Effect Coding and the Grand Mean

Why Use Effect Coding?

A factor-only generalized linear model assigns a separate linear predictor to each level of a categorical factor. Reference coding expresses those predictors relative to one selected level. Effect coding instead centers the level-specific predictors around their unweighted mean. This makes the intercept a “grand mean” across levels, while the coefficients describe deviations from it.

The word “mean” needs care here. In a generalized linear model, the intercept is generally an average on the linear-predictor scale, not an arithmetic average of the expected responses. The link function connects those two scales, and a nonlinear inverse link does not preserve averages.

Definition and Coding Matrix

Suppose a factor has \(K\) levels, labelled \(1,\ldots,K\), and the model contains no other predictors or offset. Let \(\eta_j\) denote the linear predictor for level \(j\). For an observation \(i\) in level \(j\), its expected response \(\mu_i=E[Y_i]\) is related to the predictor by \(g(\mu_i)=\eta_j\), where \(g\) is the GLM link.

A common coding matrix uses \(K-1\) columns. The first \(K-1\) levels receive the rows of the identity matrix, and the final level receives \(-1\) in every column. If \(\boldsymbol{\gamma}=(\alpha_1,\ldots,\alpha_{K-1})^\mathsf{T}\), the model is \(\eta_j=\beta_0+\mathbf{c}_j^\mathsf{T}\boldsymbol{\gamma}\), where \(\mathbf{c}_j\) is the coding row for level \(j\). The implied deviation for level \(K\) is \(\alpha_K=-\sum_{j=1}^{K-1}\gamma_j\).

The final row is not a reference level with zero effect. It represents the level whose deviation is the negative sum of the other deviations. This is what enforces the constraint.

Why the Intercept Is the Unweighted Mean

The interpretation follows directly from the constraint. It does not depend on the distribution family or link function: those determine how the linear predictors relate to the response distribution, but not how the factor levels are coded.

To verify the result, substitute \(\eta_j=\beta_0+\alpha_j\) and use \(\sum_j\alpha_j=0\):

Conversely, for any given set of \(K\) level-specific predictors, define \(\beta_0=K^{-1}\sum_j\eta_j\) and \(\alpha_j=\eta_j-\beta_0\). Then \(\sum_j\alpha_j=\sum_j\eta_j-K\beta_0=0\), so these parameters satisfy the effect-coding representation. Thus the constraint gives a valid representation for every set of level predictors.

The \(K-1\) coefficients in the displayed coding matrix are identifiable as a parameterization of these predictors. In fact, if \(\beta_0\mathbf{1}+C\boldsymbol{\gamma}=\mathbf{0}\), the mean of the \(K\) entries on the left is \(\beta_0\), because every column of \(C\) sums to zero. Therefore \(\beta_0=0\). The first \(K-1\) entries then equal the respective components of \(\boldsymbol{\gamma}\), so \(\boldsymbol{\gamma}=\mathbf{0}\). The columns for the intercept and the \(K-1\) effects are linearly independent, and the parameterization has full rank. \(\square\)

This is a parameterization of the level predictors, not a claim about how the predictors will be estimated. Estimation still depends on the response data and likelihood. When a factor-only model allows a separate predictor for each level, effect coding and reference coding describe the same set of possible level predictors; they differ in the coordinates used to express them.

Unequal Group Sizes Do Not Change the Coding

The grand mean in this parameterization gives equal weight to each level, not to each observation. This distinction matters when the observed group sizes differ. The likelihood may give levels with more observations more influence on the estimates, but that does not change the definition of the effect-coded intercept.

What the Coefficients Mean

For each of the first \(K-1\) levels, its coefficient is the difference between that level’s linear predictor and the unweighted mean across all levels. The last level’s deviation is not separately estimated as an additional free coefficient; it is the negative sum of the displayed coefficients. Individual coefficients therefore have a direct deviation interpretation, but they are not pairwise differences between levels.

For example, in the Poisson example, \(\gamma_A=\log(1/2)\) says that level A’s log mean is \(\log(1/2)\) below the average log mean. Exponentiating gives the ratio of A’s mean to the geometric mean across levels: \(e^{\gamma_A}=1/2\). It does not say that A’s mean is half the mean of a chosen reference group. Pairwise comparisons require subtracting the corresponding level deviations.

Common Mistake: Averaging Responses Instead of Predictors

A common mistake is to say that the intercept is the average response across levels, or to average the level predictors using group sizes. Neither statement describes the coding constraint. The intercept is the unweighted average of the \(K\) level-specific linear predictors. For a nonlinear link, its inverse-link value need not be the arithmetic mean of the level-specific expected responses. Unequal numbers of observations also do not turn the coding constraint into an observation-weighted average.

Another tempting error is to treat the final level as the reference level because it has a row of negative ones. Under this coding, that level generally has a nonzero deviation. Its contribution is determined by the sum-to-zero constraint; it is not fixed at zero as a reference-level predictor would be.