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Regression to the mean

Regression to the mean artifacts are present whenever repeated measures are employed. In simple English, regression to the mean refers to the fact that those with extreme scores on any measure at one point in time will probably have less extreme scores the next time they are tested for purely statistical reasons.

See Regression to the mean: wikipedia for an excellent description of the concept, with fascinating real world examples of consequences for failure to understand regression artifacts. If nothing else, the excerpt from psychologist Daniel Kahneman's acceptance speech when he won the 2002 Bank of Sweden prize for economics (also known as the Nobel prize for economics) is worth reading.

  • “ I had the most satisfying Eureka experience of my career while attempting to teach flight instructors that praise is more effective than punishment for promoting skill-learning. When I had finished my enthusiastic speech, one of the most seasoned instructors in the audience raised his hand and made his own short speech, which began by conceding that positive reinforcement might be good for the birds, but went on to deny that it was optimal for flight cadets. He said, "On many occasions I have praised flight cadets for clean execution of some aerobatic maneuver, and in general when they try it again, they do worse. On the other hand, I have often screamed at cadets for bad execution, and in general they do better the next time. So please don't tell us that reinforcement works and punishment does not, because the opposite is the case." This was a joyous moment, in which I understood an important truth about the world: because we tend to reward others when they do well and punish them when they do badly, and because there is regression to the mean, it is part of the human condition that we are statistically punished for rewarding others and rewarded for punishing them. I immediately arranged a demonstration in which each participant tossed two coins at a target behind his back, without any feedback. We measured the distances from the target and could see that those who had done best the first time had mostly deteriorated on their second try, and vice versa. But I knew that this demonstration would not undo the effects of lifelong exposure to a perverse contingency. ”

The best book I have found on the subject is A Primer on Regression Artifacts; Campbell & Kenny, 1999

Because scores on outcome measures are correlated over time, the change scores will also be correlated with the intake score. This means that patients with high levels of distress will average more change than patients with low levels of distress. Case mix adjustment through use of the Generalized Linear Model is necessary to account for these regression artifacts.

How do we know if the change we measure exceeds regression to the mean? One method that is strongly recommended by Campbell and Kenny is the use of a time reversed regression analysis.

While this may seem nonsensical, the principle is sound. Pure regression to the mean is based on random error, and is "time symmetrical". If the regression formula obtained by using the first score to predict the last score is essentially the same formula obtained by using the last score to predict the first score, then the change observed is probably simply regression to the mean even if scores trended downwards from the first to last administration. If the two regression lines are different,however, then the change exceeded regression to the mean.

The following graph shows the results of a time reversed regression using OQ-30 data from the PBH ALERT system. In this case, the effect size for the regression analysis going forward is much larger than the reverse analysis at every level of severity, demonstrating the measured pre-post change exceeded regression to the mean.

TimeReversedRegression.jpg

-- JebBrown - 27 Feb 2007

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