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Reparameterization and the conditional inverse of a balanced factorial experiment with three factors | ||
| International Journal of Nonlinear Analysis and Applications | ||
| دوره 13، شماره 1، خرداد 2022، صفحه 3733-3747 اصل مقاله (457.03 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.6150 | ||
| نویسندگان | ||
| Wisam Wadullah Saleem* 1؛ Asmaa GHalib Jaber2 | ||
| 1Department of Statistics and Informatics, Mosul University, Iraq University, Iraq | ||
| 2Department of Statistics, Baghdad University, Iraq | ||
| تاریخ دریافت: 12 دی 1400، تاریخ بازنگری: 07 بهمن 1400، تاریخ پذیرش: 23 بهمن 1400 | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Three factors؛ balanced؛ estimable؛ treatment؛ general linear model؛ testable؛ less than full rank؛ conditional inverse؛ ANOVA؛ full rank؛ test؛ statistic؛ generalization؛ reparameterization | ||
| مراجع | ||
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[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, 2017. | ||
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