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So I'm analysing a 2-level factorial design, and get the residual vs fits plot below. I don't understand why it's symmetrical around 0. In any form of linear regression I've learned that the plot in question should be randomly scattered (given the assumptions in the model are correct).

However, I've read that this symmetry in fact is the correct pattern for a factorial design, but the source doesn't explain why. (See page 6 of this source: http://www.calpoly.edu/~pan/teaching/Minitab%20DOE%20Tutorial.pdf.)

Can you explain why the residuals vs fits plot for a 2-factorial design is symmetrical around 0?

enter image description here

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Consider a simpler model: two observations and one parameter. The residuals of that model would be two points symmetric about 0.

If you run a two-level factorial with 2k observations so that two observations are at each design point, and if you have k+1 terms (k non-intercept terms) in your model, then you get the same kind of thing. There's enough free parameters to generate a hyperplane that passes through the center of each of the pairs of observations sharing factor levels.

If you reduce the model or increase the number of observations at each factor level setting then you'll get something less regular looking.

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  • $\begingroup$ thanks! On a side note, this is a split-plot design (i.e. some of the factors are blocked), meaning there are two levels of factors and hence two levels of error, say $\sigma_W$ at the whole-plot level and $\sigma_S$ at the split-plot level. Which error do you think the residuals explain? $\endgroup$
    – harisf
    Apr 21, 2015 at 18:36
  • $\begingroup$ Both. I think software will decompose the residual error into the two components, either with something like maximum likelihood or the related REML, or via a method of moments estimator and the differing residual sums of squares... $\endgroup$ Apr 22, 2015 at 13:57

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