Moderators and Curvilinearity

1. Moderators and Curvilinearity John Elias & Katelyn Procci March 17, 2010

2. Intro by way of mediation • Mediation – relationship between IV and DV (x and y) cannot be adequately explained without reference to mediator z • Part of the mechanism or pathway whereby x impacts or influences y • Mediator needs to be there in order for x to affect y – necessary for the relationship between x and y • Personality  Motivation  Performance

3. Moderation on the other hand… • Magnitude or strength of the relationship between x and y depends on the value of the moderator z • Relationship between x and y exists independently of moderator; makes sense to explain the relationship between x and y without reference to the moderator • But relationship between x and y is amplified or diminished by the value of z • Distinction arguable in certain cases, depending on theory?

4. Moderation Testing the Moderating (Interaction) Effect: Step 1: Model 1: Y = b0 + b1X + b2Z Step 2: Model 2: Y = b’0 + b’1X + b’2Z + b’3XZ The product term XZ represents the interaction effect. XZ is created by simply multiplying X and Z. Examining b’3 provides a test for the moderating effect (which is equivalent to comparing Model 2 and Model 1) Important: - Models should be tested hierarchically (i.e., Model 1 is tested first, then Model 2). - Never test model with only the product term (XZ) - In Model 2, the main effects (b’1 and b’2) are arbitrary, depending on the zero points of the scales for X and Z. Model 2 can be rewritten as: Y = b’0 + (b’1 +b’3Z)X + b’2Z = b’0 + B*X + b’2Z

5. Problems… • Multi-Colinearity: • XZ is highly correlated with X and Z • Adding XZ in model 2 will inflate the standard errors for X and Z • Low power for detecting moderation (interaction) effect

6. Briefly, curvilinearity • Curvilinearity – non-linear relationship between x and y • Can be seen as akin to interaction effect, in that the strength of the relationship between x and y depends on x – thus, x in a sense interacts with itself, in that its impact on y depends on its value

7. To test for curvilinear effect: Step 1: Model 1: Y = b0 + b1X Step 2: Model 2: Y = b’0 + b’1X + b’2X2 The quadratic term X2 represents the interaction effect. X2 is created by simply squaring X. Examining b’2 provides a test for the curvilinear effect (which is equivalent to comparing Model 2 and Model 1) Models should be tested hierarchically (Model 1 is tested first, then Model 2).

8. Problems, as always… • Curvilinear relationship can be mistaken for interaction effect. • If a third variable z is positively correlated with x, then xz will be highly correlated with x2 • Thus, if we hypothesize an interaction effect between x and z, the interaction term xz will serve as a proxy for x2 term absent in our model • Therefore, include and try to account for curvilinear effects when hypothesizing and testing for interaction effects

9. Testing for Moderation in SPSS Version 18

10. Example Study: Subjects play a game on a PC. Researchers would like to know whether or not how much a person plays games in general affects the score they are able to achieve in the game, regardless of game content. The literature suggests that the strength of this relationship may be affected by how comfortable they are playing games on a computer, where people who are more comfortable will have high scores compared to those who are less comfortable when they play games frequently. • IV(X): Game Frequency • Moderator(Z): Computer Comfort • DV(Y): Score

11. Step 1: Create an interaction variable between your IV and your Moderator. • Click on Transform > Compute

12. In the Target Variable box, name your interaction variable • Double click your IV to add it to the Numeric Expression Box • Click on the multiplication symbol • Double click your Moderator to add it to the Numeric Expression box • Click OK

13. Step 2: Run a Linear Regression with Two Models • Click on Analyze > Regression > Linear

14. Move your DV to the Dependent Box •  To set up the firstmodel, put both the IV and the Moderator in the Independent(s) box • To set up the secondmodel, click on Next

15. Place your new interaction variable in the Independent(s) box • Click on Statistics and check the following boxes: • Estimates • Confidence Intervals • Model fit • R squared change • Descriptives • Click Continue • Click OK

16. Step 3: Examine the SPSS Output • Significance of F-Change • If less than .05, the model is significant • R Square • % of variance in DV due to variables in the model • Model 1: 41.10% • Model 2: 52.70%

17. Model 1 • Red Box: Significance of IV and Moderator on DV with consideration of each-other • Model 2 • Orange Box: Significance of IV and Moderator on DV (now held constant) • Green Box: Significance of Interaction on DV above and beyond the effect of the IV and the Moderator • Want the Interaction in Model 2 to be significant • Expect the IV and Moderation to decrease in significance in Model 2

18. Tolerance: The % of variance in the Moderator that is unique to itself and not shared by the IV and the Moderator alone. • 9.20% of variation in the moderator is unique to the moderator above and beyond what it shares with the other variables in the model

19. Step 4: Write the Line’s Equation • Dependent Variable = [Unstandardizedβ for Model 2 Constant] + [Unstandardizedβ for Model 2 IV]IV + [Unstandardizedβ for Model 2 Moderator]Moderator + [Unstandardizedβ for Interaction]IV*Moderator • Y= β+ β(X) +β(Z) + β(X)(Z) • Score = 4.089 – 0.198(Game Frequency) - 0.333(Computer Comfort) + 0.152(Game Frequency)(Computer Comfort)

20. Step 5: Report Your Findings • In the second step of the regression analysis, the interaction term between the X and the Z explained a significant increase in variance in the Y,  ΔR2 = .XX,F(k-1, N-k) = XX.XX,p < .05. Thus, Z was a significant moderator of the relationship between X and Y. • In the second step of the regression analysis, the interaction term between game frequency and computer comfort explained a significant increase in variance in score,  ΔR2 = .12, F(1, 24) = 5.88, p < .02. These results suggest that computer comfort was a significant moderator of the relationship between game frequency and score.

21. Step 6: Graphing the Moderation • Choose two very different values of the moderator • Plug each into the equation to generate the equation of a line • Plug in values of X to generate data points • Graph • by hand • in Excel

22. Score = 4.089 – 0.198(X) - 0.333(Z) + 0.152(X)(Z) • Z = 2 • Score = 4.089 – 0.198(X) – 0.333(2) + 0.152(X)(2) • Score = 4.089 – 0.198(X) - 0.666 + 0.304(X) • Score = 3.423 + .106(X) • If X = 2, y = 3.635 • If x = 6, y = 4.059 • Z = 8 • Score = 4.089 – 0.198(X) – 0.333(8) + 0.152(X)(8) • Score = 4.089 – 0.198(X) – 2.664 + 1.216(X) • Score = 1.425 + 1.018(X) • If X = 2, y = 3.461 • If x = 6, y = 7.533

24. Additional SPSS Tips & Tricks • Centering your IV and Moderator • Changes the 0-point • Eliminates inflated standard error, decreasing the effect of colinearity • Does not affect power • Looking at the extremes

25. Centering your Variables • Step 1: Run descriptives on your IV and Moderator • Analyze > Descriptive Statistics > Descriptives • Move the IV and Moderator to the Variable(s) box • Click OK

26. Step 2: Make note of the means for the IV and the Moderator • IV Mean: 4.0000 • Moderator Mean: 4.6786

27. Step 3: Compute centered variables for the IV and the Moderator • Click on Transform > Compute • In the Target Variable box, name your new variable • Double click your Variable to add it to the Numeric Expression Box • Click on the subtraction symbol • Write in the mean for your variable • Click OK • Repeat for each IV and Moderator • Step 4: Recalculate interaction and rerun regression

28. Looking at the Extremes • Step 1: Split the data into 3 different sections based on the moderator using cut points. • Determine the range of the Moderator • Analyze > DescriptiveStatistics > Frequencies • Move Moderator to Variable(s) box • Click OK • Make note of the range. Divide by 3 to determine the sections.

29. Step 2: Select the cases at the extremes • Data > Select Cases • Select the If condition is satisfied button • Click on If… • Double click on your Moderator to move it to the formula box • Click on the less-than-or- equal-to symbol (<=) • Type in the lower range cut- off value for the Moderator • Click Continue

30. Step 2 Continued… • Click on If… again • Delete everything in the equation box • Double click on your Moderator to move it over • Click on the greater-than-or-equal-to symbol (>=) • Type in the upper range cut-off point • Click Continue • Click OK • Step 3: Rerun the regression

31. Testing for Curvilinearity in SPSS Version 18

32. Example Study: Subjects play a game on a PC. Researchers would like to know whether or not how much a person plays games in general affects the score they are able to achieve in the game, regardless of game content. The literature suggests that the score will depend on the how much the individual plays game and that this relationship is not linear, but curvilinear. • IV(X): Game Frequency • DV(Y): Score

33. Step 1: Create the Quadratic IV term • Click on Transform > Compute

34. In the Target Variable box, name your new variable • Double click your IV to add it to the Numeric Expression Box • Click on the multiplication symbol • Double click your IV to add it to the Numeric Expression Box • Click OK

35. Step 2: Run a Linear Regression with Two Models • Click on Analyze > Regression > Linear

36. Move your DV to the Dependent Box •  To set up the firstmodel, put the IV in the Independent(s) box • To set up the secondmodel, click on Next

37. Place your new quadratic IV variable in the Independent(s) box • Click on Statistics and check the following boxes: • Estimates • Confidence Intervals • Model fit • R squared change • Descriptives • Click Continue • Click OK

38. Step 3: Examine the SPSS Output • Significance of quadratic term • If less than .05, there is a curvilinear relationship

39. Tolerance: The % of variance in the quadratic term that is unique to itself beyond the linear effect • 4.50% of variation in the quadratic term is due to itself beyond what is explained by the linear term

40. Step 4: Write the Line’s Equation • Dependent Variable = [Unstandardizedβ for Model 2 Constant] + [Unstandardizedβ for Model 2 IV]IV + [Unstandardizedβ for Model 2 Moderator] IV2 • Y= β+ β(X) +β(X2) • Score = 1.071 + 1.506(Game Frequency) - 0.119(GameFrequency2)

41. Step 5: Report Your Findings • In the second step of the regression analysis, the quadratic term explained a significant increase in variance in the Y,  ΔR2 = .XX,F(k-1, N-k) = XX.XX,p < .05. Thus, the relationship between X and Y is curvilinear. • In the second step of the regression analysis, the quadratic term explained a significant increase in variance in score,  ΔR2 = .04, F(1, 25) = 1.48, p < .05. These results suggest that the relationship between game frequency and score is curvilinear.

42. Step 6: Graphing the Curvilinear Relationship • Click on Analyze > Regression > Curve Estimation • Move the dependent variable to the Dependent(s) box • Move the independent variable to the Independent box • Uncheck Linear box • Check Quadratic box • Click OK

44. Rational decision-making and firm performance: The moderating role of environment Goll and Rasheed, 1997 Strategic Management Journal, 18 (7), 583-591

45. The study investigates the moderating roles of certain environmental factors in the relationship between rational decision-making and organizational performance. • But what does it mean to be rational, to be a rational decision-maker?

46. Brief Intro to Decision Theory • Basic insight of decision theory: actions are like gambles, a kind of bet on the capacity of an action to deliver what you want, given certain potential risks in the environment • Even the seemingly simplest choices can be viewed as decision problems involving various contingencies • Making up your mind involves weighing these various factors, desires against risks

47. Brief Intro to Decision Theory • Actions of others provide evidence about the kinds of contingencies they recognize, the sort of risks they consider, and what goals they have • Frank Ramsey: behavior essentially has two causal factors: desires or goals, the strength of the pull towards those goals; and beliefs, assumptions and assessments the environment

48. Brief Intro to Decision Theory • Decision Theory as a means of quantifying and teasing these factors apart • Strength of pull of goals will be revealed by the extent to which a person is willing to take risks in order to attain them • Set up balance between rewards and risks - addition and subtraction of payoff, to the point at which risks involved no longer make it worthwhile to the agent • Way to work out as well how strongly people believe in the possible risk factors that they face in the environment

49. Article attempts to consider aspects of the environment beyond uncertainty or contingency… Three moderating relationships: • Moderating role of environmental munificence… • Moderating role of environmental dynamism… • Joint effect of these… On the relationship between rational decision making and performance • “Munificence refers to an environment's ability to support sustained growth of an organization”

50. Three Hypotheses: • There is a positive relationship between decision rationality and firm performance. • Environmental munificence and dynamism moderate the relationship between process rationality and organizational performance. • Organizational rationality is more strongly associated with performance in high-discretion environments (that is, high munificence and high dynamism) than in other types of environments.