Building Models: Mediation and Moderation Analysis Dr. Yang Hu yang.hu@Lancaster.ac.uk Dr. Stuart Bedston s.bedston@lancaster.ac.uk

Session overview From separate main effects to the relationship between independent variables—mediation and moderation What is mediation? How to test for mediation? How to interpret and report mediation effects? What is moderation? How to test for moderation? How to interpret and report moderation effects?

Why “building” models? Understand the inter-relations between independent variables Explore the relative importance of independent variables in relating to the outcome dependent variable Testing/detecting explanatory mechanisms Testing the contingencies of relationship between two independent variable based on a third variable …

What is “mediation”? What are the mechanisms underlying a pre-established statistical relationship? A pre-established relationship between two variables (X, Y) is theorised to exist due to an intermediate third variable (“mediator”)

What is “mediation”? An example An example: the relationship between ethnicity and feeling unsafe walking in the dark Model 0 (baseline): Black (British) and Asian (British) groups are more likely to feel unsafe walking in the dark victim of personal crime.

What is “mediation”? An example Is the feeling of unsafety due to their worry about Model 1: Having car stolen? (unlikely) Model 2: Being raped? (might be) Model 3: Being attacked because of skin colour? (likely)

What is “mediation”? An example

What is “mediation”? An example

“Full” vs. “partial” mediation “Full”/”complete” mediation: When the (a priori established) relationship between X and Y reduces in size (and falls to insignificance) when the mediator is considered “Partial” mediation: When the (a priori established) impact of X on Y remains statistically significant when the mediator is considered

Test for mediation There are as many as 14 (!) statistical methods to test hypotheses of Mediation (see Hayes, 2009) Commonly used methods: Multiple regression + Sobel test (http://quantpsy.org/sobel/sobel.htm) Structural equation modelling (SEM)

Test for mediation Conceptual work Step 1: Establish a priori relationship between the independent and dependent variables Step 2: Identify moderator in addition to independent and dependent variables to “explain” the a priori relationship Also consider whether the explanation is partial or full Group work: based on the independent and dependent variables you have identified this morning, find a potential mediator in your list of variables.

Test for mediation Analytical work Step 3: Fit the baseline model without the mediator (a typical main-effects regression model we covered in the preceding session) Step 4: building on the baseline model, add in the moderator *Make sure the models are based on the same sample to be comparable.

Test for mediation

Test for mediation Interpretation Step 5: Difference in coefficients (evaluates the changes in coefficients and their significance when the mediator is added to the model) Step 6: Calculate and interpret the size of moderation (Sobel test: http://quantpsy.org/sobel/sobel.htm)

What is “moderation”? Under what conditions/for whom/when is a pre-established statistical relationship evident? A moderator is a “relationship-modifier”: A pre-established relationship between the IV and DV may differ by the value of the moderator

Mediator vs. moderator Me = Mediator Direct effects Mo = Moderator Moderated effect

What is “moderation”? An example An example: The relationship between worry about being raped and fear for walking in the dark. Model 0: (1) Worry about being raped positively associated with fear for walking in the dark; (2) women > men to fear walking in the dark.

What is “moderation”? An example Are women more fearful of walking in the dark because they are more worried about being raped than men? The relationship between worry about being raped and fear for walking in the dark may be conditional on gender. Gender as a moderator: Worry about being raped is moderated by/interact with gender in relating to fear for walking in the dark.

What is “moderation”? An example Are women more fearful of walking in the dark because they are more worried about being raped than men?

What is “moderation”? An example Predicted probability of fear for walking in the dark, by the interaction between gender and worry about being raped

What is “moderation”? An example Predicted probability of fear for walking in the dark, by the interaction between gender and worry about being raped

What is “moderation”? An example Interaction can involve continuous variables.

Types of Interactions Enhancing Increasing moderator further increases the effect of predictor Buffering Increasing moderator decreases the effect of predictor Antagonistic Increasing moderator reverses the effect of predictor

Test for moderation Moderation is often tested through the inclusion of interaction terms in models Commonly used methods: Full interaction approach Separate-slope approach

Test for moderation Conceptual work Step 1: Establish a priori relationship between the independent and dependent variables Step 2: Identify moderator by which the relationship between independent and dependent variables may vary Also consider the type of moderation Group work: based on the independent and dependent variables you have identified this morning, find a potential mediator in your list of variables.

Test for moderation Analytical work Moderation is often tested through the inclusion of interaction terms in models Step 3: Fit the baseline model without the moderator (a typical main- effects regression model we covered in the preceding session) Step 4: building on the baseline model, add in the interaction term between the moderator and the predictor of interest *Make sure the models are based on the same sample to be comparable.

Test for moderation—How to specify an interaction term Most statistical packages allow for the direct specification of interaction terms using math symbols such as *(R) and # (Stata). But you can also create your own. Mean-centre your predictor [X] and moderator [Z] variables Construct a new ‘interaction’ variable of the form: predictor x (multiply) moderator [XZ] Use this variable as a predictor of your outcome(s) [Y] along with the original variables [X, Z] (Y=X+Z+XZ [+e])

Test for moderation Interpretation Step 5: Interpret the effects of predictor and moderator variables Step 6: Test the magnitude and statistical significance of moderator effect Look at whether adding the interaction leads to a significant improvement in how well the regression is performing (e.g. r2) The F-test to see whether the improvement is significant where f is the number of parameters in the full model (i.e. with interaction effects), r is the number of parameters in the reduced model (i.e. without interaction effects) and N is sample size

Test for moderation Plotting Step 7: If the interaction is significant then we can look at the effect of our predictor variable at representative levels of the moderator variable

Moderated mediation and mediated moderation A variable can be both a mediator and a moderator Mediation analysis and moderation analysis can be used in conjunction, it’s not a matter of one or the other.