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. Commentaries are informative essays dealing with viewpoints of statistical practice, statistical education, and other topics considered to be of general interest to the broad readership of The American Statistician. Commentaries are similar in spirit to Letters to...

[1]  J. Herniter.  A Probablistic Market Model of Purchase Timing and Brand Selection , 1971 .

[2]  A. Atkinson,et al.  Inference for Sichel's Compound Poisson Distribution , 1982 .

[3]  H. S. Sichel,et al.  On a Distribution Representing Sentence‐Length in Written Prose , 1974 .

[4]  W. A. Ericson.  On the Posterior Mean and Variance of a Population Mean , 1970 .

[5]  G. Maddala,et al.  A Function for Size Distribution of Incomes , 1976 .

[6]  M. R. Novick,et al.  Statistical Theories of Mental Test Scores. , 1971 .

[7]  Anne Anastasi,et al.  Differential Psychology: Individual and Group Differences in Behavior , 1958 .

[8]  H. Raiffa,et al.  Applied Statistical Decision Theory. , 1961 .

[9]  David C. Schmittlein,et al.  Technical Note---Why Does the NBD Model Work? Robustness in Representing Product Purchases, Brand Purchases and Imperfectly Recorded Purchases , 1985 .

[10]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[11]  M. Degroot.  Optimal Statistical Decisions , 1970 .

[12]  W. A. Ericson.  A Note on the Posterior Mean of a Population Mean , 1969 .

[13]  G. J. Goodhardt,et al.  A Consumer Purchasing Model with Erlang Inter-Purchase Times , 1973 .

[14]  C. Morris.  Parametric Empirical Bayes Inference: Theory and Applications , 1983 .

[15]  Andrew Ehrenberg,et al.  Conditional Trend Analysis: A Breakdown by Initial Purchasing Level , 1967 .

[16]  S. Stigler,et al.  Regression Toward the Mean and the Study of Change , 1980 .

[17]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[18]  P. Diaconis,et al.  Conjugate Priors for Exponential Families , 1979 .

[19]  Howard B. Lee,et al.  Foundations of Behavioral Research , 1973 .

[20]  H. Sichel.  On a Distribution Law for Word Frequencies , 1975 .

[21]  David C. Schmittlein,et al.  Predicting Future Random Events Based on Past Performance , 1981 .

[22]  T. Cook,et al.  Quasi-experimentation: Design & analysis issues for field settings , 1979 .

[23]  I. James,et al.  Products of Independent Beta Variables with Application to Connor and Mosimann's Generalized Dirichlet Distribution , 1972 .

[24]  David C. Schmittlein,et al.  Jobs, strikes, and wars: Probability models for duration , 1980 .

[25]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[26]  A. W. Kemp,et al.  The Dirichlet: A comprehensive model of buying behaviour , 1984 .