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Regression Towards the Mean David A. Kenny University of Connecticut http://davidakenny.net/old/series/primer.htm

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3 An Opening Quote “The list of studies in which the regression factor has been neglected grows monotonous as well as distressing” (Rulon, 1941).

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6 Galton Squeeze Diagram Two vertical axes. left line the pretest right line the posttest A line from the the pretest score to the mean on the posttest.

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9 Definition A person’s score on a variable that is extreme (in the sense of being far away from the mean) tends to be less extreme in standard deviation units when that person is measured on another variable.

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10 Common Misconceptions RTM results in mediocrity. RTM is unidirectional. RTM is entirely due to measurement error. RTM is some mysterious force.

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11 Illustrations The Sophomore Jinx The Gifted and Not So Gifted Spontaneous Remission

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12 Is There a Sophomore Jinx? 39 Hitting Winners from 1970 and 1994 Rookie average.285 Sophomore average.266 28 declined and 11 improved

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13 Jinx?: Scatter-plot

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14 The Gifted and Not So Gifted Extreme Score Used for Treatment Selection educational programs medical treatments What is the probability that the score will remain extreme after re-testing?

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15 A Computer Simulation Specifications randomly pick a score from a normal distribution consider only scores greater than two standard deviations above the mean. reliability of.8 What is the probability that the score will remain extreme (> 2sd’s) after re-testing?

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16 How Much? only 43% research dementia: 39% ADHD: 45% depression: 40%

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17 True Score Correction The most important but forgotten formula in psychometrics. Given a score with a reliability of , what is the best guess of the true score or X T ʹ ? X T ʹ = (X - M X ) + M X Correlation of X T ʹ with X is one.

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18 Spontaneous Remission Change in the control group using Cohen’s d (over-time mean difference divided by the pretest standard deviation). depression: -0.35 pain: -0.58

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19 Final Points RTM is not so mysterious: change happens. RTM is bidirectional. Whether the score is above or below the mean in applications may not be clear.

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20 Concluding Quote from Francis Galton (1889) “Some hate the very name of statistics but I find them full of beauty and interest. Whenever they are not brutalized, but handled by higher methods, and warily interpreted, their power of dealing with complicated phenomena is extraordinary.”

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Math 3680 Lecture #19 Correlation and Regression.

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