Moderation: Assumptions

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# Moderation: Assumptions - PowerPoint PPT Presentation

Moderation: Assumptions. David A. Kenny. What Are They?. Causality Linearity Homogeneity of Variance No Measurement Error. Causality. X and M must both cause Y.

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Presentation Transcript
What Are They?

Causality

Linearity

Homogeneity of Variance

No Measurement Error

Causality
• X and M must both cause Y.
• Ideally both X and M are manipulated variables and measured before Y. Of course, some moderators cannot be manipulated (e.g., gender).
Causal Direction
• Need to know causal direction of the X to Y relationship.
• As pointed out by Irving Kirsch, direction makes a difference!
Surprising Illustration
• Judd & Kenny (2010, Handbook of Social Psychology), pp. 121-2 (see Table 4.1).
• A dichotomous moderator with categories A and B
• The X  Y effect can be stronger for the A’s than the B’s.
• The Y  X effect can be stronger for the B’s than the A’s.
Direction of Causality Unclear
• In some cases, causality is unclear or the two variables may not even be a direct causal relationship.
• Should not conduct a moderated regression analysis.
• Tests for differences in variances in X and Y, and if no difference, test for differences in correlation.
Crazy Idea?
• Assume that either X  Y or Y  X.
• Given parsimony, moderator effects should be relatively weak.
• Pick the causal direction by the one with fewer moderator effects.
Proxy Moderator
• Say we find that Gender moderates the X  Y relationship.
• Is it gender or something correlated with gender: height, social roles, power, or some other variable.
• Moderators can suggest possible mediators.
Graphing
• Helpful to look for violations of linearity and homogeneity of variance assumptions.
• M is categorical.
• Display the points for M in a scatterplot by different symbols.
• See if the gap between M categories change in a nonlinear way.
Linearity
• Using a product term implies a linear relationship between M and X to Y relationship: linear moderation.
• The effect of X on Y changes by a constant amount as M increases or decreases.
• It is also assumed that the X  Y effect is linear: linear effect of X.
Alternative to Linear Moderation
• Threshold model: For X to cause Y, M must be greater (lesser) than a particular value.
• The value of M at which the effect of X on Ychanges might be empirically determined by adapting an approach described by Hamaker, Grasman, and Kamphuis (2010).
Second Alternative to Linear Moderation
• Curvilinear model: As M increases (decreases), the effect of X on Y increases but when M gets to a particular value the effect reverses.
Testing Linear Moderation
• Add M2 and XM2 to the regression equation.
• Test the XM2 coefficient.
• If positive, the X  Y effect accelerates as M increases.
• If negative, then the X  Y effect de-accelerates as M increases.
• If significant, consider a transformation of M.
The Linear Effect of X
• Graph the data and look for nonlinearities.
• Add X2 and X2M to the regression equation.
• Test the X2 and X2M coefficients.
• If significant, consider a transformation of X.
Nonlinearity or Moderation?
• Consider a dichotomous moderator in which not much overlap with X (X and M highly correlated).
• Can be difficult to disentangle moderation and nonlinearity effects of X.
Nonlinear Relationship

Y

X

Moderation

Y

X

Homogeneity of Variance
• Variance in Moderation Analysis
• X
• Y (actually the errors in Y)
Different Variance in X for Levels of M
• Not a problem if regression coefficients are computed.
• Would be a problem if the correlation between X and Y were computed.
• Correlations tend to be stronger when more variance.
Equal Error Variance
• A key assumption of moderated regression.
• Visual examination
• Plot residuals against the predicted values and against X and Y
• Rarely tested
• Categorical moderator
• Bartlett’s test
• Continuous moderator
• not so clear how to test
• The category with the smaller variance will have too weak a slope and the category with the larger variance will too strong a slope.
• Separately compute slopes for each of the groups, possibly using a multiple groups structural equation model.
• No statistical solution that I am aware of.
• Try to transform X or M to create homogeneous variances.
Variance Differences as a Form of Moderation
• Sometimes what a moderator does is not so much affect the X to Y relationship but rather alters the variances of X and Y.
• A moderator may reduce or increase the variance in X.
• Stress  Mood varies by work versus home; perhaps effects the same, but much more variance in stress at work than home.
Measurement Error
• Product Reliability (X and M have a normal distribution)
• Reliability of a product: rxrm(1 + rxm2)
• Low reliability of the product
• Weaker effects and less power
• Bias in XM Due to Measurement Error in X and M
• Bias Due to Differential X Variance for Different Levels of M
Differential Reliability
• categorical moderator
• differential variances in X
• If measurement error in X, then reliability of X varies, biasing the two slopes differentially.
• Multiple groups SEM model should be considered