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What Makes Variables Random: Probability for the Applied Researcher provides an introduction to the foundations of probability that underlie the statistical analyses used in applied research. By explaining probability in terms of measure theory, it gives the applied researchers a conceptual framework to guide statistical modeling and analysis, and to better understand and interpret results.

The book provides a conceptual understanding of probability and its structure. It is intended to augment existing calculus-based textbooks on probability and statistics and is specifically targeted to researchers and advanced undergraduate and graduate students in the applied research fields of the social sciences, psychology, and health and healthcare sciences.

Materials are presented in three sections. The first section provides an overall introduction and presents some mathematical concepts used throughout the rest of the text. The second section presents the basic structure of measure theory and its special case of probability theory. The third section provides the connection between a conceptual understanding of measure-theoretic probability and applied research. This section starts with a chapter on its use in understanding basic models and finishes with a chapter that focuses on more complicated problems, particularly those related to various types and definitions of analyses related to hierarchical modeling.



Table of Contents
Part 1 - Preliminaries Chapter 1

Introduction

Additional Readings

Mathematical Preliminaries

Set Theory

Functions

Additional Readings

Part 2-Measure and Probability

Measure Theory

Measurable Spaces

Measures and Measure Spaces

Measurable Functions

Integration

Additional Readings

Probability

Conditional Probabilities and Independence

Product Spaces

Dependent Observations

Random Variables

Cumulative Distribution Functions

Probability Density Functions

Expected Values

Random Vectors

Dependence Within Observations

Dependence Across Observations

Another View of Dependence

Densities Conditioned on Continuous Variables

Statistics

What's Wrong with the Power Set?

Do We Need to Know P to get PX?

Its Just Mathematics-The Interpretation is Up to You

Additional Readings

Part 3-Applications

Basic Models

Experiments with Measurement Error Only

Experiments with Fixed Units and Random Assignment

Observational Studies with Random Samples

Experiments with Random Samples and Assignment

Observational Studies with Natural Data Sets

Population Models

Data Models

Connecting Population and Data Generating Process Models

Connecting Data Generating Process Models and Data Models

Models of Distributions and Densities

Arbitrary Models

Additional Readings

Common Problems

Interpreting Standard Errors

Notational Conventions

Random v Fixed Effects

Inherent Fixed Units, Fixed Effects, and Standard Errors

Inherent Fixed Units, Random Effects, and Standard Errors

Treating Fixed Effects as Random

Conclusion

Additional Readings

Bibliography