Why Contrast Coding Does Not Change Fitted Means

Introduction

A regression model can represent the same factor levels in more than one way. One coding might make a chosen level the reference; another might use contrasts that compare several levels. The coefficient values, and what each coefficient means, can change with that choice. The model’s fitted mean for each factor level need not change at all.

This distinction is useful when reading model output or comparing software settings. The coefficients are coordinates for expressing the fitted pattern across levels. To check whether two codings describe the same fitted pattern, compare the fitted means—not the coefficient values one by one.

A Factor-Only Model

Consider a response \(Y_i\) and a factor with \(K\) levels. Write \(g_i\) for the level observed for case \(i\). A factor-only model for the conditional mean is

Here \(\mu_g\) is the conditional mean for level \(g\), and \(\varepsilon_i\) is the difference between the observed response and that level’s mean. For this tutorial, the fitted model uses ordinary least squares (OLS) and includes enough factor columns to represent an arbitrary mean for every observed level. OLS chooses coefficients to minimize the sum of squared residuals. No normality assumption is needed for the coding-invariance result below.

With reference-level coding, one level is represented by the intercept, and the other levels have indicator columns. For three levels A, B, and C, taking A as the reference gives

Thus \(\beta_0\) is the fitted mean for A, while \(\beta_1\) and \(\beta_2\) are differences from A. These coefficient meanings follow from this coding choice; they are not definitions that apply to every coding.

Re-expressing the Same Three Means

Now represent the same three levels with two contrast columns. Assign the values \((-1,-1)\) to A, \((1,-1)\) to B, and \((0,2)\) to C. Let the coefficients be \(\gamma_0,\gamma_1,\gamma_2\), so the fitted means are

These equations are a different representation of the three fitted means. Solving them for the coefficients gives

The values assigned to the levels determine the coefficient meanings. Here \(\gamma_0\) is the equally weighted average of the three fitted level means. The other coefficients encode contrasts among those means. The predictions can be recovered from the equations regardless of how the coefficients themselves are interpreted.

Why the Fitted Values Stay the Same

The general reason is a property of OLS under an invertible change of predictor coordinates. Let \(X\) be the design matrix for one coding, and let \(\boldsymbol{\beta}\) be its coefficient vector. Suppose another coding has design matrix \(Z=XA\), where \(A\) is a nonsingular square matrix. “Nonsingular” means that \(A\) has an inverse. If the original coefficients are \(\boldsymbol{\beta}\), set the new coefficients to \(\boldsymbol{\theta}=A^{-1}\boldsymbol{\beta}\). Then

So any fitted-value vector available under the first coding is also available under the second. The reverse holds as well, because the change can be undone with \(A^{-1}\). The two codings therefore allow exactly the same fitted-value vectors. OLS minimizes the same sum of squared residuals over that same set, so the fitted values are identical. If the design matrix has full column rank, the coefficient estimates are uniquely related by \(\widehat{\boldsymbol{\theta}}=A^{-1}\widehat{\boldsymbol{\beta}}\).

For a factor with all \(K\) levels represented, an intercept and \(K-1\) suitable factor columns can express any set of \(K\) level means. Changing from one such coding to another changes the coordinates used to express those means, but not which sets of means can be represented. The derivation concerns the model’s fitted values; it does not require a probability distribution for the errors.

Coefficient Meaning Depends on Coding

Even a two-level factor illustrates why coefficient values should not be compared without checking the coding. Suppose a hypothetical outcome has fitted means of 30 units for group A and 36 units for group B. With A as the reference, the intercept is 30 and the indicator coefficient for B is \(36-30=6\). The coefficient describes B minus A.

Instead, assign a single predictor \(x=-1\) to A and \(x=1\) to B, and write the fitted mean as \(\gamma_0+\gamma_1x\). To reproduce the two means, solve

Adding the equations gives \(2\gamma_0=66\), so \(\gamma_0=33\). Subtracting the first equation from the second gives \(2\gamma_1=6\), so \(\gamma_1=3\). The predictions are \(33-3=30\) for A and \(33+3=36\) for B. The slope coefficient is now half the difference between the groups, not the full difference. Both codings yield the same fitted means of 30 and 36 units.

A coefficient is therefore not a fitted mean unless its coding makes it one, as the reference-level intercept does. Nor is a coefficient necessarily the difference between two particular levels; that depends on how the contrast values were assigned. To interpret any coefficient, first identify the coding and substitute its values into the model equation.

Common Mistake and Practical Check

A common mistake is to conclude that changing the reference level has changed the model’s predictions because the intercept or factor coefficients changed. That reasoning confuses the model’s representation with its fitted values. A changed coefficient can simply reflect a new baseline or a different scale for a contrast.

A reliable check is to calculate the fitted mean for every factor level under each coding. If all levels receive the same fitted means, the two representations agree on those predictions. For an OLS model using an intercept and a full set of contrasts for the factor, an invertible recoding preserves the available fitted values and hence the fitted values obtained by minimizing squared residuals. This statement assumes the recoding represents the same model; changing the model’s terms or restrictions is not merely changing the coding.