Moderation: Assumptions

David A. Kenny

What Are They?CausalityLinearityHomogeneity of VarianceNo 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 Y changes 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

Moderation

X

X

Y

Y

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

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

• 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 + rxm

2)–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

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