Coding Factorial Fixed Effects in Mixed Models

Why Coding Matters in a Mixed Model

A factorial mixed model separates the structure of the observations from the way its fixed treatment means are labelled. The random-effects part describes dependence, such as shared variation among observations in the same block. The fixed-effects part represents the treatment combinations. Coding determines how the coefficients in that fixed part combine to produce each treatment-combination mean.

The same set of cell means can be represented using a reference cell or using sum-to-zero contrasts. The fitted means do not change when the coding is changed, but individual coefficients do. This matters when interpreting output or stating a hypothesis: a single reference-coded coefficient may describe an effect at one level of the other factor, not an overall main effect.

The Mixed Model and Reference-Cell Coding

Consider two fixed factors, \(A\) and \(B\). Let \(A\) have levels \(A_0,A_1\), and let \(B\) have levels \(B_0,B_1,B_2\). Suppose observations are collected in blocks, with each block containing observations for all six treatment combinations. A random-intercept block model is

Here \(i\) indexes blocks, \(j\) indexes the levels of \(A\), and \(k\) indexes the levels of \(B\). The random block effects are independent across blocks; the errors are independent across observations and independent of the block effects. The model assumes normally distributed random effects and errors with the stated variances, and a common residual variance. The factors are treated as fixed: their levels are the specific levels whose means are being compared. Coding changes only the fixed-mean representation, not these random-effects assumptions.

Choose \(A_0\) and \(B_0\) as reference levels. The reference-cell model is

The symbol \(\mu_{jk}\) denotes the fixed mean for cell \((A_j,B_k)\). Thus \(\beta_0\) is the reference-cell mean, \(\beta_A\) is the difference between \(A_1\) and \(A_0\) at \(B_0\), and \(\beta_{B1}\) and \(\beta_{B2}\) are differences from \(B_0\) at \(A_0\). Each interaction coefficient is a difference-in-differences. For example, \(\beta_{AB1}\) measures how much the \(A_1\)-versus-\(A_0\) difference at \(B_1\) departs from that difference at \(B_0\).

Sum-to-Zero Coding

For \(A\), assign scores \(-1\) to \(A_0\) and \(+1\) to \(A_1\). For \(B\), use two contrast columns with level scores \((1,0)\) for \(B_0\), \((0,1)\) for \(B_1\), and \((-1,-1)\) for \(B_2\). Each contrast column sums to zero across its factor levels. Denote the \(A\) score by \(x_A\), and the two \(B\) scores by \(z_1,z_2\). The fixed mean is

The interaction columns are products of the corresponding factor scores. Because the scores sum to zero, \(\gamma_0\) is the equally weighted average of the six cell means. The coefficient \(\gamma_A\) is half the difference between the equally weighted \(A_1\) and \(A_0\) means. The \(B\) coefficients describe marginal departures from the grand mean for \(B_0\) and \(B_1\); the departure for \(B_2\) is \(-\gamma_{B1}-\gamma_{B2}\). These equal-weight interpretations concern the model’s cell means, not the raw proportions of observations in each cell.

The interaction coefficients measure departures from an additive pattern. For example, at \(B_0\), the \(A_1\)-versus-\(A_0\) difference is \(2(\gamma_A+\gamma_{AB1})\). At \(B_2\), it is \(2(\gamma_A-\gamma_{AB1}-\gamma_{AB2})\). Thus the interaction terms allow the \(A\) difference to vary across the levels of \(B\).

Why the Fitted Means Do Not Depend on the Coding

A coding scheme is a way to express the same six cell means using six coefficients. Each scheme has one intercept, one coefficient for \(A\), two for \(B\), and two for the interaction. The mapping from coefficients to cell means is one-to-one: the reference coefficients can be recovered by taking cell differences, and the sum-to-zero coefficients can be recovered by taking averages and contrasts. Consequently, if the same full factorial fixed-effects model is fitted with either coding, changing coding is a reparameterization, not a change in the set of possible fitted cell means.

For the example, the reference coefficients can be recovered from the cell means as follows:

Conversely, the sum-to-zero intercept and coefficients are determined by the equal-weight averages and contrasts just calculated. Substituting them into the sum-to-zero equation reproduces all six means. For instance, at \(A_0,B_0\), the scores are \(-1,1,0\), giving \(13.5-\frac{7}{6}-2.5+\frac{1}{6}=10\). This agreement is a direct check that the conversion has preserved the fitted cell mean.

State Hypotheses in Terms of the Question

A coefficient test has a meaning only after its coding and its target comparison are specified. Under reference coding, \(H_0:\beta_A=0\) tests whether the \(A_1\)-versus-\(A_0\) difference is zero specifically at \(B_0\). It does not by itself test whether \(A\) has no effect at every level of \(B\). The latter question requires the joint null

The interaction null asks whether the \(A\) difference is the same at all levels of \(B\). In reference coding it is

This is equivalent to the sum-to-zero interaction null \(H_0:\gamma_{AB1}=\gamma_{AB2}=0\). In either coding, the alternative is that at least one of the interaction contrasts is nonzero.

Common Mistake: Treating a Main-Effect Coefficient as Universal

A common error is to read \(\beta_A\) from a reference-coded model as “the effect of \(A\)” without qualification. With an interaction in the model, \(\beta_A\) is the \(A\) difference only at the reference level \(B_0\). The differences at \(B_1\) and \(B_2\) are \(\beta_A+\beta_{AB1}\) and \(\beta_A+\beta_{AB2}\). To test an average difference, state the corresponding average contrast; to test no \(A\) difference at any \(B\) level, test the relevant coefficients jointly.

A second error is to assume that coding changes the model’s treatment predictions. It changes the coefficient values and their direct interpretations, but a complete factorial model with both main effects and their interaction still represents the same six cell means. If fitted means change after recoding, check whether the fixed-effects terms, data, or model-fitting choices also changed.