Interactions Between Quantitative Predictors

Why Let a Slope Change?

In an additive multiple regression model, the fitted change in the response associated with one predictor is the same at every fixed value of the other predictors. That may be too restrictive. For example, an additional hour of study might be associated with a larger score increase for students with stronger preparation than for students with weaker preparation. An interaction between two quantitative predictors represents this kind of changing slope.

The interaction is represented by adding the product of the two predictors to the regression equation. The product term does not merely add another independent predictor: it changes how the fitted response varies with each of the original predictors.

The Model and Its Coefficients

Let \(X_1\) and \(X_2\) be quantitative predictors and \(Y\) the response. A multiple regression model with their interaction is

Here, \(\beta_0,\beta_1,\beta_2,\beta_3\) are model coefficients and \(\varepsilon\) is the error: the difference between an observed response and the conditional mean specified by the model. Equivalently, the model says

The conditional mean is linear in the coefficients, but the relationship between \(Y\) and one predictor depends on the other predictor. In practice, the coefficients are estimated from data; the fitted model uses estimates such as \(b_0,b_1,b_2,b_3\) in place of the corresponding population coefficients.

To interpret the slopes, hold one predictor fixed and examine the conditional mean as the other changes. Holding \(X_2=x_2\) fixed gives

Thus, the fitted slope for \(X_1\), at a particular value \(x_2\) of \(X_2\), is \(\beta_1+\beta_3x_2\). Reversing the roles of the predictors shows that the fitted slope for \(X_2\), at \(X_1=x_1\), is \(\beta_2+\beta_3x_1\). The two slopes are conditional: each describes a change in fitted mean response while the other predictor is held fixed.

The coefficients \(\beta_1\) and \(\beta_2\) are not generally the slopes across all predictor values. Specifically, \(\beta_1\) is the fitted slope for \(X_1\) when \(X_2=0\), and \(\beta_2\) is the fitted slope for \(X_2\) when \(X_1=0\). If zero is outside the observed or meaningful range of a predictor, these main-effect coefficients may be difficult to interpret directly. Centering predictors can help, as shown below.

Worked Examples

Why the Product Term Changes the Slope

The slope formulas follow by collecting the terms involving one predictor. For fixed \(X_2=x_2\), the parts of the conditional mean involving \(X_1\) are \(\beta_1X_1+\beta_3X_1x_2\). Factoring out \(X_1\) gives \((\beta_1+\beta_3x_2)X_1\), so the coefficient of \(X_1\)—and thus its fitted slope—is \(\beta_1+\beta_3x_2\). The same reasoning with \(X_1=x_1\) fixed gives the slope \(\beta_2+\beta_3x_1\) for \(X_2\).

A finite-change version makes the interpretation especially clear. If \(X_2\) increases by \(\Delta x_2\), the fitted slope for \(X_1\) changes by

This is why \(\beta_3\) is described as a change in slope per unit of the other predictor. It is not, by itself, the slope for either predictor or the predicted response change for an observation.

Centering Predictors

Centering replaces each predictor with its difference from a chosen reference value. It can make the main-effect coefficients more useful when a predictor’s zero is not meaningful. In the study-time example, let \(U=X_1-5\) and \(V=X_2-60\), so \(U\) measures hours relative to 5 and \(V\) measures preparation points relative to 60. Substituting \(X_1=U+5\) and \(X_2=V+60\) into the fitted equation gives

For example, the intercept is 66 points: the predicted score at 5 study hours and 60 preparation points. The coefficient 7.2 is now the fitted study-time slope at a preparation score of 60, rather than at a preparation score of zero. The coefficient 0.70 is the fitted preparation-score slope at 5 study hours. The interaction coefficient remains 0.08, and centering does not change fitted predictions: the new variables simply express the same predictor values relative to reference points.

Assumptions and Common Pitfalls

A regression interpretation depends on the model being appropriate for the question and data. The interaction equation assumes that, at each fixed value of one predictor, the conditional mean is linear in the other predictor, with its slope changing linearly according to the interaction term. Interpretations should be restricted to predictor values for which the model is supported by data; a computed slope far beyond the observed range may be an unreliable extrapolation.

For usual regression inference, additional conditions are needed for the error term. In particular, the errors should have conditional mean zero and be independent with constant variance; normal errors are used for exact small-sample \(t\)-based inference. A fitted product-term coefficient alone does not establish that an interaction is statistically distinguishable from zero. Testing that coefficient is a separate inferential question.

• Misreading a main effect: Saying “the study-time slope is 2.4 points per hour” in the first example ignores the product term. The correct slope is \(2.4+0.08X_2\), so it must be stated at a preparation score.
• Misreading the product coefficient: The value 0.08 is not the score increase from one more study hour. It is the change in the study-time slope per additional preparation point.
• Forgetting units: A slope has response units per predictor unit; the interaction coefficient has response units per product of predictor units.
• Claiming causation: A fitted interaction describes how an association changes across predictor values. Without an appropriate causal design and assumptions, it does not show that changing a predictor causes the response to change.