Moderation: Assumptions David A. Kenny

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 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 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

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 + 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

Additional Webinars Effect Size and Power ModText