Surprising Inferences from Unsurprising Observations: Do Conditional Expectations Really Regress to the Mean?

Abstract

Most social science descriptions of the statistical regression effect envision the effect as occurring toward the population mean. If individuals that initially had extreme values regress back toward the mean on subsequent observations, then as a corollary individuals who were at the population mean initially are expected to stay at the population mean. Both the regression to the mean statement and its corollary are generally false for models exhibiting a regression effect. For commonly used probability mixture models, conditional expectations of subsequent observations based on previous observations regress not to the mean, but to some other value. Examples are presented using mixtures of normal, Poisson, and binomial random variables.

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Key Words:

• Prior distributions
• Probability mixture models
• Regression to the mean