PSY 307 – Statistics for the Behavioral SciencesRegression LineDemoLeast Squares EquationFormula for Regression LineError Bars show the Standard Error of the Estimate (Regression Line)Predictive Error for a Value of XStandard Error of the EstimateCalculating Predictive ErrorKinds of Errors for ALEKSComparing the Regression Line to the MeanZ Score ApproachSquared Correlation CoefficientInterpretation of r2Regression Toward the MeanRegression FallacyTesting for Regression FallacyPSY 307 – Statistics for the Behavioral SciencesChapter 7 – RegressionRegression 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.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.htmlLeast 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.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’.Error Bars show the Standard Error of the Estimate (Regression Line)Predictive Error for a Value of XX = 50Y’ = 137Error of Y’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.Calculating Predictive Error2)(22nYYnSSsxyxyDefinition Formula:Computation Formula:2)1(2nrSSsyxyKinds 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.Comparing the Regression Line to the MeanMean of YZ Score ApproachPrediction using Z scores:Zy = (Zx) 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.Squared Correlation Coefficientr2 – 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.Interpretation of r2r2 – not r – is the true measure of strength of association and the proportion of a perfect relationship.Large values of r2 are unusual in behavioral research.Large values of r2 do not indicate causation.“Explained variance” refers to predictability not causality.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.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.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