# Interpreting main effect with significant interaction term in continuous by continuous multiple regression

I have the classic multiple regression model $$Z=β_0+β_1X+β_2Y+β_3XY$$ that produces output with a significant interaction term. There's a lot of advice out there (and the formal principle of marginality) that says I shouldn't interpret the main (marginal) effects with a significant interaction term. However, in my case both X and Y are continuous variables and I'm struggling to understand what the main effect terms mean in this case and why I shouldn't interpret them.

I do understand why interpreting the main effect (if an interaction is significant) doesn't make sense for categorical data: the "average category" is meaningless because categorical data isn't ordered and discussing an "average bin" is nonsensical. With continuous variables though, it does seem like the main effect term is still a meaningful value because continuous variables have meaningful averages.

This seems to hold up under simulation as well - I can specify a model with both main effect and interaction terms, run a linear regression on it, and recover reasonable estimates for the parameters of both the main effect and the interaction effect. R code below. This implies that the main effect does have some meaning and can be interpreted if the variables are continuous. I do also realize that I cannot report the main term alone, and my discussion of this would be something like "the effect of X alone was estimated as <value> but varied between <upper> and <lower> depending on its interaction with Y". Is this an acceptable way to conclude things from the main term even though there's a significant interaction effect? Or is there something I'm missing?

set.seed(123)
x_vals <- (-10):10
y_vals <- (-10):10
mr_data <- expand.grid(x_vals, y_vals)
mr_data <- setNames(mr_data, c("x_vals", "y_vals"))
beta_1 <- 2
beta_2 <- 3
beta_3 <- 1
mr_data$z_vals <- beta_1*mr_data$x_vals + # Main effect of x
beta_2*mr_data$y_vals + # Main effect of y beta_3*mr_data$x_vals*mr_data$y_vals + # Interaction effect rnorm(441) # Some noise so lm() doesn't complain lm(z_vals~x_vals*y_vals, data = mr_data) # Coefs are ~same as betas above  Coefficients: (Intercept) x_vals y_vals x_vals:y_vals 0.01216 1.99236 2.99894 0.99953  • What your model and coefficients seem to be suggesting is that$z$seems to be positively associated with both$x$and$y$, and increasing$x$by$1$tends to increase$z$by about$2+y$, while increasing$y$by$1$tends to increase$z$by about$3+x$. But since$x$and$y$can be as negative as$-10$, this may be a confusing way of putting it Commented Jun 2, 2022 at 18:46 • The problem is that the most negative$z$happens not when$x=-10,y=-10$but when$x=+10, y=-10$; this is because when$y$is very negative the relationship between$z$and$x\$ is not positive Commented Jun 2, 2022 at 19:00
• The “main effect” reported for X is its association with outcome when Y = 0, and vice-versa. So re-centering Y will affect the “main effect” of X, and vice-versa. Any interpretation must be informed by that fact.
– EdM
Commented Jun 2, 2022 at 20:56