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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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Moderation assumptions
Moderation: Assumptions

David A. Kenny

What are they
What Are They?



Homogeneity of Variance

No Measurement Error


  • 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
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
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
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
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
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.


  • 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.


  • 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
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
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
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
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
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






Homogeneity of variance
Homogeneity of Variance

  • Variance in Moderation Analysis

    • X

    • Y (actually the errors in Y)

Different variance in x for levels of m
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
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

Violation of equal error variance assumption categorical moderator
Violation of Equal Error Variance Assumption: Categorical Moderator

  • 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.

Violation of equal error variance assumption continuous moderator
Violation of Equal Error Variance Assumption: Continuous Moderator

  • 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
Variance Differences as Moderatora 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
Measurement Error Moderator

  • 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
Differential Reliability Moderator

  • 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

Additional webinars
Additional Webinars Moderator

  • Effect Size and Power

  • ModText