Recovering Group Means from Any Coding

Why Recovering the Means Matters

A regression table reports coefficients, but the quantities most useful for understanding a categorical predictor are often the fitted outcomes for its categories. Those fitted means are not always individual coefficients: depending on the coding, a category mean may be an intercept, an intercept plus one or more coefficients, or a coefficient from a model without an intercept.

The goal here is to make that translation explicit. We will use one categorical predictor with three categories, called A, B, and C. The same reasoning applies to any number of categories: identify the coding row for a category, substitute it into the fitted regression equation, and simplify.

Model and Conditions

Let \(Y_{ig}\) be the outcome for observation \(i\) in category \(g\). A one-factor categorical model can be written as

where \(\mu_g\) is the mean outcome for category \(g\), and \(\varepsilon_{ig}\) is the observation’s error. To estimate these category means by ordinary least squares, the model must represent each category, and each category considered must have observations. The coding must also be full rank: its columns must allow the category means to be uniquely represented.

For the usual linear-model interpretation, errors are assumed to have mean zero and to be independent; a common error variance is often assumed for standard error calculations. These error assumptions support statistical inference, but the arithmetic for recovering fitted means is determined by the fitted equation and the coding. Normality is not needed to substitute category codes into that equation.

The formulas below concern a model with one categorical predictor and no additional predictors or interactions. If other terms are present, the fitted outcome may also depend on their values; then the fitted category mean must be calculated at specified values of those other predictors.

Reference Coding: Start at the Reference Category

Under reference coding, one category is the reference, and the model has an intercept plus an indicator for each other category. Let A be the reference. With coefficients \(\beta_0,\beta_B,\beta_C\), the fitted equation is

Here \(I(B)=1\) for an observation in B and \(0\) otherwise; \(I(C)\) is defined in the same way. For A, both indicators are zero. For B, only \(I(B)\) is one; for C, only \(I(C)\) is one. Thus the category means are recovered by substitution:

Sum Coding: Substitute the Category’s Codes

For three categories, one common sum-coding arrangement assigns A the codes \((1,0)\), B the codes \((0,1)\), and C the codes \((-1,-1)\). With intercept \(\beta_0\) and two coded coefficients \(\beta_1,\beta_2\), the fitted equation is \(\hat{Y}=\hat{\beta}_0+\hat{\beta}_1 x_1+\hat{\beta}_2 x_2\). The category means come from substituting each category’s pair of codes.

In this parameterization, the intercept alone is generally not any one category’s fitted mean. The code values for the category must be used to recover its prediction.

Cell-Means Coding: Read Each Mean Directly

In cell-means coding, the model has no intercept and includes one indicator for every category:

Exactly one of these indicators equals one for each observation. Consequently, the fitted mean for a category is its corresponding coefficient:

Why the Same Means Can Have Different Coefficients

The coefficients depend on the coding, but the fitted category means need not. To see how this works, start with any three fitted means \(m_A,m_B,m_C\). Under reference coding with A as the reference, the coefficients that reproduce them are \(\beta_0=m_A\), \(\beta_B=m_B-m_A\), and \(\beta_C=m_C-m_A\). Substitution gives \(m_A\), \(m_A+(m_B-m_A)=m_B\), and \(m_A+(m_C-m_A)=m_C\).

For the sum-coding arrangement above, choose \(\beta_0=(m_A+m_B+m_C)/3\), \(\beta_1=m_A-\beta_0\), and \(\beta_2=m_B-\beta_0\). The fitted values for A and B are then \(\beta_0+\beta_1=m_A\) and \(\beta_0+\beta_2=m_B\). For C,

Thus this sum-coded equation reproduces all three means too. In cell-means coding, set \(\gamma_A=m_A\), \(\gamma_B=m_B\), and \(\gamma_C=m_C\), which reproduces them directly. This algebra shows why coefficient values cannot be compared across coding schemes without first accounting for how each scheme defines the fitted equation.

A Common Pitfall

A frequent mistake is to read the intercept as the fitted mean for every category—or to assume it always represents the reference category. That is correct for the reference-coded equation shown here, but not for the sum-coded equation, where the category’s coded terms must be added, or cell-means coding, which has no intercept. Another error is to treat an indicator coefficient as a category mean when it is actually a difference from the reference mean.

A reliable check is to write the fitted equation and the category’s indicator values before doing arithmetic. If the resulting predictions do not match the intended means, check the coding values, the signs, and whether an intercept is present. Also keep the averaging target clear: the average of three category means weights categories equally, while an overall observation-level mean weights categories according to their sample sizes.