Reference Levels in Two-Factor Interaction Models

Why Two Reference Levels Matter

When a model includes two categorical predictors and their interaction, interpreting a coefficient requires knowing the reference level of each predictor. Together, those two reference levels identify one baseline cell: the combination of categories represented by the intercept. Main-effect coefficients compare cells that differ from this baseline along one factor, while an interaction coefficient describes how a difference along one factor changes across levels of the other.

The same fitted cell means can be represented with different reference levels and different coefficients. The predictions do not change; the comparisons attached to the coefficients do. This distinction is useful when explaining a model to a business audience and when checking whether a reported coefficient answers the intended question.

The Two-Factor Treatment-Coded Model

First consider two factors with two levels each: \(A\) has reference level \(a_0\) and alternative level \(a_1\); \(B\) has reference level \(b_0\) and alternative level \(b_1\). Define \(I_A=1\) when \(A=a_1\), and \(I_B=1\) when \(B=b_1\); otherwise each indicator is zero. The treatment-coded linear model is

This is an identity-link model for a numerical response. The fitted means and coefficient interpretations below follow from the model’s coding. For ordinary least-squares inference, the usual assumptions concern the errors: conditional mean zero, independent observations, and constant error variance for the standard variance estimates. Those assumptions matter for inference, not for the algebra that translates coefficients into fitted means. With a different link function, the same coding applies to the linear predictor, but coefficients generally describe changes on the link scale rather than direct differences in response means.

The four fitted means follow by substituting the indicator values:

Thus, \(\beta_A\) is the \(A\)-level difference when \(B=b_0\), and \(\beta_B\) is the \(B\)-level difference when \(A=a_0\). Neither main effect is automatically an overall or average difference across all levels of the other factor.

Why the Interaction Is a Difference in Differences

The interaction coefficient can be derived directly from the four cell means. At \(B=b_0\), the difference between the \(A\) levels is \(\widehat{\mu}_{a_1b_0}-\widehat{\mu}_{a_0b_0}=\beta_A\). At \(B=b_1\), the corresponding difference is \(\widehat{\mu}_{a_1b_1}-\widehat{\mu}_{a_0b_1}=\beta_A+\beta_{AB}\). Subtracting the first difference from the second cancels \(\beta_A\), leaving the interaction:

This difference-in-differences form is a useful check: if the effect of moving from \(a_0\) to \(a_1\) is the same at both levels of \(B\), the interaction is zero. For factors with more than two levels, each non-reference level has its own indicator; each interaction coefficient compares the relevant pair of level changes with the baseline cell.

More Than Two Levels

If a factor has several levels, treatment coding creates one indicator for each non-reference level. In a two-factor model, every combination of non-reference indicators also has an interaction term. The intercept remains the baseline cell mean. A main effect for a particular level is the difference from the reference level of that factor when the other factor is at its reference level. An interaction coefficient is the additional adjustment for the joint combination beyond adding those two baseline comparisons.

Changing the Reference Levels

Changing either reference level changes what the intercept and main effects compare. It does not change the fitted values for the cells when the model is refit with the same interaction structure. In a two-by-two table, changing just one reference factor reverses the sign of the corresponding interaction coefficient; changing both reference factors leaves its sign unchanged. These statements follow from the difference-in-differences expression: reversing one contrast reverses the expression’s sign, while reversing both contrasts preserves it.

Using Reference Levels in R

In R, a formula such as lm(y ~ A * B, data = d) includes the two main effects and their interaction. For unordered factors using treatment contrasts, the factor reference levels determine the baseline cell. Set them deliberately before fitting, for example with relevel(A, ref = "East") and relevel(B, ref = "Standard"). The exact level names must match the data.

When interpreting output, first confirm the factor levels and their references, then identify the baseline combination. Do not infer the baseline from the order in which categories appear in a report or plot. The model’s design matrix determines the coding; a changed reference can alter coefficient labels and values without changing the fitted cell predictions.

Common Exam Trap

A frequent error is to call the difference between a non-baseline cell and the baseline cell an interaction. That is a total cell difference: it includes both main effects and the interaction. For the first worked example, West/Premium minus East/Standard is \(79-80=-1\). The interaction is not \(-1\); it is \(-2\), found by comparing the West-versus-East difference across Premium and Standard: \((79-75)-(86-80)=4-6=-2\).

Another error is to interpret a main effect as applying uniformly across the other factor. Here, the West effect is \(6\) under Standard but \(4\) under Premium. The coefficient \(6\) is specifically the West effect at the Standard reference level; the interaction supplies the adjustment at Premium. State the reference levels when describing a main effect, and verify an interaction with the relevant four cells.