Reparameterization and the conditional inverse of a balanced factorial experiment with three factors

In this research, a factorial experiment 23 was studied through a balanced mathematical model applied in a complete random design (CRD) to show the effect of the main factors and the interactions between the factors through the use of the general linear model in which the design matrix (X′X) has less than full rank and thus the parameters vector (β) is neither estimable nor testable. Therefore, the re-parameter method and conditional inverse were used to transform the design matrix (X′X) to a full-rank matrix, so that the parameters vector (β) is capable of estimable and testable, after analyzing the experiment data and testing hypotheses it was found that the interactions (αβγ)∗ijk and (αβ)∗ij are not significant, while the factors (α)∗i, (β)∗j, (γ)∗k and the interactions (αγ)∗ik and (βγ)∗jk have significant effects.