Reading Sum-Coded Regression Output

Reading the Intercept and Coefficients

Sum-coded regression output can be unfamiliar because no category is assigned a coefficient of zero. Instead, the intercept represents the unweighted average of the fitted category means, and each category’s effect represents its difference from that average. To recover fitted means, combine the intercept with the appropriate category effect—including the implied effect for the category whose coefficient is not shown.

The precise interpretation depends on the coding convention used by the software. This tutorial uses the common sum-coding convention in which a factor with \(K\) categories has \(K-1\) displayed columns: the first \(K-1\) categories each receive a 1 in their own column and 0 in the others, while the last category receives \(-1\) in every column. Check the output’s contrast labels or model specification before decoding it; other conventions can use different coefficient scales.

The Three-Category Coding Pattern

For three categories, call them A, B, and C, the usual sum-coding pattern is:

Write the model as \(\hat{y}=\hat{\beta}_0+\hat{\beta}_A x_A+\hat{\beta}_B x_B\), where \(x_A\) and \(x_B\) are the two coding columns. Substituting each category’s codes gives:

Thus, although software usually displays only two coefficients for this three-category factor, you can calculate all three category effects. The effect for C is \(-\hat{\beta}_A-\hat{\beta}_B\), not zero. Add each effect to the intercept to get that category’s fitted mean.

Why the Coefficients Sum to Zero

The zero-sum constraint follows directly from the coding. For three categories, the category effects are \(\hat{\beta}_A\), \(\hat{\beta}_B\), and \(-\hat{\beta}_A-\hat{\beta}_B\). Their sum is:

Each category’s effect is therefore its fitted mean minus the average of the \(K\) fitted category means. To see this for three categories, average the three fitted means from the coding equations:

So the intercept is the unweighted arithmetic average of the three fitted category means. This is a property of the coding and fitted category means; it does not say that every observation is equally represented in the observed data.

The same result holds for \(K\) categories. Under the stated convention, the fitted mean for each of the first \(K-1\) categories is \(\hat{\beta}_0+\hat{\beta}_j\). The final category’s effect is minus the sum of the displayed effects, so its fitted mean is \(\hat{\beta}_0-\sum_{j=1}^{K-1}\hat{\beta}_j\). Adding all \(K\) fitted means cancels the displayed effects, leaving \(K\hat{\beta}_0\). Dividing by \(K\) shows that the intercept is their unweighted average, and the \(K\) category effects consequently sum to zero.

Unequal Category Sizes and the Meaning of the Intercept

The intercept is an unweighted average of category means, not generally the overall sample mean. When category sizes differ, the overall sample mean gives more weight to categories with more observations. This distinction is particularly useful when checking output from a categorical-only ordinary least squares model: its fitted category means are the within-category sample means, but their unweighted average need not equal the sample mean across all observations.

A Reliable Way to Decode the Output

Before calculating, confirm that the model has an intercept and that the factor really uses the sum-coding pattern described above. Read the category order from the software output or model specification: the category assigned the all-\(-1\) row is the one whose effect must be reconstructed. Then use this sequence.

Common Mistake: Treating a Sum-Coded Category as the Baseline

A common error is to assume that the category without a displayed coefficient has an effect of zero, just as a reference category does under reference coding. That would make its fitted mean equal to the intercept, but it is not how the sum-coding pattern here works. The final category has an effect equal to the negative sum of the displayed effects; it is generally not zero.

For example, with three categories and displayed effects of 4 and \(-1\), the omitted effect is \(-3\), not 0. The resulting fitted mean for that category is the intercept minus 3 response units. Another mistake is to call the intercept the sample-size-weighted grand mean without checking the model and category sizes. Under the categorical-only model here, the intercept is the unweighted average of the fitted category means. Always distinguish that average from the overall mean of the observations.