Document Type : Research Paper

Authors

• Wisam Wadullah Saleem ¹
• Asmaa GHalib Jaber ²

¹ Department of Statistics and Informatics, Mosul University, Iraq University, Iraq

² Department of Statistics, Baghdad University, Iraq

Abstract

In this research, a factorial experiment $2^3$ 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 $(X^{{}^{\prime }}X)$  has less than full rank and thus the parameters vector $(\beta )$ is neither estimable nor testable. Therefore, the re-parameter method and conditional inverse were used to transform the design matrix $(X^{{}^{\prime }}X)$ to a full-rank matrix, so that the parameters vector $(\beta )$  is capable of estimable and  testable, after analyzing the experiment data and testing hypotheses it was found that the interactions ${(\alpha \beta \gamma )}_{ijk}^{\ast }$ and ${(\alpha \beta )}_{ij}^{\ast }$ are not significant, while the factors ${(\alpha )}_i^{\ast }$, ${(\beta )}_j^{\ast }$, ${(\gamma )}_k^{\ast }$ and the  interactions ${(\alpha \gamma )}_{ik}^{\ast }$ and ${(\beta \gamma )}_{jk}^{\ast }$ have significant effects.

Keywords

• Three factors
• balanced
• estimable
• treatment
• general linear model
• testable
• less than full rank
• conditional inverse
• ANOVA
• full rank
• test
• statistic
• generalization
• reparameterization