1. Reminders  HW2 due today  Exam 1 next Tues (9/27) – Ch 1-5 –3 sections: Short answers (concepts, definitions) Calculations (you’ll be given the formulas) SPSS output interpretation

2. Chapter 4 Prediction, Part 3 Sept 20, 2005

3. The Regression Line  Relation between predictor variable and predicted values of the criterion variable  Slope of regression line –Equals b, the raw-score regression coefficient  Intercept of the regression line (where line crosses y axis) –Equals a, the regression constant

4. Drawing the Regression Line 1.Draw and label the axes for a scatter diagram 2.Figure predicted value on criterion for a low value on predictor variable You can randomly choose what value to plug in.. Y hat = -.271 +.4 (x) Y hat = -.271 +.4 (20) = 7.73 3.Repeat step 2. with a high value on predictor Y hat = -.271 +.4 (80) = 31.73 4.Draw a line passing through the two marks 5.Hint: you can also use (Mx, My) to save time as one of your 2 points. Reg line always passes through the means of x and y.

5. Drawing the Regression Line

6. Regression Error  Now that you have a regression line or equation, can find predicted y scores… –Then, assume that you later collect a new sample of x & y scores You can compare how the accuracy of predicted ŷ to the actual y scores Sometimes you’ll overestimate, sometimes underestimate…this is ERROR. –Can we get a measure of error? How much is OK?

7. Error and Proportionate Reduction in Error  Error –Actual score minus the predicted score  Proportionate reduction in error –Squared error using prediction (reg) model = SS Error =  (y - ŷ) 2 –Compare this to amount of error w/o this prediction (reg) model. If no other model, best guess would be the mean. –Total squared error when predicting from the mean is SS Total =  (y – My) 2

8. Error and Proportionate Reduction in Error  Formula for proportionate reduction in error: compares reg model to mean baseline Want reg model to be much better than mean (baseline) – fewer prediction errors

9. Example – Hrs. Slept & Mood See Tables 4-5 and 4-6  Reg model was ŷ = -6.57 + 1.33(x)  Use mean model to find error (y-My) 2 for each person & sum up that column  SStot  Find prediction using reg model: –plug in x values into reg model to get ŷ –Find (y-ŷ) 2 for each person, sum up that column  SSerror  Find PRE

10. Error and Proportionate Reduction in Error (cont.)  If our reg model no better than mean, SS error = SS total, so (0/ SS tot ) = 0. –Using this regression model, we reduce error over the mean model by 0%….not good prediction.  If reg model has 0 error (perfect), SS tot -0/SS tot = 1, or 100% reduction of error.  Proportionate reduction in error = r 2  aka “Proportion of variance in y accounted for by x”, ranges between 0-100%.

11. Multiple Regression  Bivariate prediction – 1 predictor, 1 criterion  Multiple regression – use multiple predictors –Reg model/equations are same, just use separate reg coefficients (  ) for each predictor –Ex) Z-score multiple regression formula with three predictor variables –Note that here,  does not equal r due to overlap among predictors.

12. Mult Reg (cont.)  How to judge the relative importance of each predictor variable in predicting the criterion?  Consider both the rs and the βs –Not necessarily the same rank order of magnitude for rs and βs, so check both. –βs indicate unique relationship betw a predictor and criterion, controlling for other predictors –r’s indicate general relationship betw x & y (includes effects of other predictors)

13. Prediction in Research Articles  Bivariate prediction models rarely reported  Multiple regression results commonly reported –Note example table in book, reports r’s and βs for each predictor; reports R 2 in note at bottom.

14. SPSS Reg Example –Analyze  Regression  Linear –Note that terms used in SPSS are “Independent Variable”…this is x (predictor) –“Dependent Variable”…this is y (criterion) –Climate data, IV = exclusion experiences DV = likelihood of choosing ISU again What to look for: –“Model Summary” section - shows r2 –ANOVA section – 1 st line gives ‘sig value’, if <.05  signif –Coefficients section – 1 st line gives ‘constant’ = a »2 nd line gives ‘standardized coefficients’ = b or beta

15. Group Activity  Use climate data to find regression model using views of ISU climate (as IV) to predict likelihood of attending ISU again (as DV).  1) No need to print the output, just write out the regression model on your paper.  2) What is the r 2 value you get? What does it mean here?

16. Group Activity…  Finishing the patient satisfaction / therapist empathy problem from Thurs. (remember, r =.9) Pair Ther. Patient Predicted Empathy(x) Satisf (y) Y Y - Predicted Y…..(see board). 1704 2945 3 362 4481 M = 62M = 3 SD = 22.14SD = 1.58Reg Equation = ……… a)Figure error & squared error for each prediction, then find proportion of reduction in error over SStotal b)Does it match r 2 ?