Bayesian Moderation Analysis

Summary

Moderation analysis tests whether the relationship between a predictor $x$ and outcome $y$ changes as a function of a third variable $m$ (the moderator). It is implemented as multiple regression with an interaction term. The Bayesian approach yields a posterior over the moderation coefficient ${\beta }_2$, enabling probabilistic statements about the moderation effect.

What is Moderation?

A moderator $m$ changes the slope of $x$ on $y$ — not the mean level of $y$ (that would be a main effect). Schematically:

m ──→ [x → y]   (m moderates the x-y relationship)

Moderation vs. Mediation

• Moderation: $m$ changes the strength of the $x\to y$ relationship. No causal path $x\to m\to y$ implied.
• Mediation: $x$ affects $y$ (partly) through $m$. Requires causal DAG reasoning.

See the PyMC mediation analysis example for contrast.

The Model

$$y_i\sim \mathcal{N}({\mu }_i,{\sigma }^2)$$ $${\mu }_i={\beta }_0+{\beta }_1{x}_i+{\beta }_2(x_i\cdot m_i)+{\beta }_3{m}_i$$

Interpretation of parameters:

Parameter	Interpretation
${\beta }_0$	Intercept (value of $y$ when $x=0$, $m=0$)
${\beta }_1$	Effect of $x$ on $y$ when $m=0$
${\beta }_2$	Moderation coefficient: how much the slope ${\beta }_1$ changes per unit of $m$
${\beta }_3$	Main effect of $m$ on $y$ (controlling for $x$)
$\sigma$	Residual SD

The total effect of $x$ on $y$ at a given level of moderator $m$ is:

$$f(m)={\beta }_1+{\beta }_2\cdot m$$

PyMC Implementation

with pm.Model() as model:
    x = pm.ConstantData("x", training_hours)
    m = pm.ConstantData("m", age)
 
    β0 = pm.Normal("β0", mu=0, sigma=10)
    β1 = pm.Normal("β1", mu=0, sigma=10)
    β2 = pm.Normal("β2", mu=0, sigma=10)
    β3 = pm.Normal("β3", mu=0, sigma=10)
    σ  = pm.HalfCauchy("σ", 1)
 
    mu = β0 + β1*x + β2*x*m + β3*m
    pm.Normal("y", mu=mu, sigma=σ, observed=muscle_pct)

Visualisation: Spotlight Graph

The spotlight graph plots $f(m)={\beta }_1+{\beta }_2\cdot m$ as a function of $m$, at selected percentiles of the moderator. This directly visualises how the slope changes:

• If ${\beta }_2<0$: slope decreases as $m$ increases (training less effective with age)
• If ${\beta }_2\approx 0$: no moderation — the $x$-$y$ relationship is constant across $m$

# Posterior estimate of the moderation effect
xi = np.linspace(min(age), max(age), 20)
rate = posterior.β1 + posterior.β2 * xi  # how β1 varies with m

Multicollinearity and the Interaction Term

Including $x\cdot m$ alongside $x$ and $m$ introduces multicollinearity — the interaction term is correlated with its component variables. Options:

• Mean-centering $x$ and $m$ before computing the product reduces multicollinearity
• Despite common concern, multicollinearity in the interaction term does not bias estimates of the moderation effect ${\beta }_2$ — it only increases uncertainty (widens credible intervals)
• McClelland et al. (2017): multicollinearity is a “red herring in the search for moderator variables”

Applied Example: Training × Age on Muscle Mass

• $x$ = weekly training hours
• $m$ = age (moderator)
• $y$ = muscle percentage

Finding: ${\beta }_2<0$ (credibly), meaning training becomes less effective at building muscle mass in older individuals.

Connections

• Spurious Association and Confounds — interaction effects and multivariate regression
• Bayesian Linear Regression — priors as regularization for correlated predictors
• Generalized Linear Models — moderation extends naturally to logistic/Poisson regression
• Bayesian Non-parametric Causal Inference — non-parametric alternative when the interaction form is unknown

Source

• Bayesian moderation analysis — PyMC example by Benjamin T. Vincent (2021–2023)
• Hayes (2017): Introduction to Mediation, Moderation, and Conditional Process Analysis