If we were to assume a linear relationship between x and y described by the model y = a + βx + e it is unlikely that we would consider writing the model as y = a + bx + cx + e. It is even more unlikely that we would apply the least squares principle by minimizing Σe2 with respect to a, b, and c. Yet a similar thing happens in experimental design. In fact, it is common practice to use less than full-rank models where the parameters are not defined and, in cases where they are defined, to minimize Σe2 with respect to the full set of parameters which are not functionally independent.
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Folks, J. Leroy
"Experimental Design Models,"
Journal of the Graduate Research Center: Vol. 33:
1, Article 5.
Available at: https://scholar.smu.edu/journal_grc/vol33/iss1/5