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Multiple Regression Analysis: Part 4

Multiple Regression Analysis: Part 4. More Extensions of MRA. Background. What we have seen so far: MRA to describe, predict, and/or explain Issues related to cross-validation Using MRA with categorical variables Where this is leading Flexibility of MRA

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Multiple Regression Analysis: Part 4

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  1. Multiple Regression Analysis: Part 4 More Extensions of MRA

  2. Background • What we have seen so far: • MRA to describe, predict, and/or explain • Issues related to cross-validation • Using MRA with categorical variables • Where this is leading • Flexibility of MRA • Today’s let’s introduce more extensions: • Examining interactions and Mediation using MRA Let’s start by looking at Mediation 

  3. Q: What is mediation? • A: It is not moderation! No Mediation X Y Mediation M b a c X Y

  4. Steps to conducting mediation analysis • Step 1:  Show that the initial variable is correlated with the outcome. Use Y as the criterion variable in a regression equation and X as a predictor (estimate and test path c). This step establishes that there is an effect that may be mediated. • Step 2: Show that the initial variable is correlated with the mediator. Use M as the criterion variable in the regression equation and X as a predictor (estimate and test path a). This step essentially involves treating the mediator as if it were an outcome variable. • Step 3:  Show that the mediator affects the outcome variable. Use Y as the criterion variable in a regression equation and X and M as predictors (estimate and test path b). It is not sufficient just to correlate the mediator with the outcome; the mediator and the outcome may be correlated because they are both caused by the initial variable X. Thus, the initial variable must be controlled in establishing the effect of the mediator on the outcome. • Step 4:  To establish that M completely mediates the X-Y relationship, the effect of X on Y controlling for M should be zero (estimate and test path c').  The effects in both Steps 3 and 4 are estimated in the same regression equation. These steps were copied directly from David A. Kenny’s website: http://users.rcn.com/dakenny/kenny.htm

  5. In our example… • Does “rockingness” mediate the effect of marketing expenditures on sales? Step 1:

  6. Continuing… Step 2: Step 3:

  7. Mediation Analysis Results Mediation # of Plays 1.81 (.556) .082 (.009) Marketing CD Sales .105 / -.043 (.029) / (.052) Indirect effect = .083*1.81 = .150. Test significance via Sobel (or other) test. Web Reference: http://www.psych.ku.edu/preacher/

  8. Continuous Variable Interaction Example • Sidestepping Multicollinearity via centering • Testing for significance • Interpreting • Example • Job satisfaction as a function of Hygiene and Caring Atmosphere

  9. Without Centering R2 = .542

  10. With Centering R2 = .542

  11. Interpretation of interaction • Implication: • A continuous variable interaction implies the regression line takes on a different slope depending upon values of X1 & X2. • Methods of visual interpretation • Graphing regression line at different values of X1 & X2 • Categorizing continuous variables and plotting interaction via GLM procedure

  12. Graphing slope with different X1&X2 values 1) Choose a high, medium, and low value for X2 and solve the following: Example: Low value of Caring atmosphere might be -2.62 (after centering) 2) Next, solve for two reasonable (but extreme) values of X1 Example: Doing so for -2 and +2 for basics gives us 3.46 and 5.04 3) Repeat for medium & high values of X2

  13. Then create a plot to examine

  14. Recall our Workplace Deviance Example • Y’ = 2.569 + .391*DC + (-.019)MN + (-.10)*DCxMN • Where: • DC = Dummycode for condition (0=control, 1=experimental) • MN = Maintaining Norms score • DCxMN = Interaction Term Note: data based on an actual study, interaction effect is manufactured for purpose of illustration.

  15. Non-identified group (DC=0) Intercept = a Slope = bx Identified group (DC=1) Intercept = a + bDC Slope = bx + bDCxX Graphing the interaction:Two groups x one continuous variable Need to find the slope and intercept for each group: First, let DC = Dummy Code and X = continuous variable Y’control = 2.569 + (-0.019)MN Y’experimental = 2.960 + (-0.119)MN

  16. Graphing the interaction (Cont’d) • To draw the line, we need two points for each equation. • Choose a reasonably high and reasonably low value for each group. • Control Group = -16.15 and 17.85 • Experimental Group = -16.15 and 13.85 Control Predicted Low = 2.876 = 2.569 + (-0.019)(-16.15) Predicted High = 2.230 = 2.569 + (-0.019)(17.85) Experimental Predicted Low = 4.882 = 2.960 + (-0.119)(-16.15) Predicted High = 1.312 = 2.960 + (-0.119)(13.85)

  17. Graph of Interaction Note: data based on an actual study, interaction effect is manufactured for purpose of illustration.

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