Centering Continuous Predictors in Interaction Models

Why Center a Predictor in an Interaction Model?

In a factor-by-continuous interaction model, the coefficient for a factor describes a comparison at a particular value of the continuous predictor: the value coded as zero. If zero is far outside the data or has no useful scientific meaning, that comparison can be hard to interpret. Centering replaces zero with a chosen reference value, such as a clinically meaningful measurement or the sample mean.

Centering is a change of coordinates, not a change to the modeled relationship. The factor-specific slopes, their differences, and the fitted linear predictors can all be preserved. What changes is the interpretation—and numerical value—of the intercept and factor main-effect coefficients.

Coefficient Transformation

Consider a GLM with link function \(g\), where \(g(\mu_i)=\eta_i\). Let \(z_i\) be a vector of coded columns for a factor, and let \(x_i\) be a continuous predictor. A factor-by-continuous interaction model can be written as

Here \(\alpha\) is the intercept, \(\gamma\) contains factor main-effect coefficients, \(\beta\) is the continuous-predictor coefficient for the reference factor profile, and \(\delta\) contains interaction coefficients. For a binary factor under reference coding, \(z_i\) is a single indicator: it is zero for the reference group and one for the other group.

Since \(x_i=x_{c,i}+c\), substitution gives

Thus the centered coefficients are \(\alpha_c=\alpha+c\beta\), \(\gamma_c=\gamma+c\delta\), \(\beta_c=\beta\), and \(\delta_c=\delta\). In particular, the factor main-effect coefficients generally change because the factor comparison is now evaluated at \(x=c\). The interaction coefficients do not change.

What the Coefficients Mean After Centering

Under reference coding, the reference group has \(z_i=0\), so its linear predictor is \(\alpha+\beta x_i\). A group represented by \(z_i\) has linear predictor \(\alpha+z_i^{\mathsf T}\gamma+(\beta+z_i^{\mathsf T}\delta)x_i\). Its slope is therefore \(\beta+z_i^{\mathsf T}\delta\), and the slope difference from the reference group is \(z_i^{\mathsf T}\delta\).

At the centering value \(x_i=c\), the difference between that group’s and the reference group’s linear predictors is \(z_i^{\mathsf T}(\gamma+c\delta)\). This is why the centered factor coefficient is \(\gamma+c\delta\): it describes the factor contrast at the chosen reference point on the linear-predictor scale. In a logistic model, for example, this is a contrast in log odds, not a constant difference in probabilities.

Why the Fitted Linear Predictors Are Preserved

Proof. By definition, \(x_i=x_{c,i}+c\). Substituting this identity into the original predictor and collecting terms gives

The final expression is precisely the centered model with the stated transformed coefficients. It equals the original \(\eta_i\) for each observation. Therefore the model’s implied mean, as specified by its link, is also unchanged. The transformation is reversible: \(\alpha=\alpha_c-c\beta_c\) and \(\gamma=\gamma_c-c\delta_c\), while \(\beta=\beta_c\) and \(\delta=\delta_c\). Thus centering re-expresses the same linear predictors rather than restricting the model. \(\square\)

This equality also explains why fitting the same GLM with \(x\) or \(x_c\) gives the same attainable fitted linear predictors, provided the model includes the same factor terms and interactions and the coefficient space is unrestricted. The statement is about reparameterization; it does not assert that numerical algorithms must follow identical computational paths.

Derivation. For factor code \(z\), the coefficient of \(x_i\) in its linear predictor is \(\beta+z^{\mathsf T}\delta\). After centering, the coefficient of \(x_{c,i}\) is \(\beta_c+z^{\mathsf T}\delta_c=\beta+z^{\mathsf T}\delta\). For two factor profiles \(z_a\) and \(z_b\), their slope difference is

The centered model has the same expression because \(\delta_c=\delta\). Hence both slopes and their difference are preserved. \(\square\)

Common Mistake and Practical Use

A frequent error is to update only the intercept when centering, while leaving the factor main-effect coefficients unchanged. That misses the contribution of the interaction at the new zero point. In the binary-factor example, changing the intercept from \(10\) to \(20\) but keeping the factor coefficient at \(4\) would claim that the groups differ by \(4\) at \(x=5\). The actual difference there is \(4+3(5)=19\). The correct centered factor coefficient must include the interaction coefficient multiplied by the centering constant.

Centering is useful when zero is not a meaningful predictor value, when a comparison at a scientifically relevant value is desired, or when a predictor’s numerical scale makes the intercept difficult to read. It does not remove the interaction, change the link, or make the factor contrast constant across values of \(x\). It simply states the main effects at a different reference value.