3003PSY Mini-lecture - Variance Explained in Multiple Regression PDF

3003PSY Survey Design and Analysis in Psychology MULTIPLE REGRESSION: VARIANCE EXPLAINED  uIn correlation, r2 = proportion of variance shared by X and Y, i.e., the two correlates—overlap uIn regression, this still holds of PROPORTIONS co...

3003PSY Survey Design and Analysis in Psychology MULTIPLE REGRESSION: VARIANCE EXPLAINED  uIn correlation, r2 = proportion of variance shared by X and Y, i.e., the two correlates—overlap uIn regression, this still holds of PROPORTIONS course, but… OF VARIANCE u… now, we speak of the amount of variance in Y accounted for by X (either a single predictor or the set of predictors)  PROPORTIONS OF VARIANCE Multiple R2 is variance in Y accounted for by Xk uusually referred to simply as R2 uNot specific to any one single X, it is for the set of them. uThe regression equation is also referred to as the Model uwe check the value of R2 in the Model Summary box in the SPSS output  IN EQUATIONS—WHAT IS R2? SSY = å (Y - Y ) 2 SS reg = å (Y ¢ - Y ) 2 SS res = å (Y - Y ) ¢ 2 R = 2 SS reg = å (Y - Y ) ¢ 2 SSY å (Y - Y ) 2  IN EQUATIONS—WHAT IS R2? SSY = å (Y - Y ) 2 SS reg = å (Y ¢ - Y ) 2 SS res = å (Y - Y ) ¢ 2 R = 2 SS reg = å (Y - Y ) ¢ 2 SSY å (Y - Y ) 2 R = 2 SS reg = å (Y - Y ) ¢ 2 SSY å (Y - Y ) 2  uSemipartial correlations—sr and sr2 uRelationship between Xk and Y when all other predictors partialled EXTENDING OUR out of only Xk. INTERPRETATION OF uGood indicator of the unique REGRESSION PREDICTORS relationship between Xk and Y uAlways squared (i.e., sr2) to give proportion of unique variance in Y accounted for by Xk  RELATING THIS TO SPSS AGAIN REGRESSION var = GRE_Q Attendance Stats_Exam /descriptives=def /stat=def zpp /dep = Stats_Exam /enter = GRE_Q Attendance.  R2 = SSreg/SSY 783.925/2430.55 =.323 SSreg = 783.925 SSres = 1646.625 SSY = 2430.550  GRE_Q (X1) → Stats_Exam (Y) unique GRE_Q (X1) + Attendance (X2) → Stats_Exam (Y) shared Attendance (X2) → Stats_Exam (Y) unique GRE_Q (X1) ↔ Attendance (X2) unique—external R2 = Red + Green + X1 Yellow Red is the effect of GRE_Q on Stats_Exam, accounting for Attendance Y Green is vice versa Yellow is shared effect X2 Blue is outside the Based on Tabachnick regression completely and Fidell’s (2007) infamous Fig. 5.3! Calculate the unique variance of GRE-Q X1 Y After variance shared with Attendance (X2) has been removed from Y, GRE_Q (X1) accounts for 19% for additional variance in stats exam performance X1 Y Calculate the unique variance of Attendance Y X2 After variance shared with GRE_Q (X1) has been removed from Y, Attendance (X2) accounts for 29% of additional variance in stats exam performance. Y X2 Shared Variance is the difference between the combined unique variance (sr2GRE_Q +sr2attendance) and total variance explained (R2) Shared variance between the two predictor variables on stats exam performance was 13.5%  SUMMARY u R square refers to the variance in the Y variable explained by the linear composite (aka regression equation or model) u In multiple regression we want the variance explained by each predictor variable after controlling for the other predictor variables uWe call this unique variance and is represented by squared semipartial correlations uThe difference between total variance explained (R2) and unique variance is shared variance uWe can look at the unique variance to better understand the strengths of the relationships between our predictor variables and the outcome