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.
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
Whats 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
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