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?

  • 1
    $\begingroup$ Where did you find such information? There is no such constraint. $\endgroup$
    – Tim
    May 15, 2019 at 14:15
  • $\begingroup$ @Tim Consider two-way ANOVA for one example (whether main effects or with interaction); there will have to be some form of regularization or constraint on the parameters or the model is not identifiable; in the usual regression parameterization, a baseline level of each factor will be omitted (constraining its parameter to 0). In another parameterization, sum-to-zero constraints are used. There are other parameterizations still, but they all introduce enough constraints to make the model identifiable. $\endgroup$
    – Glen_b
    May 16, 2019 at 3:18
  • $\begingroup$ @Glen_b I meant that this is not generally true. $\endgroup$
    – Tim
    May 16, 2019 at 4:02
  • $\begingroup$ Related: stats.stackexchange.com/q/257778/119261. $\endgroup$ Apr 5, 2020 at 21:09

1 Answer 1


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.


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