Course Level: Intermediate
Format Requirements: This class is a seminar format. Participants should have a basic understanding of probability and statistics. Previous exposure to basic definitions in set theory are helpful, although I go over them in class.
Background: This course provides applied researchers a conceptual understanding of measure theory, which provides the basis of probability and statistics. Although applied researchers do not typically use measure theory as their technical language for statistical analysis, it provides the basis for a much clearer understanding of study design and interpretation of results. An understanding of measure theory allows practitioners to better avoid the misapplication of such models as hierarchical models, and the misinterpretation of standard errors.
Description and Objectives: In this age of multi-level/hierarchical modeling, the use of varied study designs, and naturally generated data sets, conceptual mistakes are easy to make but also easy to remedy. Some common mistakes are the beliefs that data constitute random variables, that data can be nested, that observations are dependent because they share a common feature (e.g. patients are dependent because they have the same physician), and that standard errors must account for random effects. This workshop will provide the basis to avoid these and other errors by presenting a conceptual overview of measure theory, probability spaces, measurable functions, and their relevance to applied research.
Measure theory is used to generate probability theory, and a conceptual understanding of the measure-theoretic basis for probability can be very helpful to applied researchers. Although the language of mathematics is used in this workshop, this is a conceptual-level workshop; it is not a rigorous mathematical treatment of the topic. This workshop will focus on understanding basic concepts and their direct connection to application. Such an understanding can help researchers identify whether their observations are dependent, what their standard errors really mean, and whether random effects or multi-level models are appropriate. Many researchers think they understand these principles, but the literature indicates that so many are mistaken. This workshop will clarify the issues and help correct mistaken beliefs.
NOTE that measure theory is not measurement theory: this is not a workshop about measurement, scales, or psychometrics.
Objectives:
• the basic concepts of measure theory,
• what variability is represented by reported standard errors,
• whether observations are dependent (e.g., are patients nested within physician or do they merely share the characteristic of having the same physician?),
• whether a random effects specification or multi-level model is applicable,
• the difficulty in interpreting standard errors from analyses of administrative and naturally occurring data.