A new method for obtaining T3SSs in the T-WFE model based on P matrices

Articles in Press

Document Type : Original Article

Author
Sajjad Piradl
Department of Statistics, Payame Noor University, Tehran, Iran
10.22034/maco.6.1.1
Abstract
Type 3 sum of squares (T3SS) is used to test model comparisons that violate the marginality principle when testing main effects and are not dependent on the order of the model terms, and also, the sum of the individual effect SS is not equal to the total effect SS. This paper presents a new method to obtain T3SSs based on projection (P) matrices in two-way fixed effects (T-WFE) model. Previous research has shown that in analysis of variance (ANOVA) for unbalanced data, the total SS is not obtained by summing the T3SSs of the components considered as factors of variation. In fact, in such a case, there is a significant difference between the values of these two quantities. The method proposed in this paper uses P matrices to identify where differences occur and the magnitude of their differences, while the traditional ANOVA method cannot clearly explain these. This paper also discusses how to use the eigenvalues and eigenvectors of P matrices to obtain T3SSs. Then, a study was conducted on a real dataset based on an unbalanced dataset for the proposed model fitting method to calculate TESSs based on P matrices.

Keywords

Subjects

[1] N. Huntington-Klein, The effect: An introduction to research design and causality, Routledge / Taylor & Francic, UK, 2022.
[2] L. K. Skipper, P. J. A. Skipper, Experimental design and scientific data analysis, Routledge / Taylor & Francic, UK, 2024.
[3] G. P. Quinn, M. J. Keough, Experimental design and data analysis for biologists, 2th edition , Cambridge University Press, UK, 2023.
[4] R. A. Johnson, R. G. K. Bhattacharyya, Statistics: Principles and methods, 8th edition, John Wiley & Sons, USA, 2019.
[5] C. Hirotsu, Advanced analysis of variance, John Wiley & Sons, USA, 2017.
[6] G. A. Milliken, D. E. Johnson, Analysis of Messy data volume 1: Designed experiments, 2th edition, Chapman and Hall / CRC, USA, 2009.
[7] L. Durell, N. Mastin, L. M. Lovin, W. Baylor Steele IV, B. W. Brooks, A. S. Hering, Balanced two-way functional ANOVA: A case study for toxicological photolocomotor response studies, Journal of Agricultural, Biological and Environmental Statistics, 1–30, 2025.
[8] L. R. LaMotte, Testing ANOVA effects: A resolution for unbalanced models, Communications in Statistics– Theory and Methods, 53(22):7860-7870, 2024.
[9] X. Cao, J. Fang, Z. Yao, Strong sum of projections in type II factors, Journal of Functional Analysis, 281(5):1–11, 2021.
[10] S. Goldstein, S. A. Paszkiewicz, Linear Combinations of Projections in Type III Factors, Integral Equations and Operator Theory, 94(4):1–32, 2022.
[11] W. Yu, Q. Zhang, W. Li, High-dimensional projection-based ANOVA test, Journal of Multivariate Analysis, 207:1–15, 2025.
[12] J. Tyssedal, S. Hussain, Factor screening in nonregular two-level designs based on projection-based variable selection, Journal of Applied Statistics, 43(3):490–508, 2016.
[13] K. Aishima, Consistent estimation with the use of orthogonal projections for a linear regression model with errors in the variables, Linear Algebra and its Applications, 681(1):101–126, 2024.
[14] J. Aguilar-Loyo, A comparative analysis of two-way fixed effects estimators in staggered treatment designs, Journal of Econometrics, 251:1–16, 2025.
Mathematical Analysis and Convex Optimization

Articles in Press, Accepted Manuscript
Available Online from 31 December 2025