# What is the definition of a "main effect" of a factor when it has more than two levels?

In a factorial experiment the definition of the main effect of a factor A with two levels, 0 and 1, is the difference in outcome means between the two level averaging over all other factors, $\bar{y}_{A=0} - \bar{y}_{A=1}$.

What is the extension of this definition to factors with more than two levels?

Some possible ideas are

• The difference between a factor level and the grand mean giving us one main effect for each factor level.
• The difference in mean outcome between each pair of factor levels.
• The anova sum of squares associated with the factor giving us one main effect per factor.

Is there a formal definition of 'main effect' when factors have more than two levels?

• The Wikipedia page on main effects is surprisingly sparse. https://en.wikipedia.org/wiki/Main_effect According to Wikipedia the term "main effect" refers to the overall factor and not specific factor levels, i.e. one main effect per factor. Nov 12, 2017 at 19:26