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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 Chapter 4 Prediction, Part 3 Sept 20, 2005 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 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. Drawing the Regression Line 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? Error and Proportionate Reduction in Error Error – Actual score minus the predicted score 2 ˆ Error (Y Y ) 2 Proportionate reduction in error – Squared error using prediction (reg) model = SSError = (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 SSTotal = (y – My)2 Error and Proportionate Reduction in Error Formula for proportionate reduction in error: compares reg model to mean baseline SS Total SS Error Proportion ate reduction in error SS Total Want reg model to be much better than mean (baseline) – fewer prediction errors 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 Error and Proportionate Reduction in Error (cont.) If our reg model no better than mean, SSerror = SStotal, so (0/ SStot) = 0. – Using this regression model, we reduce error over the mean model by 0%….not good prediction. If reg model has 0 error (perfect), SStot-0/SStot = 1, or 100% reduction of error. Proportionate reduction in error = r2 aka “Proportion of variance in y accounted for by x”, ranges between 0-100%. 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 ZˆY (1 )( Z X1 ) (2 )( Z X 2 ) (3 )( Z X 3 ) – Note that here, does not equal r due to overlap among predictors. 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) 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 R2 in note at bottom. 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 – 1st line gives ‘sig value’, if < .05 signif – Coefficients section – 1st line gives ‘constant’ = a » 2nd line gives ‘standardized coefficients’ = b or beta 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 r2 value you get? What does it mean here? Group Activity… Finishing the patient satisfaction / therapist empathy problem from Thurs. (remember, r = .9) Pair Ther. Empathy(x) 1 2 3 4 70 94 36 48 M = 62 SD = 22.14 Patient Satisf (y) Predicted Y Y - Predicted Y…..(see board). 4 5 2 1 M=3 SD = 1.58 Reg Equation = ……… a) Figure error & squared error for each prediction, then find proportion of reduction in error over SStotal b) Does it match r2?