1. PSY 307 – Statistics for the Behavioral Sciences Chapter 7 – Regression

2. Regression Line  A way of making a somewhat precise prediction based upon the relationships between two variables. Predictor variable & criterion variable  The regression line is placed so that it minimizes the predictive error.  When based upon the squared predictive error the line is called a least squares regression line.

3. Demo  This demo from the textbook’s student website shows how different lines result in different MSE’s (mean square error): http://www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/index.html

4. Least Squares Equation  Y’ = bX + a  To obtain Y’: Solve for b and a using the data from the correlation analysis Substitute b and a into the regression equation and solve for Y’.  To find points along the line, substitute X values into the regression equation and calculate Y.

5. Formula for Regression Line  Solving for b:  Solving for a:  Then insert both into formula: Y’ = bX + a  Plug in values of X and solve for Y’.

6. Error Bars show the Standard Error of the Estimate (Regression Line)

7. Predictive Error for a Value of X X = 50 Y’ = 137 Error of Y’

8. Standard Error of the Estimate  The average amount of predictive error. Average amount actual Y values deviate from predicted Y’ values. No predictive error when r = 1 Extreme predictive error when r = 0  Again, formulas vary.

9. Calculating Predictive Error Definition Formula: Computation Formula:

10. Kinds of Errors for ALEKS  Difference between the predictions of the regression line and the mean (used as a predictor).  Difference between the predictions of the regression line and the observed values. Predictive error  The difference between these two kinds of errors.

11. Comparing the Regression Line to the Mean Mean of Y

12. Z Score Approach  Prediction using Z scores: Z y = (Z x ) where  = r  is called the standardized regression coefficient because it is being used for prediction.  Prediction using raw scores: Change the person’s raw score to a z- score using the z-score formula. Multiple by , then change the resulting z-score back to a raw score.

13. Squared Correlation Coefficient  r 2 – the square of the correlation coefficient Also called coefficient of determination  Measures the proportion of variance of one variable predictable from its relationship with the other variable.  It is the variance of the errors from repetitively predicting the mean, minus error variance using least squares, expressed as a proportion.

14. Interpretation of r 2  r 2 – not r – is the true measure of strength of association and the proportion of a perfect relationship.  Large values of r 2 are unusual in behavioral research.  Large values of r 2 do not indicate causation. “Explained variance” refers to predictability not causality.

15. Regression Toward the Mean  The mean is a statistical default – use the mean to predict when r is 0 or unknown. Smaller values of r move the prediction toward the mean. The smaller r is, the greater the predictive error, hedged by moving toward the mean.  Chance results in a regression to the mean with repeated measures.

16. Regression Fallacy  The statistical regression of extreme values toward the mean occurs due to chance. Israeli pilots praised for landings do worse on next landing.  It is a mistake (fallacy) to interpret this regression as a real effect. Praise did not cause the change in landings.

17. Testing for Regression Fallacy  Divide the group showing regression into two groups: (1) manipulation, (2) control without manipulation.  Underachievers could show improvement due to regression upward to mean. Always include a control group for regression to the mean.