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Regression to the Mean and Unusual Observations

Understand how standardized residuals, leverage values, and Cook's D can help identify unusual observations in regression analysis. Explore robust regression and the pitfalls of regression fallacy.

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Regression to the Mean and Unusual Observations

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  1. Stat 302 – Day 27 Regression to the mean

  2. Last Time – Unusual Observations • Standardized residuals = how far the observed response is from predicted, standardized • Larger than 2 or 3 is considered substantial • Leverage values = how far the observed x-value is from the average x-value • Larger than 4/n or 6/n is considered substantial • Cook’s D = how much all the predicted values would change if the observation was removed • Larger than 0.5 or 1.0 is considered substantial

  3. Question (e) • Where could you put an observation in this dataset, so that it is not an outlier in either the x direction or the y direction but is an outlier in the regression? Do you think this observation will have much influence on the regression line?

  4. Question (e) • Montana is a pretty good example

  5. Question (e) • What about over here?

  6. Robust regression

  7. Quiz

  8. Technology Exploration (p. 402)

  9. Continued • Recall: The formula for the least squares estimate for the intercept simplifies (l) Substitute this in for in the regression equation rewrite and rewrite with all the “y” terms on the left and all the “x” terms on the right.

  10. Continued (m) Suppose that we choose a golfer whose first round score is above average (x > ). Do we predict the second round score ( ) will be above average or below average ( )? By the same amount that the first round score was above average or by less? What if a golfer is 1 standard deviation below the mean (x = - sx)?

  11. Regression Fallacy • Attributing poor performance to something other than the regression effect • … one of the instructors who relayed that in his experience praising a cadet for executing a clean maneuver is typically followed by a lesser performance, whereas screaming at a cadet for bad execution is typically followed by improved performance.

  12. Regression Fallacy • Attributing poor performance to something other than the regression effect • To appear on the cover, a player had a transcendent season… expect some decline in performance as the player returns back to his natural production

  13. To Do • Quiz 15* • No Quiz 14 • Finish HW 5 • Look for HW 6 • Be working on project • Progress Report due Monday!

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