Three-Way Interactions in Group-Specific Slopes

When a Slope Difference Depends on Another Group

Suppose an outcome is related to a quantitative predictor, such as a treatment dose, and the relationship may differ across two categorical factors, such as treatment protocol and age group. It may not be enough to ask whether the dose slope differs by protocol. The protocol-related slope difference might itself be different for younger and older participants. A model needs a term that represents this extra change.

This is the role of a three-way interaction between a quantitative predictor and two categorical factors. It describes whether the difference in slopes across one factor changes across levels of the other factor. It does not, by itself, describe the slope for a particular group or establish that one variable causes another.

Model and Reference Coding

For a clear starting point, let \(x\) be a quantitative predictor and let \(A\) and \(B\) be binary indicators for two categorical factors. Code the chosen reference category of each factor as 0 and its comparison category as 1. For example, \(A=1\) might indicate a new protocol rather than a standard protocol, while \(B=1\) indicates an older rather than younger age group.

The full conditional-mean model is

Equivalently, an individual outcome can be written as this conditional mean plus an error term \(\varepsilon\). For interpreting the model, assume that the mean outcome is represented adequately by the stated equation: within each combination of \(A\) and \(B\), the conditional mean is linear in \(x\). The usual regression conditions for inference also include independent errors with mean zero and constant variance given the predictors; normal errors are used for exact small-sample inference. This tutorial focuses on interpreting the model, not carrying out an inferential test.

Including all lower-order terms matters. The \(xA\) and \(xB\) terms allow the slope to vary with either factor, while \(xAB\) allows the change in slope associated with one factor to vary across the other. The \(A\), \(B\), and \(AB\) terms allow group differences in the conditional mean that do not depend on \(x\). These terms have different roles: the three-way coefficient concerns slopes, not just group mean differences.

How the Three-Way Coefficient Enters the Slope Difference

For fixed values of \(A\) and \(B\), the conditional mean is linear in \(x\). Its slope is the coefficient multiplying \(x\), after accounting for the indicator values. Differentiating the conditional mean with respect to \(x\) gives

To compare the slope for \(A=1\) with the slope for \(A=0\), hold \(B=b\) fixed and subtract. When \(A=0\), the slope is \(\beta_x+\beta_{xB}b\). When \(A=1\), it is \(\beta_x+\beta_{xA}+\beta_{xB}b+\beta_{xAB}b\). Therefore,

This shows precisely how the difference depends on the second factor. At the reference level \(B=0\), the slope difference is \(\beta_{xA}\). At \(B=1\), it is \(\beta_{xA}+\beta_{xAB}\). The difference between these two slope differences is

This subtraction is the key interpretation. The coefficient \(\beta_{xAB}\) is a difference between slope differences—a change in how the slope varies across \(A\), comparing the two levels of \(B\). It is not the slope for any one of the four \(A,B\) combinations.

Worked Interpretations of the Three-Way Term

A Short Derivation: Why This Is a Difference of Differences

Let \(D_A(b)\) denote the slope for \(A=1\) minus the slope for \(A=0\) at \(B=b\). The slope expression derived above gives \(D_A(b)=\beta_{xA}+\beta_{xAB}b\). Evaluate this once at each value of \(B\):

Thus, the three-way coefficient is a difference of two slope differences. The same coefficient can be interpreted in the reverse order: it is the change across \(A\) in the slope difference for \(B\). These are two descriptions of the same contrast because both compare the four predictor slopes using the same alternating subtraction.

Common Mistakes and Why the Model Matters

The three-way term is useful when a single summary of the slope difference across \(A\) would conceal meaningful variation across \(B\). Reporting only \(\beta_{xA}\) as “the slope difference between the \(A\) groups” is incomplete when \(xAB\) is in the model: \(\beta_{xA}\) describes that difference specifically at the reference category \(B=0\). At \(B=1\), the difference includes both coefficients.

• Wrong: “The three-way coefficient is the slope for the \(A=1,B=1\) group.” Correction: It is a difference between slope differences; a group’s slope also depends on the other predictor-slope coefficients.
• Wrong: “\(\beta_{xA}\) is the \(A\)-group slope difference for everyone.” Correction: It is the difference at \(B=0\); at \(B=1\), the difference is \(\beta_{xA}+\beta_{xAB}\).
• Wrong: “A positive three-way coefficient means all slopes are positive.” Correction: Its sign indicates how a slope difference changes across strata, not the signs of the individual slopes.
• Wrong: “The model proves that the factor changes the effect of dose.” Correction: Regression describes conditional associations unless the study design and assumptions support a causal interpretation.

Reference coding determines which stratum-specific slope difference is represented directly by \(\beta_{xA}\), but it does not change fitted predictions. If the reference category changes, the coefficient interpretation changes accordingly; the conditional comparisons can still be calculated from the appropriately coded model.