26. Moderation

Author

Affiliation

Johnny van Doorn

University of Amsterdam

Published

November 7, 2025

In this lecture we aim to:

Look at interactions between continuous predictors
Introduce the PROCESS module
Demonstrate in JASP

Reading: Chapter 10

Moderation

In statistics and regression analysis, moderation occurs when the relationship between two variables depends on a third variable. The third variable is referred to as the moderator variable or simply the moderator. The effect of a moderating variable is characterized statistically as an interaction.

Source WIKIPEDIA

Model

\(\definecolor{red}{RGB}{255,0,0} \definecolor{black}{RGB}{0,0,0} \color{black}Out_i = b_0 + b_1 Pred_i + b_2 Mod_i + \color{red}b_3 Pred_i \times Mod_i \color{black}+ \epsilon_i\)

Model

View data

Correlations

cor(data)

            hours.awake   ml.coffee wakefulness
hours.awake  1.00000000 -0.05495344   0.6935652
ml.coffee   -0.05495344  1.00000000   0.6175905
wakefulness  0.69356517  0.61759050   1.0000000

Scatterplots

3D plot

Take it for a spin (does not work on tablet)

1 SD planes

sds       <- c(mean(ml.coffee)+(sd(ml.coffee)*c(-1,0,1)))
quantiles <- as.vector(quantile(ml.coffee,seq(.1,.9,.1)))

Fit model

fit <- lm(wakefulness ~ hours.awake + ml.coffee + hours.awake*ml.coffee); summary(fit)$coefficients

                        Estimate   Std. Error   t value     Pr(>|t|)
(Intercept)           61.6811406 12.103312777  5.096220 1.734165e-06
hours.awake           -3.6114686  1.123014658 -3.215869 1.772655e-03
ml.coffee              0.2656703  0.042873015  6.196679 1.436438e-08
hours.awake:ml.coffee  0.1024285  0.004037068 25.371988 2.357675e-44

Regression equation

\(\definecolor{red}{RGB}{255,0,0} \definecolor{black}{RGB}{0,0,0} \color{black}\widehat{Out_i} = b_0 + b_1 Pred_i + b_2 Mod_i + \color{red}b_3 Pred_i \times Mod_i \color{black}\)

\(\definecolor{red}{RGB}{255,0,0} \definecolor{black}{RGB}{0,0,0} \color{black}\widehat{Out_i} = 61.68 + -3.61 \times Pred_i + 0.27 \times Mod_i + \color{red} 0.1 \times Pred_i \times Mod_i \color{black}\)

\(\definecolor{red}{RGB}{255,0,0} \definecolor{black}{RGB}{0,0,0} \color{black}\widehat{Out_i} = 61.68 + -3.61 \times 15 + 0.27 \times 213 + \color{red} 0.1 \times 15 \times 213 \color{black} \approx 391.36\)

Expected surface

Expected values

\(r^2\) is the proportion of blue to orange, while \(1 - r^2\) is the proportion of red to orange

Expected vs. observed

Turn quantitative into categorical?!

## Round to number of cups of coffee, always rounding down
data$cups.coffee <- data$ml.coffee %/% 150

Demonstration

Moderation in Process module
Moderation in regression module
Equivalences in regression

Closing

Recap

Interactions still matter with continuous predictors
Continuous interactions can be assessed with:
- Linear regression
- PROCESS module

Recommended Exercises

Exercise 10.1, Exercise 10.2

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