Moderation: Interpretation

David A. Kenny

The Moderated Regression Equation

• Y = aX + bM + cXM + Ea = “main effect” of Xb = “main effect” of Mc = interaction between X and

M • Important to include both X and M

in the model.

Interpretation of the Two Main Effects

a: “main effect” of XThe effect of X when M is zero.

b: “main effect” of MThe effect of M when X is zero.

For these effects to be meaningful, zero must be a meaningful value.

Centering

• Make sure zero in interpretable for X and M.

• If not then center: Compute the grand mean of X and M and subtract it (or something close).

Interpretation of the Interaction Effect (Moderation)

c = the effect of XM

For every one unit change of M, the effect of X (i.e., increases (or decreases) by c units.

Path c represents then how much a changes.

The Moderated Effect of X on Y

• Y = aX + bM + cXM + E• Y = (a + cM)X + bM + E• The effect of X on Y is equal to a +

cM.• If we set a + cM to zero, then X has

no effect on Y when M = -a/c.

Brief Examplestress level or Shappiness or Hsocio-economic status (C)

H = -2S + 1C + 0.1SC + E

For every one unit change of socioeconomic status, the effect of Stress levels on Happiness weakens by 0.1 units. Note that for C = 20, the effect of stress on happiness is zero.

Other Strategies for Interpretation

•Computing simple effects•Graphing the moderation

Simple Effects

• The effect that X has on Y at different levels of M.

• Sometimes called “pick a point.”• Two ways to compute:• Plug into a + cM the value for M

and solve.• Complicated to obtain a standard

error for this value.

Re-centering

• Mʹ is desired value of M for which a simple effect is desired.

• Re-estimate the moderated regression equation using M - Mʹ for M.

• The new value of a will be the simple effect of X when M = Mʹ.

• Note this gives a p value and a confidence interval.

What To Do If M Is Continuous?

• Pick two values• Typically one standard deviation

above and below the mean of M.• Make sure the values are possible

values.

GraphingBetter than simple effects because it displays all of the effects.How?

Compute the least-squares means for each of the four combinations of

two values of Xtwo values of M

Place in the graph.Connect the sets of two points with

a common value of M.

Graphing Programs

Paul Jose’s ModGraph http://www.victoria.ac.nz/psyc/paul-jose-files/modgraph/

Andrew Hayes’s ModProbehttp://www.afhayes.com/spss-sas-and-mplus-macros-and-code.html#modprobe

Regions of Significance

• Johnson-Neyman• Potthoff extension

Regions of Significance

• Useful if M is measured at the interval level of measurement.

• Tells you at what intervals of M is the X Y effect significant.

• It might tell you the when M is at least 5, the effect is positive and significant, but when M is less than 3, the effect is negative and significant.

• Can flip X and M.

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Additional Webinars

• Assumptions• Effect Size and Power• ModText