Chapter 4 Prediction

Predictor and Criterion Variables  Predictor variable (X)  Criterion variable (Y)

Prediction Using Z Scores  Prediction model –Predicted Z score (on criterion variable) = standardized regression coefficient multiplied by Z score on predictor variable –Formula

Prediction Using Z Scores  The standardized regression coefficient (β) –In bivariate prediction, β = r

Raw-Score Prediction Using the Z-Score Prediction Model 1.Change raw score on predictor to a Z score 2.Multiply β by the predictor variable Z score

Raw-Score Prediction Using the Z-Score Prediction Model 3.Change the predicted Z score on the criterion variable to a raw score

Raw Score Prediction Using the Direct Raw-Score Prediction Model  Direct raw-score prediction model –Predicted raw score (on criterion variable) = regression constant plus the result of multiplying a raw-score regression coefficient by the raw score on the predictor variable –Formula

Raw Score Prediction Using the Direct Raw-Score Prediction Model  The regression constant ( a ) –Predicted raw score on criterion variable when raw score on predictor variable is 0  Raw-score regression coefficient ( b ) –How much the predicted criterion variable increases for every increase of 1 on the predictor variable

Raw Score Prediction Using the Direct Raw-Score Prediction Model 1.Figure the regression constant ( a ) 2.Figure the raw-score regression coefficient ( b ) 3.Find predicted raw score on the criterion variable

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 –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 variable for a low value on predictor variable – mark point on graph 3.Repeat step 2. with a high value on predictor variable 4.Draw a line passing through the two marks

Drawing the Regression Line

Error and Proportionate Reduction in Error  Error –Actual score minus the predicted score  Proportionate reduction in error –Squared error using prediction model = SS Error –Total squared error when predicting from the mean = SS Total

Error and Proportionate Reduction in Error  Formula for proportionate reduction in error:  Proportionate reduction in error = r 2  Proportion of variance accounted for

Multiple Regression  Bivariate prediction  Multiple correlation  Multiple regression

Multiple Regression  Multiple regression prediction models –Each predictor variable has its own regression coefficient –e.g., Z-score multiple regression formula with three predictor variables:

Limitations of Regression  Regression inaccurate if –Correlation is curvilinear –Restriction in range –Unreliable measures

Controversies and Limitations  Controversy about how to judge the relative importance of each predictor variable in predicting the dependent variable  Consider both the rs and the βs

Prediction in Research Articles  Bivariate prediction models rarely reported  Multiple regression results commonly reported