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Moderation: Assumptions

David A. Kenny

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.

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

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