Changing the Reference Level in an Interaction Model

Why Change the Reference Level?

In a factor-numeric interaction model, each factor category can have its own intercept and its own slope with respect to the numeric predictor. Treatment coding expresses those differences relative to one chosen reference category. Changing that reference changes the coefficient values, but it does not change the group-specific lines represented by the model.

This distinction matters when a model’s original reference category is not the most useful comparison for interpretation. For example, an analyst may want to describe every category relative to a newly selected benchmark. The key is to calculate each new effect as a difference from the new reference, not to copy the old effects or change their signs indiscriminately.

Model and Coefficients Under Treatment Coding

Consider a response \(Y\), a numeric predictor \(x\), and a factor \(G\) with categories \(A\), \(B\), and \(C\). Assume the conditional mean is linear in \(x\) within each category. With \(A\) as the reference category, write the model as

Here \(I(G=B)\) equals 1 when an observation belongs to category \(B\), and 0 otherwise; \(I(G=C)\) is defined similarly. The conditional-mean lines are therefore

The reference category \(A\) has no indicator terms, so its intercept is \(\beta_0\) and its slope is \(\beta_x\). Category \(B\)'s intercept differs from \(A\)'s by \(\beta_B\), and its slope differs by \(\beta_{Bx}\). The same interpretation applies to \(C\). These are comparisons at the chosen coordinate \(x=0\) for intercepts, and comparisons of change per one-unit increase in \(x\) for slopes.

This algebra applies directly to a linear model for the conditional mean. For a generalized linear model, the same reparameterization applies to the linear predictor \(\eta=g(\mu)\), where \(\mu=E(Y\mid G,x)\) and \(g\) is the link function. In that setting, the transformed coefficients are differences on the link scale; the response-scale means are obtained by applying the inverse link. The formulas below do not claim that response-scale differences remain constant for a nonlinear inverse link.

Deriving the New Coefficients

Suppose category \(B\) becomes the reference. Its line under the original parameterization is \((\beta_0+\beta_B)+(\beta_x+\beta_{Bx})x\), so those two quantities become the new baseline intercept and slope. Each other category’s new effects are its old group-line coefficients minus the corresponding coefficients of \(B\).

The signs follow from subtraction: for instance, \(A\)'s intercept relative to \(B\) is \(\beta_0-(\beta_0+\beta_B)=-\beta_B\). Its slope relative to \(B\) is \(\beta_x-(\beta_x+\beta_{Bx})=-\beta_{Bx}\). Likewise, \(C\)'s intercept relative to \(B\) is \((\beta_0+\beta_C)-(\beta_0+\beta_B)=\beta_C-\beta_B\); its slope difference is \((\beta_x+\beta_{Cx})-(\beta_x+\beta_{Bx})=\beta_{Cx}-\beta_{Bx}\). This is why an effect relative to a new reference is generally a difference between two old effects, not merely one old coefficient with a possible sign change.

More generally, let the line for category \(g\) be \(\alpha_g+\gamma_g x\). If category \(r\) becomes the reference, its new baseline is \(\alpha_r+\gamma_r x\), and category \(g\)'s new intercept and slope effects are \(\alpha_g-\alpha_r\) and \(\gamma_g-\gamma_r\), respectively. This form works for any number of factor levels, provided the same linear-in-\(x\) model is being re-expressed.

What Changes and What Does Not

Re-referencing changes the numerical values and interpretations of individual treatment-coded coefficients. It does not change the category-specific intercepts and slopes when reconstructed, the fitted conditional means, or (for an otherwise identical fitted model) its residuals and overall fit. In estimation, changing the reference is a reparameterization of the same model space: the columns used to describe that space may differ, but the represented group lines do not.

A coefficient test tied to one particular treatment-coded coefficient also changes its comparison when the reference changes. For example, a main-effect coefficient compares intercepts at \(x=0\), and an interaction coefficient compares slopes, each against the current reference. A test of whether all category-specific slopes are equal, however, concerns the same substantive restriction regardless of which category is chosen as reference. Do not confuse a changed coefficient interpretation with a changed fitted model.

The interpretation of an intercept difference depends on the origin of \(x\). If \(x=0\) is outside the range of practical interest, the intercept comparison may be an extrapolation. Centering \(x\) can make that comparison more useful, but centering and changing a factor reference are distinct transformations: centering changes the point at which intercepts are compared, while re-referencing changes the category used for comparison. This tutorial focuses on the latter.

In R, changing a factor's reference with a function such as relevel changes treatment contrasts when the model is fitted again. Provided the model terms and data are otherwise unchanged, the re-expressed model should give the same fitted values. It is good practice to verify a transformed coefficient set by reconstructing the line for each category, rather than relying only on coefficient labels. This topic is aligned with the SOA Exam PA (Predictive Analytics) syllabus.

A Reliable Re-referencing Procedure

Common Exam Traps

• Changing signs without calculating differences. Only a former reference category gets effects that are the negatives of the old reference's effects. For a third category, subtract the new reference's effects from its own.
• Mixing intercept and slope effects. A new main effect is an intercept difference; a new interaction coefficient is a slope difference. Keep their units and calculations distinct.
• Adding the new effects twice. The new baseline already represents the new reference line. To obtain another category's effects, calculate differences from that line, then add those differences to the new baseline once.
• Assuming the coefficients themselves are invariant. Their values depend on the reference coding. The reconstructed category-specific lines and fitted predictions are the invariant quantities.
• Forgetting the scale in a GLM. With a non-identity link, these coefficient differences describe the linear predictor, not generally additive differences in the response mean.