Treatment Effects in ANOVA

The one-way ANOVA model can be written as

$$ y_{ij} = \mu + \tau_i + \varepsilon_{ij}, $$

where $\mu$ is the baseline level and $\tau_i$ is the effect of treatment $i$.

The hypothesis of equal treatment means can then be written as

$$ H_0: \tau_1 = \tau_2 = \cdots = \tau_k = 0 $$

under an appropriate identifiability constraint such as a reference-category constraint or a sum-to-zero constraint.

This is the bridge between ANOVA and regression treatment coding.