7.2 Symmetry of the linear interaction

Interactions are symmetric. If the effect of $x$ depends on $z$ , then the effect of $z$ also depends on $x$ . This symmetry has important consequences for interpretation and visualization.

7.2.1 The symmetry

The linear interaction model is:

$\mu_i = \alpha + \beta_x x_i + \beta_z z_i + \beta_{xz} x_i z_i$

The slope of $x$ is:

$\frac{\partial \mu}{\partial x} = \beta_x + \beta_{xz} z$

And the slope of $z$ is:

$\frac{\partial \mu}{\partial z} = \beta_z + \beta_{xz} x$

The same interaction term $\beta_{xz}$ appears in both. The interaction is symmetric: it is equally true that "the effect of $x$ depends on $z$ " and "the effect of $z$ depends on $x$ ."

# Demonstrate the symmetry of interactions
set.seed(42)
n <- 200
study <- runif(n, 0, 10)
sleep <- runif(n, 4, 9)
score <- 50 + 3 * study + 2 * sleep + 0.5 * study * sleep + rnorm(n, sd = 5)

m_int <- lm(score ~ study * sleep)
bx <- coef(m_int)[2]
bz <- coef(m_int)[3]
bxz <- coef(m_int)[4]

cat("Interaction model:\n")
cat(" beta_study:", round(bx, 3), "\n")
cat(" beta_sleep:", round(bz, 3), "\n")
cat(" beta_study:sleep:", round(bxz, 3), "\n")

cat("\nSlope of study at sleep = 4:", round(bx + bxz _ 4, 3), "\n")
cat("Slope of study at sleep = 6:", round(bx + bxz _ 6, 3), "\n")
cat("Slope of study at sleep = 8:", round(bx + bxz * 8, 3), "\n")

cat("\nSlope of sleep at study = 0:", round(bz + bxz _ 0, 3), "\n")
cat("Slope of sleep at study = 5:", round(bz + bxz _ 5, 3), "\n")
cat("Slope of sleep at study = 10:", round(bz + bxz * 10, 3), "\n")

cat("\nThe interaction is symmetric:\n")
cat(" Effect of study depends on sleep\n")
cat(" Effect of sleep depends on study\n")
cat(" Same coefficient (beta_xz) governs both\n")

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7.2.2 Visualizing the symmetry

We can visualize the interaction from either perspective: plot the effect of $x$ at different values of $z$ , or plot the effect of $z$ at different values of $x$ . Both show the same interaction.

# Visualize the symmetry of interactions
set.seed(42)
n <- 200
study <- runif(n, 0, 10)
sleep <- runif(n, 4, 9)
score <- 50 + 3 * study + 2 * sleep + 0.5 * study * sleep + rnorm(n, sd = 5)

m_int <- lm(score ~ study * sleep)

par(mfrow = c(1, 2))

# Left: effect of study at different sleep levels

study_seq <- seq(0, 10, length.out = 100)
colors <- c("firebrick", "steelblue", "darkgreen")
sleep_vals <- c(4, 6, 8)

plot(NULL, xlim = c(0, 10), ylim = c(40, 130),
xlab = "Study hours", ylab = "Score",
main = "Effect of study at different sleep levels")
for (i in 1:3) {
pred <- coef(m*int)[1] + coef(m_int)[2] * study*seq +
coef(m_int)[3] * sleep*vals[i] +
coef(m_int)[4] * study*seq * sleep_vals[i]
lines(study_seq, pred, lwd = 2, col = colors[i])
}
legend("topleft", legend = paste("Sleep =", sleep_vals), col = colors, lwd = 2)

# Right: effect of sleep at different study levels

sleep_seq <- seq(4, 9, length.out = 100)
study_vals <- c(0, 5, 10)

plot(NULL, xlim = c(4, 9), ylim = c(40, 130),
xlab = "Sleep hours", ylab = "Score",
main = "Effect of sleep at different study levels")
for (i in 1:3) {
pred <- coef(m*int)[1] + coef(m_int)[2] * study*vals[i] +
coef(m_int)[3] * sleep*seq +
coef(m_int)[4] * study*vals[i] * sleep_seq
lines(sleep_seq, pred, lwd = 2, col = colors[i])
}
legend("topleft", legend = paste("Study =", study_vals), col = colors, lwd = 2)

par(mfrow = c(1, 1))

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Rethinking

The symmetry is not a choice — it is a mathematical fact. The interaction term $x \cdot z$ is the same as $z \cdot x$ . There is no "moderator" and "moderated" variable — both moderate each other equally. When we say "the effect of $x$ depends on $z$ ," we are also saying "the effect of $z$ depends on $x$ ." The choice of which variable to plot on the x-axis is a presentation choice, not a statistical one. Both perspectives show the same interaction.

Summary of Section 7.2

Interactions are symmetric: the effect of $x$ depends on $z$ and vice versa.
The same coefficient $\beta_{xz}$ governs both perspectives.
The slope of $x$ is $\beta_x + \beta_{xz} z$ ; the slope of $z$ is $\beta_z + \beta_{xz} x$ .
Visualizing from either perspective shows the same interaction.
There is no "moderator" vs "moderated" — both moderate each other.

Building an interaction

Continuous interactions