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LECTURE 5

LECTURE 5. MULTIPLE REGRESSION TOPICS SQUARED MULTIPLE CORRELATION B AND BETA WEIGHTS HIERARCHICAL REGRESSION MODELS SETS OF INDEPENDENT VARIABLES SIGNIFICANCE TESTING SETS POWER ERROR RATES. SQUARED MULTIPLE CORRELATION. Measure of variance accounted for by predictors

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LECTURE 5

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  1. LECTURE 5 • MULTIPLE REGRESSION TOPICS • SQUARED MULTIPLE CORRELATION • B AND BETA WEIGHTS • HIERARCHICAL REGRESSION MODELS • SETS OF INDEPENDENT VARIABLES • SIGNIFICANCE TESTING SETS • POWER • ERROR RATES

  2. SQUARED MULTIPLE CORRELATION • Measure of variance accounted for by predictors • Always increases (or stays same) with additional predictors • Always >= 0 in OLS • More stable than individual predictors (compensatory effect across samples)

  3. Multiple regression analysis • The test of the overall hypothesis that y is unrelated to all predictors, equivalent to • H0: 2y123… = 0 • H1: 2y123… = 0 • is tested by • F = [ R2y123… / p] / [ ( 1 - R2y123…) / (n – p – 1) ] • F = [ SSreg / p ] / [ SSe / (n – p – 1)]

  4. SSreg ssx1 SSy SSe ssx2 Fig. 8.4: Venn diagram for multiple regression with two predictors and one outcome measure

  5. SSreg ssx1 SSy SSe ssx2 Fig. 8.4: Venn diagram for multiple regression with two predictors and one outcome measure

  6. Type I ssx1 SSx1 SSy SSe SSx2 Type III ssx2 Fig. 8.5: Type I and III contributions

  7. B and Beta Weights • B weights • are t-distributed under multinormality • Give change in y per unit change in predictor x • “raw” or “unstandardized” coefficients

  8. B and Beta Weights • Beta weights • are NOT t-distributed- no correct significance test • Give change in y in standard deviation units per standard deviation change in predictor x • “standardized” coefficients • More easily interpreted

  9. PATH DIAGRAM FOR REGRESSION – Beta weight form  = .5 X1 .387 r = .4 Y e X2  = .6 R2 = .742 + .82 - 2(.74)(.8)(.4)  (1-.42) = .85

  10. Depression e .471 .4 LOC. CON. -.345 -.448 -.317 DEPRESSION SELF-EST .399 R2 = .60 -.186 SELF-REL

  11. PATH DIAGRAM FOR REGRESSIONS – Beta weight form X1  = .2 .387 r = .35* R2y= .2 Y1 e1  = .3 X2  = .2  = .5  = .3 Y2 e2 R2y= .6

  12. HIERARCHICAL REGRESSION • Predictors entered in SETS • First set either causally prior, existing conditions, or theoretically/empirically established structure • Next set added to decide if model changes • Mediation effect • Independent contribution to R-square

  13. HIERARCHICAL REGRESSION • Sample-focused procedures: • Forward regression • Backward regression • Stepwise regression • Criteria may include: R-square change in sample, error reduction

  14. STATISTICAL TESTING – Single additional predictor • R-square change: F-test for increase in SS per predictor in relation to MSerror for complete model: F (1,dfe) = (SSA+B – SSA )/ MSeAB A B Y A byB SSe B Y t= byB / sebyB

  15. STATISTICAL TESTING –Sets of predictors • R-square change: F-test for increase in SS per p predictors in relation to MSerror for complete model: F (p,dfe) = ((SSA+B – SSA )/p)/ MSeAB Y A B is a set of p predictors SSe B

  16. Experimentwise Error Rate • Bonferroni error rate: ptotal <= p1 + p2 + p3 + … • Allocate error differentially according to theory: • Predicted variables should have liberal error for deletion (eg. .05 to retain in model) • Unpredicted additional variables should have conservative error to add (eg. .01 to add to model)

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