Everything You Wanted to Know about Moderation (but were afraid to ask) Jeremy F. Dawson University of Sheffield
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1 Everything You Wanted to Know about Moderation (but were afraid to ask) Jeremy F. Dawson University of Sheffield Andreas W. Richter University of Cambridge
2 Resources for this PDW Slides SPSS data set SPSS syntax file Excel templates Available at
3
4 Everything You Wanted to Know about Modera<on Many theories are concerned with whether, or to which extent, the effect of an independent variable on a dependent variable depends on another, so called moderator variable
5 Everything You Wanted to Know about Modera<on Examples: Hoever et al. (2012, JAP): The relationship between team diversity and team creativity depends on the level of perspective taking. Baer (2012, AMJ): The relationship between the generation of ideas and their implementation depends on both employees motivation and their ability to network.
6 Session organizer 1. Testing and probing two-way and three-way interactions using MRA 2. Curvilinear interactions 3. Interactions with non-normal outcomes 4. Extensions of MRA
7 Tes<ng two- way interac<ons Ŷ = b 0 + b 1 X + b 2 Z + b 3 XZ Predicted Y Intercept First order effects Interaction term
8 Probing two- way interac<ons Hypothesis: The relationship between team diversity and team creativity is moderated by perspective taking (cf. Hoever et al., 2012, JAP). Scenario 1: disordinal Scenario 2: ordinal Creativity Low Perspective Taking High Perspective Taking Creativity Low Perspective Taking High Perspective Taking Low High Team Diversity Low High Team Diversity Y = 0.00X Z *XZ Y = 0.00X Z *XZ
9 Scenario 1: buffering Creativity Low Perspective Taking High Perspective Taking Low High Team Diversity Scenario 2: interference/ antagonistic Scenario 3: synergistic/ enhancing Creativity Low Perspective Taking High Perspective Taking Creativity Low Perspective Taking High Perspective Taking Low High Low High Team Diversity Team Diversity
10 Tes<ng two- way interac<ons in SPSS Example data set of 424 employees Independent variables/moderators: Training, Autonomy, Responsibility, Age (all continuous) Dependent variables: Job satisfaction, well being (continuous) Receiving bonus (binary) Days absence in last year (count) H1: Training has a more positive effect on job satisfaction for younger workers than for older workers
11 Tes<ng two- way interac<ons in SPSS IV: TRAIN_C Moderator: AGE_C DV: JOBSAT compute TRAXAGE = TRAIN_C*AGE_C. 1. Compute interaction term regression /statistics = r coeff bcov /dependent = JOBSAT /method = enter TRAIN_C AGE_C TRAXAGE. 2. Run regression to test moderation
12 PloBng two- way interac<ons way with options template
13 Probing two- way interac<ons: Simple slope tests (Aiken & West, 1991) Low Perspective Taking Creativity Medium Perspective Taking High Perspective Taking Low High Team Diversity
14 Simple slope tests: Direct method These figures should be taken from the coefficient covariance matrix (acquired using the BCOV keyword in SPSS). Note that the variance of a coefficient is the covariance of that coefficient with itself! These are then produced automatically: here they tell us that the slope is positive and statistically significant at both 25 and 55 (although less at 55) See Aiken & West (1991) or Dawson (2014) for formula
15 Simple slope tests: Indirect method Principle: The coefficient of the IV gives the slope when the moderator = 0 Method: Center the moderator around the testing value; re-calculate interactions and run the regression Interpretation: The coefficient and p-value of the IV in the new analysis give the result of the simple slope test compute AGE_55 = AGE-55. compute TRAXAGE_55 = TRAIN_C*AGE_55. regression /statistics=r coeff bcov /dependent=jobsat /method=enter TRAIN_C AGE_55 TRAXAGE_55.
16 Simple slope tests: Some thoughts Simple slope tests are far more meaningful when meaningful values of the moderator are used Ensure correct values are chosen after centering decision is made! Here, for example, AGE was centered around the mean (41.55), so ages of 25 and 55 are actually and respectively Choosing values 1 SD above and below the mean is arbitrary and should generally be avoided Remember, statistical significance merely indicates a difference from zero it says nothing about the size or importance of an effect
17 J- N regions of significance and confidence bands (Bauer & Curran, 2006)
18 Tes<ng three- way interac<ons Ŷ = b 0 + b 1 X + b 2 Z + b 3 W + b 4 XZ + b 5 XW + b 6 ZW + b 7 XZW Lower order effects 3-way interaction term
19 Probing three- way interac<ons: Simple slope tests (Aiken & West, 1991) Hypothesis: The relationship between team diversity and team creativity is moderated by perspective taking for managerial teams. Low Perspective Taking/Mgt Team Creativity Low High Team Diversity High Perspective Taking/Mgt Team Low Perspective Taking/Action Team High Perspective Taking/Action Team
20 Probing three- way interac<ons: Simple interac<on tests (Aiken & West, 2000) Hypothesis: The relationship between team diversity and team creativity is moderated by perspective taking for managerial, but not for action teams. Creativity Low High Low Perspective Taking High Perspective Taking Creativity Low High Low Perspective Taking High Perspective Taking Team Diversity Team Diversity Managerial Teams Action Teams
21 Probing three- way interac<ons: Slope difference tests (Dawson & Richter, 2006) Hypothesis: Team diversity predicts team creativity most strongly if teams use perspective taking and are managerial rather than action teams. Low Perspective Taking/Mgt Team Creativity Low High Team Diversity High Perspective Taking/Mgt Team Low Perspective Taking/Action Team High Perspective Taking/Action Team
22 Tes<ng three- way interac<ons H2: The positive effect of training on job satisfaction for younger workers is strengthened when autonomy is higher compute TRAXAUT = TRAIN_C*AUTON_C. compute AUTXAGE = AUTON_C*AGE_C. compute TRXAUXAG = TRAIN_C*AUTON_C*AGE_C. regression /statistics=r coeff bcov /dependent=jobsat /method=enter TRAIN_C AUTON_C AGE_C TRAXAUT TRAXAGE AUTXAGE TRXAUXAG. 1. Compute remaining interaction terms 2. Run regression to test moderation
23 PloBng three- way interac<ons way with options template
24 Slope difference test These figures should be taken from the coefficient covariance matrix (acquired using the BCOV keyword in SPSS) Be careful about the order: SPSS sometimes switches this around! These are then produced automatically: here we find that slope 3 (age 25, high autonomy) is significantly greater than the other three slopes It is important to hypothesize which slopes should be different from each other! See Dawson & Richter (2006) or Dawson (2014) for formulas
25 End of sec<on 1: Ques<ons?
26 Session organizer 1. Testing and probing two-way and three-way interactions using MRA 2. Curvilinear interactions 3. Interactions with non-normal outcomes 4. Extensions of MRA
27 Curvilinear interac<ons Examples: Baer & Oldham (2006, JAP): The curvilinear relationship between employees experienced creative time pressure and creativity is moderated by amount of support for creativity. Zhou et al. (2009, JAP): The curvilinear relationship between number of weak ties and creativity is moderated by conformity value.
28 Curvilinear effects Ŷ = b 0 + b 1 X + b 2 X 2
29 Tes<ng curvilinear interac<ons Hypothesis: a curvilinear relationship between team diversity and team creativity moderated by perspective taking (cf. Hoever et al., 2012, JAP). Creativity Low Perspective Taking High Perspective Taking Low Team Diversity High Ŷ = b 0 + b 1 X + b 2 X 2 + b 3 Z + b 4 XZ + b 5 X 2 Z + r
30 Tes<ng a curvilinear rela<onship H3: The relationship between responsibility and wellbeing is an inverted U shape: well-being is highest when responsibility is moderate compute RESP_C2 = RESP_C*RESP_C. 1. Compute quadratic (squared) term regression /statistics=r coeff bcov /dependent=wellbeing /method=enter RESP_C RESP_C2. 2. Run regression to test effect
31 PloBng a curvilinear rela<onship - Quadratic regression template
32 Tes<ng a curvilinear interac<on H4: The relationship between responsibility and wellbeing is stronger when training is low compute RESXTRA = RESP_C*TRAIN_C. compute RES2XTRA = RESP_C2*TRAIN_C. regression /statistics=r coeff bcov /dependent=wellbeing /method=enter RESP_C RESP_C2 TRAIN_C RESXTRA RES2XTRA. 1. Compute two interaction terms 2. Run regression to test interaction Note: Evidence of curvilinear interaction if and only if RES2XTRA coefficient is significant
33 PloBng a curvilinear interac<on - Quadratic two-way interactions
34 Probing curvilinear interac<ons Simple slope (or curve) test analogous to linear interactions, but with two versions: i. Testing whether there is a curvilinear effect at a particular value of the moderator ii. Testing whether there is any effect at a particular value of the moderator
35 Probing curvilinear interac<ons (i) Testing whether there is a curvilinear effect at a particular value of the moderator: Use indirect method of simple slope test and check IV 2 term e.g. for TRAIN = 4: compute TRAIN_4=TRAIN-4. compute RESXTRA_4 = RESP_C*TRAIN_4. compute RES2XTRA_4 = RESP_C2*TRAIN_4. regression /statistics=r coeff bcov /dependent=wellbeing /method=enter RESP_C RESP_C2 TRAIN_4 RESXTRA_4 RES2XTRA_4. Check value/significance of this term
36 Probing curvilinear interac<ons (ii) Testing whether there is any effect at a particular value of the moderator: Use indirect method of simple slope test and check for variance explained jointly by IV and IV 2 terms e.g. (having computed terms as on previous slide): regression Need this keyword in /statistics=r coeff bcov change syntax to give F-test /dependent=wellbeing /method=enter TRAIN_4 RESXTRA_4 RES2XTRA_4 /method = enter RESP_C RESP_C2. IV and IV 2 terms entered in separate (latter) step
37 End of sec<on 2: Ques<ons?
38 Session organizer 1. Testing and probing two-way and three-way interactions using MRA 2. Curvilinear interactions 3. Interactions with non-normal outcomes 4. Extensions of MRA
39 Interac<ons with Non- Normal outcomes Hypothesis: The relationship between team diversity and receiving a team creativity bonus is moderated by perspective taking (cf. Hoever et al., 2012, JAP). Probability of bonus Low High Team Diversity Low Perspective Taking High Perspective Taking
40 Tes<ng interac<ons with binary outcomes Binary logistic regression Logit (Ŷ) = b 0 + b 1 X + b 2 Z + b 3 XZ Logit link function Note: Logit(Ŷ) = ln[ŷ/(1- Ŷ)]
41 Tes<ng an interac<on with a binary outcome H5: Employees with more responsibility are more likely to receive a bonus when they are older logistic regression variables BONUS /method = enter RESP_C AGE RESP_C*AGE. Logistic regression syntax: no need to compute interaction term separately!
42 PloBng an interac<on with a binary outcome way logistic interactions
43 Probing interac<ons with non- normal outcomes Simple slope tests need to be done using the indirect method e.g. for AGE = 25: compute AGE_25 = AGE-25. logistic regression variables BONUS /method = enter RESP_C AGE_25 RESP_C*AGE_25. Check value/significance of this term
44 Tes<ng interac<ons with discrete (count) outcomes Poisson or Negative Binomial regression Log (Ŷ) = b 0 + b 1 X + b 2 Z + b 3 XZ Natural log link function
45 Tes<ng an interac<on with a count outcome H6: Employees with less responsibility are likely to have more days absence when they are younger genlin ABSENCE with RESP_C AGE /model RESP_C AGE RESP_C*AGE intercept = yes distribution = poisson link = log. Generalized linear models syntax: no need to compute interaction term separately!
46 PloBng an interac<on with a count outcome way Poisson interactions
47 End of sec<on 3: Ques<ons?
48 Session organizer 1. Testing and probing two-way and three-way interactions using MRA 2. Curvilinear interactions 3. Interactions with non-normal outcomes 4. Extensions of MRA
49 Mul<level analysis - hypotheses Hirst et al. (2008, AMJ): H1: Team learning behavior (GL) moderates the goal orientation (IL) creativity (IL) relationship Group level: Team Learning Behavior H1 Individual level: Goal Orientation Creativity
50 Cross- level interac<ons - level- 2 model specifica<on
51 Cross- level interac<ons - level- 1 model specifica<on Intercept and slope for each group j Residual Level- 1: Yij = β 0j + β 1j X ij + rij Outcome measure for individual i in group j Predictor value of individual i in group j
52 Cross- level interac<ons - level- 2 model specifica<on Level- 1: Yij = β 0j + β 1j X ij + rij Group level variable Level- 2: β 0j = γ 00 + γ 01 G j + U 0j Second stage intercept terms Slopes relating Gj to intercept and slope terms from level 1 equation Level 2 residuals β 1j = γ 10 + γ 11 G j + U 1j
53 Probing mul<level interac<ons Interactions can be plotted using the same template as relevant for single-level interactions Estimates produced in output are equivalent to unstandardized coefficients in ordinary regression Care is needed over mean & SD of variables However, in general, simple slope & slope difference tests do not work Simple slope tests can be done instead using the indirect method Slope difference tests are more complicated!
54 Interac<ons in SEM Mplus allows interactions between latent variables All latent variables have mean & SD fixed at 0 and 1 Intercept given by weighted mean of intercepts of indicator variables for DV Simple slope tests cannot be conducted, however Given the (relatively) arbitrary nature of the latent variables, it is doubtful whether they would be meaningful in any case!
55 Tes<ng mul<ple interac<ons Best to do this simultaneously Difficult to plot, however If multiple two-way interactions, but involving no more than three variables, can do it via the 3-way template, leaving unused coefficients as 0 Always consider what is necessary to test your specific hypothesis!
56 End of PDW: Ques<ons?
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