Application of Linear Side Conditions Tβ=0 on Non-Full Rank Linear Models

Meliyana Bohori (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Lampung, Bandar Lampung, 35141, Indonesia)
Mustofa Usman (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Lampung, Bandar Lampung, 35141, Indonesia)
Riza Sawitri (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Lampung, Bandar Lampung, 35141, Indonesia)
Edwin Russel (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Lampung, Bandar Lampung, 35141, Indonesia)
Jamal Ibrahim Daoud (Department of Science in Engineering, Faculty of Engineering, International Islamic University, Selangor, 53100, Malaysia)

Integra: Journal of Integrated Mathematics and Computer Science
Vol. 3 No. 2 (2026): July

Linear model is one of the foundation of statistical analysis, including analysis of variance (ANOVA). However, under non-full rank conditions, the design matrix exhibits linear depen dence, preventing the unique estimation of model parameters. This study aims to address this issue and to cope the problem by applying side conditions in the form of the linear constraint TB=0 without altering the model structure. The methods employed include the formulation of a restricted linear model, the derivation of parameter estimators, and simulation studies. The results show that side conditions produce unique, stable, and nearly unbiased parameter estimators. Furthermore, the results indicate that the power of test increases as differences between parameters increase and decreases as the variance of the error term increases. Thus, this approach is effective in ensuring the uniqueness of parameter estimates and improving the quality of analysis in ANOVA models with non-full rank design matrices.