Regression & Correlation

Effect Coding (Sum-to-Zero) for Categorical Factors

Maps a categorical factor with k levels onto k − 1 indicator columns; the intercept becomes the grand mean.

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Factorial > Analyze (Coded levels with reference contrasts)

Description

Sum-to-zero coding (R: contr.sum) assigns each non-reference level a unit vector (+1 in its own column, 0 elsewhere) and gives the reference level a vector of −1s. The matching coefficient reads as the deviation from the grand mean; the reference deviation is the negative sum of the others. Advantage over dummy coding: the intercept estimates the grand mean rather than just the reference cell, ANOVA main effects are orthogonal to the intercept, and balanced designs simplify.

Formulas

Reference

ref \to (- 1, - 1, \dots, - 1)

Reference level is represented by a vector of −1

Non-reference

level_{j} \to e_{j} (unit vector with +1 at position j)

Non-reference level j gets the unit vector with +1 at position j

Sum-to-zero constraint

j \sum β_{j} = 0 \Rightarrow β_{ref} = - j \neq = ref \sum β_{j}

The effect coefficients sum to zero; the reference effect follows from that

Assumptions

At least 2 levels per factor
Reference level is one of the supplied levels

Limitations

In extremely unbalanced designs the sum-to-zero coefficients may be hard to interpret — consider treatment (dummy) coding instead

References

R Core Team (2024). contr.sum() in stats::contrasts
Montgomery, D.C. (2017). Design and Analysis of Experiments, 9th Ed., Wiley — Chapter 3.5 (Effect Coding)