On the Analysis of Variance for Time Sequenced Samples

The 3 way nested ANOVA model y_ijk = μ + a_i + B_ij + γ_k + (αγ)_ik + epsilon_ijk With α (treatment or group effects) and γ (time) both being fixed effects and B (the individual effects) random and nested within α, is introduced and explored. The problems associated with the usual approach are explained. The alternative model Is developed and a method of evaluation via the method of linear contrasts is recomended. The test statistic has the distribution of a convolution of F-distributions. Further, a method of investigating the assumptions of the model is offered and a further generalization using path spaces (of dim. K) is developed. Here again the appropriate test statistic has the distribution of a convolution of F-distributions. This combined with the method of linear contrasts offers an elegant solution to the BEHRENS-FISHER problem. y_ijk = f_ik + B_ij + epsilon_ijk