Лукашевич, СофiяЯмненко, Ростислав2024-05-132024-05-132023Лукашевич С. О. Похiднi функцiї вiрогiдностi для лiнiйної змiшаної моделi з припущенням про складну симетрiю / Лукашевич С. О., Ямненко Р. Є. // Могилянський математичний журнал. - 2023. - T. 6. - C. 24-27. - https://doi.org/10.18523/2617-70806202324-272617-70802663-0648https://doi.org/10.18523/2617-70806202324-27https://ekmair.ukma.edu.ua/handle/123456789/29512The paper explores the properties of linear mixed models with simple random effects of the form: 𝑦𝑖 = 𝑋𝑖𝛽 + 𝑍𝑖𝛾𝑖 + 𝜀𝑖, 𝑖 = 1, . . . ,𝑀, 𝛾𝑖 ∼ 𝑁(0, ), 𝜀𝑖 ∼ 𝑁(0, 𝜎2𝐼), where 𝑀 is the number of distinct groups, each consisting of 𝑛𝑖 observations. Random effects 𝛾𝑖 and within-group errors 𝜀𝑖 are independent across different groups and within the same group. 𝛽 is a 𝑝-dimensional vector of fixed effects, 𝛾𝑖 is a 𝑞-dimensional vector of random effects, and 𝑋𝑖 and 𝑍𝑖 are known design matrices of dimensions 𝑛𝑖×𝑝 and 𝑛𝑖×𝑞, of fixed and random effects respectively. Vectors 𝜀𝑖 represent within-group errors with a spherically Gaussian distribution. Assuming a compound symmetry in the correlation structure of the matrix governing the dependence among within-group errors, analytical formulas for the first two partial derivatives of the profile restricted maximum likelihood function with respect to the correlation parameters of the model are derived. The analytical representation of derivatives facilitates the effective utilization of numerical algorithms like Newton-Raphson or Levenberg-Marquardt. The restricted maximum likelihood (REML) estimation is a statistical technique employed to estimate the parameters within a mixed-effects model, particularly in the realm of linear mixed models. It serves as an extension of the maximum likelihood estimation method, aiming to furnish unbiased and efficient parameter estimates, especially in scenarios involving correlated data. Within the framework of the REML approach, the likelihood function undergoes adjustments to remove the nuisance parameters linked to fixed effects. This modification contributes to enhancing the efficiency of parameter estimation, particularly in situations where the primary focus is on estimating variance components or when the model encompasses both fixed and random effects.У роботi дослiджено властивостi лiнiйних змiшаних моделей iз простими випадковими ефектами виду 𝑦𝑖 = 𝑋𝑖𝛽 + 𝑍𝑖𝛾𝑖 + 𝜀𝑖, 𝑖 = 1, . . . ,𝑀, 𝛾𝑖 ∼ 𝑁(0, ), 𝜀𝑖 ∼ 𝑁(0, 𝜎2𝐼), де 𝛽 – 𝑝-вимiрний вектор фiксованих ефектiв, 𝛾𝑖 – 𝑞-вимiрний вектор випадкових ефектiв, 𝑋𝑖 та 𝑍𝑖 – вiдомi матрицi регресорiв розмiрностей 𝑛𝑖 ×𝑝 i 𝑛𝑖 ×𝑞, а 𝜀𝑖 – вектори внутрiшньогрупових похибок зi сферичним гауссiвським розподiлом. Припускаючи складну симетрiю кореляцiйної структури залежностi мiж внутрiшньогруповими похибками, отримано аналiтичнi формули для перших двох часткових похiдних за кореляцiйними параметрами моделi.ukзмiшана лiнiйна модельобмежена оцiнка максимальної вiрогiдностiпохiднавипадковi ефектискладна симетрiястаттяLinear mixed modelREML estimatorderivativerandom effectscompound symmetryArticle