What makes variables random : probability for the applied researcher

**__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. Cover -- Title Page -- Copyright Page -- Contents -- Preface -- Section I Preliminaries -- 1. Introduction -- Additional Readings -- 2. Mathematical Preliminaries -- Set Theory -- Functions -- Additional Readings -- Section II Measure and Probability -- 3. Measure Theory -- Measurable Spaces -- Measures and Measure Spaces -- Measurable Functions -- Integration -- Additional Readings -- 4. 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? -- It's Just Mathematics-The Interpretation Is Up To You -- Additional Readings -- Section III Applications -- 5. 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 -- 6. Common Problems -- Interpreting Standard Errors -- Notational Conventions -- Random versus 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 -- Index -- A -- B -- C -- D -- E -- F -- H -- I -- M -- N -- O -- P -- R -- S -- U -- V Content: 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