Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 27I / ii / Solution

Assume $I,J\ge 2$. The additive two-way model has dimension $1+(I-1)+(J-1)=I+J-1$; the null model, with no exam effects, has dimension $I$. Write row, column and grand means as ${\overline{X}}_{i\cdot },{\overline{X}}_{\cdot j},{\overline{X}}_{\cdot \cdot }$. Orthogonal projection onto the full model gives fitted values ${\overline{X}}_{i\cdot }+{\overline{X}}_{\cdot j}-{\overline{X}}_{\cdot \cdot }$, while the null fit is ${\overline{X}}_{i\cdot }$. The added exam sum of squares and full residual sum of squares areThe preceding orthogonal-projection result gives independent null chi-squares with $J-1$ and $(I-1)(J-1)$ degrees of freedom. ThereforeReject the no-exam-effect hypothesis for a large value. The printed alternative repeats the identifying constraint ${\sum }_j{\beta }_j=0$; a genuine alternative additionally has at least one nonzero ${\beta }_j$. No interaction term is being fitted; the independent equal-variance Gaussian error assumption is what makes this exact test valid.