The space of quadratic forms in the \(n \times m\) array \(Y\) is a vector space of dimension \(m n(m n-1) / 2\). Exhibit a basis. Show that the four quadratic forms in Exercise 1.3 are invariant with respect to row and column permutations. Deduce that every invariant quadratic form is a linear combination of these four.

Data From Exercise 1.3

The \(F\)-ratio reported by anova (...) for treatment effects is not the ratio shown above. At least one is misleading for this design. Which one? Explain your reasoning.