Mediation and Moderation It’s a multivariate world out there… Todd D. Little, Director Institute for Measurement, Methodology, Analysis and Policy

immap.educ.ttu.edu 2 Mediator: middle-person, letter carrier, delivery agent X  M  Y Mediation occurs when X  M  Y is significant regardless of whether X  Y is significant or decreased Moderator: “changer” variable that alters the strength of another relationship (i.e. an interaction!) Moderation occurs when X  Y depends on Z Mediator vs. Moderator

immap.educ.ttu.edu 3 Paths are regression weights M is the mediator, “letter carrier” Traditional Pieces of the Mediation Puzzle

immap.educ.ttu.edu 4 Depicting Moderation in Path Diagram

immap.educ.ttu.edu 5 Mediation, Moderation, Both or Neither Mediation: The transmission of a causal effect by way of a mediator variable. It answers the question "why, by what means, or through what process does X exert its effect on Y?" Moderation: A variable that influences the strength of association between two or more variables Both: A moderator’s influence can be mediated and the parameters of a mediating process can be moderated. Neither (Additive effects): When two or more variables predict an outcome, there are many pathways to the same level of the outcome – this is neither mediation nor moderation.

immap.educ.ttu.edu 6 Baron & Kenny’s Causal Steps approach [BAD] ’ Do a series of regressions and determine if C ’ is < C. Not a test of mediation (ab pathway) Product of Coefficients approach (Sobel test) [NOT SO BAD] Determine if ab is significant using a Wald test Significance test of ab is problematic. Assumes normal distribution of SEs. SEM estimation of indirect effect [NOT GREAT] Determine if ab is significant ab is a parameter in the model using the AP statement (LISREL) or MODEL CONSTRAINT (Mplus) Uses ML or MLR estimation to determine significance of indirect effect Five General Approaches to Mediation

immap.educ.ttu.edu 7 MonteCarlo approach [BETTER] Determine if ab is significant Assumes a and b are normally distributed but not ab Simulates vales of ab based on estimate and SE of a and b See quantpsy.org for a web based calculator Bootstrap approach [BEST] Determine if ab is significant Relies on resampling to determine the appropriate standard error to test for significance Draw 1000s of samples from data with replacement Run model on each bootstrapped sample Get distribution of estimates from samples Automated in Mplus, can be done in other SEM packages LISREL Five General Approaches to Mediation

immap.educ.ttu.edu 8 Cole and Maxwell Most mediation studies use cross-sectional data. Causal conclusions require temporal separation. Cole and Maxwell highlight this problem and suggest modeling strategies that help to mitigate it. The coffin: Cole, D. A., & Maxwell, S. E. (2003). Testing mediational models with longitudinal data: Questions and tips in the use of structural equation modeling. Journal of Abnormal Psychology, 112, The nail: Maxwell, S. E., & Cole, D. A. (2007). Bias in cross sectional analysis of longitudinal mediation. Psychological Methods, 12,

immap.educ.ttu.edu 9 Cole and Maxwell “...testing mediational hypotheses with cross- sectional data will be accurate only under fairly restrictive conditions.... When these conditions do not pertain, cross-sectional studies provide biased and potentially very misleading estimates of mediational processes.” (p. 560) “...the conditions under which cross-sectional data accurately reflect longitudinal mediational effects would seem to be highly restrictive and exceedingly rare.”

immap.educ.ttu.edu 10 A cross-sectional example – later supported – yet still inaccurate estimates of effects

immap.educ.ttu.edu 11 Basic Elements of Full Longitudinal Mediation

immap.educ.ttu.edu 12 Steps in Testing Mediation

immap.educ.ttu.edu 13 The New Minimum Conditions: Half Longitudinal Assuming stationarity, the minimum design for mediation is: Advantages: 1. Explicitly models change in both Y and M 2. Permits experimental manipulation of both X and M.

immap.educ.ttu.edu 14 Half Longitudinal Design Example

immap.educ.ttu.edu 15 A Longitudinal Example – Just Mediation Tested

immap.educ.ttu.edu 16 A Longitudinal Example – Final Model

immap.educ.ttu.edu 17 SEM Moderation using Orthogonalized Product Terms ← Use all orthogonalized interaction terms XZ not correlated with either X or Z Note correlated residuals among the interaction indicators

immap.educ.ttu.edu 18 Longitudinal Moderation in SEM

immap.educ.ttu.edu 19 Longitudinal Moderation - Example

Recommended Readings Little, T. D. (2013). Longitudinal structural equation modeling. New York, NY: Guilford.  What can I say… Cole, D. A., & Maxwell, S. E. (2003). Testing mediational models with longitudinal data: Questions and tips in the use of structural equation modeling. Journal of Abnormal Psychology, 112,  The coffin. Maxwell, S. E., & Cole, D. A. (2007). Bias in cross sectional analysis of longitudinal mediation. Psychological Methods, 12,  The nail in the lid of the coffin. Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process modeling: a regression based approach. New York, NY: Guildford  An integration of mediation and moderation as condition process modeling. Jose, P. E. (2013). Doing statistical mediation and moderation. New York, NY: Guilford.  An accessible first course on mediation and moderation. immap.educ.ttu.edu 20