The review includes an historical context, as well as a description of different definitions and understandings of what is meant by a superpopulation.

Overview of the Statistical Context
Background and History
Generalizing from Populations
Superpopulations: a Framework for comparing statistics
Targets of inference
Problems with the superpopulation concept
Superpopulations linked to a potentially observable population
Superpopulation Models
General models
Exchangeable Models
Ericson's Superpopulation
Stanek's Expanded Population
Extension to factorial experimental designs
The role of random sampling

A Simple Example of an Exchangeable Superpopulation Model

Ericson proposed a framework for an exchangeable superpopulation model. This framework is described and related to the expanded superpopulation of Stanek In order for the expanded superpopulation model to be related, a projection of the superpopulation is made onto a subspace where the resulting random variables are exchangeable. A simple example is given to describe the basic ideas.