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

Document Type : Research Paper


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

2 Department of Statistics, Baghdad University, Iraq


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 $(X^{'}X)$  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 $(X^{'}X)$ 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 ${(\alpha\beta\gamma)}_{ijk}^{*}$ and ${(\alpha\beta)}_{i j}^{*}$ are not significant, while the factors ${(\alpha)}_{i}^{*}$, ${(\beta)}_{j}^{*}$, ${(\gamma)}_{k}^{*}$ and the  interactions ${(\alpha\gamma)}_{ik}^{*}$ and ${(\beta\gamma)}_{jk}^{*}$ have significant effects.


[1] K.M. Al-Rawi and A.K. Khalaf Allah, Design and Analysis of Agricultural Experiments, Second Edition, Dar
Al-Kutub Directorate for Printing and Publishing, University of Mosul, 2020.
[2] M.T. Anan, Experiments Design, Directorate of University Books and Publications, University of Aleppo, 2019.
[3] F.A. Graybill, Theory and Application of the Linear Model, Wadsworth Publishing Company, Inc., Belmont,
California, 1976.
[4] K. Hinkelmann and K. Oscar, Design and Analysis of Experiments, John Wily & Sons, Inc., New York, 2005.
[5] R.E. Kirk, Completely randomized Factorial design with three or more treatments and randomized block factorial
design, SAGE Publications, Inc., Thousand Oaks, 2013.
[6] R.E. Kirk, Experimental designs: an overview, Proced. Behav. Sci. 1995.
[7] R.E. Kirk, General Linear Model Approach to ANOVA, SAGE Publications, Inc., Thousand Oaks, 2014.
[8] K.V. Mardia, J.T. Kent and J.M. Bibby, Multivariate Analysis, Academic Press, Harcourt Brace & Company,
Publishers, London San Diego, New York, Boston Sydney, and Tokyo Toronto, 1979.
[9] D.C. Montgomery, Design and Analysis of Experiments, 9th Edition, John Wily & Sons, Inc., New York, 2017.
[10] L.C. Onyiah, Design and Analysis of Experiments, Taylor & Francis Group, LLC , New York, 2008.
[11] H.M. Raymond and S. M. Janet, A First Course in the Theory of Linear Statistical Models, PWS-KENT Publishing Company, 1991.
[12] S.R. Searle and M.H.J. Gruber, Linear Models, 2th Edition, John Wily & Sons, Inc , New York, 2017.
[13] R.S. Shayle and I.K. Andre, Matrix Algebra Useful for Statistics, 2th Edition; John Wily & Sons, Inc , New York,
Volume 13, Issue 1
March 2022
Pages 3733-3747
  • Receive Date: 02 January 2022
  • Revise Date: 27 January 2022
  • Accept Date: 12 February 2022