22. Mediation

Johnny van Doorn

University of Amsterdam

28 October 2026

In this lecture we aim to:

Explore mediation analysis
Link it to regression
Introduce the Process module
Demonstrate in JASP

Reading: Chapter 10 (§10.2, §10.4–10.7)

Applet

Link

Multiple regression

\(\widehat{\text{album sales}} = b_0 + b_1 \text{airplay} + b_2 \text{adverts}\)

Boxes are variables, lines are \(b\)s
No arrows, so no causal claim

Mediation

In statistics, a mediation model is one that seeks to identify and explain the mechanism or process that underlies an observed relationship between an independent variable and a dependent variable via the inclusion of a third hypothetical variable, known as a mediator variable (also a mediating variable, intermediary variable, or intervening variable).

Source: WIKIPEDIA

Shared variance

Back to lecture 6. Do anxious students simply revise less?

Predictor anxiety, mediator revision
Total effect, regions \(A + C\)
Path \(a\), regions \(C + D\)
Path \(b\), region \(B\)
Path \(c\), region \(A\)
Shared by both predictors, region \(C\)

Example

Tal-Or et al. (2010, Study 2, \(n\) = 123): Testing causal direction in the influence of presumed media influence

Does an article about a sugar shortage on the front page increase the intention to buy sugar, or is this effect mediated by presumed media influence?

Condition: article on the front page (1) or an inside page (0), randomized (predictor)
MediaInfluence: mean of 2 questionnaire items, 1 to 7 (mediator)
- How much they think others will buy sugar after reading the article
Intention: mean of 4 questionnaire items, 1 to 7 (outcome)
- Agreement with possible reactions to the article, such as buying sugar

Only Condition was randomized, so path \(a\) is causal but path \(b\) may be confounded or reversed

Total effect

Total effect:
- \(\widehat{\text{Outcome}} = b_0 + b_t \text{Predictor}_i\)

Mediation Image

Mediation paths

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Total effect:
- \(\widehat{\text{Outcome}} = b_0 + b_t \text{Predictor}_i\)
Mediator effect (a):
- \(\widehat{\text{Mediator}} = b_0 + b_a \text{Predictor}_i\)
Combined effect (b + c):
- \(\widehat{\text{Outcome}} = b_0 + b_{c} \text{Predictor}_i + b_b \text{Mediator}_i\)

The data

Fit 3 models

model.outcome.predictor  <- lm(intention       ~ condition)
model.mediator.predictor <- lm(media.influence ~ condition)
model.combined           <- lm(intention       ~ condition + media.influence)

Total effect:
- \(\widehat{\text{Outcome}} = b_0 + b_t \text{Predictor}_i\)
Mediator effect:
- \(\widehat{\text{Mediator}} = b_0 + b_a \text{Predictor}_i\)
Combined effect:
- \(\widehat{\text{Outcome}} = b_0 + b_{c} \text{Predictor}_i + b_b \text{Mediator}_i\)

Extract beta coefficients

b.outcome.predictor       <- model.outcome.predictor$coefficients[2]   # b.t
b.mediator.predictor      <- model.mediator.predictor$coefficients[2]  # b.a
b.combined.mediator       <- model.combined$coefficients[3]            # b.b
b.combined.predictor      <- model.combined$coefficients[2]            # b.c

Total effect:
- \(\widehat{\text{Outcome}} = b_0 + b_t \text{Predictor}_i\)
Mediator effect:
- \(\widehat{\text{Mediator}} = b_0 + b_a \text{Predictor}_i\)
Combined effect:
- \(\widehat{\text{Outcome}} = b_0 + b_{c} \text{Predictor}_i + b_b \text{Mediator}_i\)

View beta coefficients

Path plot

Visual

Calculate indirect effect

\(a \times b = b_a \times b_b\)

indirect <- b.mediator.predictor * b.combined.mediator; indirect  # a * b

[1] 0.2413355

indirect + b.combined.predictor # a * b + c

[1] 0.4956897

b.outcome.predictor         # total effect

[1] 0.4956897

Effect sizes of mediation

Simplest is \(ab\) itself with its CI, in units of the outcome
Partially standardized, \(\frac{ab}{s_{Outcome}}\)
Standardized (index of mediation), \(\frac{ab}{s_{Outcome}} \times s_{Predictor}\), to compare studies
- Options → Standardized estimates

Is the indirect effect significant?

Test the indirect effect \(a \times b\) itself
JASP gives its estimate, \(p\)-value and 95% CI
- In the Direct and indirect effects table
Significant indirect effect if \(p < \alpha\)
The CI shows how precisely \(a \times b\) is estimated
Assumptions problematic? Bootstrap the indirect effect
- Options → Method → Bootstrap
- Only the CI is bootstrapped, not the \(p\)-value

					95% Confidence Interval
	Estimate	Std. Error	z-value	p	Lower	Upper
Condition → Intention	0.254	0.253	1.007	.314	-0.241	0.750
Condition → MediaInfluence → Intention	0.241	0.127	1.902	.057	-0.007	0.490

Checking the assumptions

Where to check each assumption in JASP (mediator and combined models)¹

Assumption	In JASP	Looks fine when
Multicollinearity (combined model)	Statistics → Coefficients → Tolerance and VIF	all VIFs below 10, average close to 1
Linearity, equal variances	Plots → Residuals vs. predicted	a random cloud around 0
Normality of the residuals	Plots → Q-Q plot standardized residuals → Confidence interval	points close to the line, mostly within the band
Outliers, influential cases	Statistics → Residuals → Casewise diagnostics	about 5% of std. residuals beyond \(\pm 2\) at most, Cook’s distance below 1

Caution with Process models

Rohrer et al (2022). That’s a Lot to Process! Pitfalls of Popular Path Models. Advances in Methods and Practices in Psychological Science.

When we test such hypotheses with the help of path models applied to observational data, we are in the business of causal inference on the basis of observational data