Analyzing Three Factor Experiments using Partitioned Design Matrices

Abstract

When analyzing factorial experiments, especially from just looking at the data layout, we could end up at wrong conclusion. Several possibilities of the models using three factors are being discussed. The models are characterized by experimental unit conditions and the way the treatments are allocated to the experimental units. In some experimental situation, it also depends on sampling units. Those models are : factorial experiments in completely randomized design, factorial experiments in randomized complete block design, split plot design with one factor allocated in the main plot, split plot design with two factors allocated in the main plot, and split-split plot design. Partitioned Design Matrices approach could be an alternative solution in analyzing experimental design models instead of using usual sigma notation for summation. Kronecker product is used to build the design matrix for each source of variation components used in the model. This approach is much easier and simpler to be used in explaining on how the partitioned design matrices for each source of variation being built, to calculate the degrees of freedom and sum of squares, to show that each source of variation has a chi-square distribution, and to show the independence between two sum of squares.