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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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what are they
What Are They?

Causality

Linearity

Homogeneity of Variance

No Measurement Error

causality
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
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.
graphing
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
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
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.
slide17
Nonlinear Relationship

Y

X

Moderation

Y

X

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 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
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
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
Additional Webinars
  • Effect Size and Power
  • ModText
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