What Makes Variables Random

Probability for the Applied Researcher

Peter J. Veazie

• Publisher: Chapman and Hall/CRC
• Year: 2017
• Print ISBN: 9780367848712
• eBook ISBN: 9781351720120
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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.

Section III

Applications

5

Basic Models

The preceding chapters presented a conceptual introduction to measure-theoretic mathematical probability. This chapter will introduce uses of probability spaces as models for common research designs. We will start with the conceptually simple (i.e., modeling measurement error) and progress to a study design that may defy the usefulness of probability spaces as a modeling paradigm (i.e., modeling natural data generating processes for observational data).

Experiments with Measurement Error Only

Perhaps the simplest scientific use of a probability space is to model the variability associated with measurement in the context of a fully controlled experiment of a deterministic process. Consider an experiment in which all conditions are fully controlled except for chance error in the measurement instrument itself. In other words, for each run of the experiment, the physical outcome is the same, but the measurements may vary by chance. If we run the experiment N times, we record N measurements that may differ due solely to measurement error. How do we use this set of data to draw inferences regarding the underlying process?

We can model the measurement process for the experimental setup as a probability space: let Ω denote an outcome set comprising possible states that the measurement instrument can take, let $\mathcal{A}$ be an appropriately rich sigma-algebra, and let P denote a probability measure that models the objective uncertainty associated with the state of a given measurement outcome being in the sets of $\mathcal{A}$. Then (Ω, $\mathcal{A}$, P) is a probability space representing the uncertainty of the measurement process of the experiment. Suppose we define a function, labeling it Y, from each state of the measurement instrument to numbers on the real line. Now, if ℬ is a sufficiently rich sigma-algebra on ℝ such that the range of the function Y⁻¹(B) for all B ∈ ℬ is contained in $\mathcal{A}$, then (ℝ, ℬ) is a measurable space and Y is measurable $\mathcal{A}$: Y is therefore a random variable representing variation in the measurement process. If we let p(B) = P(Y⁻¹(B)) define a measure on (ℝ, ℬ), then (ℝ, ℬ, p) is a probability space associated with the random variable Y. Characteristics of p provide information useful for making inferences about the experiment accounting for measurement error.

For N independent runs of the experiment, there is one realization from each of N random variables from the probability space. If the experimental runs are identical, it is reasonable to presume that each of these random variables has the same distribution, and we can use statistics (a function of the N random variables, which consequently is itself a random variable) to estimate properties of p and thereby provide information for making inferences. For example, the histogram of the data reflects p from the N realizations from the probability space. If we are interested in the expected value of Y taken with respect to p as a summary of the experimental outcome, then the sample mean and confidence interval provide a reasonable estimate of this quantity, as shown in any introductory statistics text. If our interest is not in the estimate itself, but instead we are interested in testing a particular hypothesis, then we can proceed with statistical hypothesis testing. For example, suppose we have a theory that implies a state corresponding to a perfect measurement of less...

Table of contents

• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Section I Preliminaries
• Section II Measure and Probability
• Section III Applications
• Bibliography
• Index