In an ANOVA model, there is a constraint that the coefficients must sum to zero. What does this actually mean? I do understand the reason why you might want to make them sum to zero, i.e. to have two degrees of freedom to estimate two parameters for instance, and not 3 df for estimating 2 parameters.

What parameters actually sum to zero?

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#### Best Answer

I think you are confusing the *coefficients* with the *contrasts*. The *contrast* refers to the specific way that the coefficient is estimated. When fitting ANOVAs we describe contrasts that sum to 0 as orthogonal. For instance, in a linear regression model the usual dummy encoding for a factor variable (say Education) is:

$$ begin{array}{l|ccc} & C_0 & C_1 & C_2 \ hline text{Less than High School} & 1 & -1 & -1 \ text{High School} & 0 & 1 & 0 \ text{Some College} & 0 & 0 & 1\ end{array}$$

So with the exception of the intercept term ($C_0$) the contrasts add up to 0 columnwise. That means that the interpretation of $C_1$ is a mean difference between high school and less than high school and $C_2$ a mean difference from some college to less than high school.