{"title":"具有复合对称性假设的线性混合模型的似然函数导数","authors":"Sofia Lukashevych, R. Yamnenko","doi":"10.18523/2617-70806202324-27","DOIUrl":null,"url":null,"abstract":"The paper explores the properties of linear mixed models with simple random effects of the form: yi = Xiβ + ZiYi + εi, i = 1, . . . ,M, Yi ∼ N(0, Ψ), εi ∼ Т(0, σ2I), where M is the number of distinct groups, each consisting of ni observations. Random effects Yi and within-group errors εi are independent across different groups and within the same group. β is a p-dimensional vector of fixed effects, Yi is a q-dimensional vector of random effects, and Xi and Zi are known design matrices of dimensions nixp and nixq, 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 Ψ 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.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":" 10","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Likelihood function derivatives for a linear mixed model with compound symmetry assumption\",\"authors\":\"Sofia Lukashevych, R. Yamnenko\",\"doi\":\"10.18523/2617-70806202324-27\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The paper explores the properties of linear mixed models with simple random effects of the form: yi = Xiβ + ZiYi + εi, i = 1, . . . ,M, Yi ∼ N(0, Ψ), εi ∼ Т(0, σ2I), where M is the number of distinct groups, each consisting of ni observations. Random effects Yi and within-group errors εi are independent across different groups and within the same group. β is a p-dimensional vector of fixed effects, Yi is a q-dimensional vector of random effects, and Xi and Zi are known design matrices of dimensions nixp and nixq, 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 Ψ 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.\",\"PeriodicalId\":404986,\"journal\":{\"name\":\"Mohyla Mathematical Journal\",\"volume\":\" 10\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-04-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mohyla Mathematical Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.18523/2617-70806202324-27\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mohyla Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.18523/2617-70806202324-27","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
本文探讨了具有简单随机效应形式的线性混合模型的性质:yi = Xiβ + ZiYi + εi, i = 1, .,M,Yi ∼ N(0,Ψ),εi ∼ Т(0,σ2I),其中 M 是不同组的数量,每个组由 ni 个观测值组成。随机效应 Yi 和组内误差 εi 在不同组之间和同一组内都是独立的。β 是 p 维的固定效应向量,Yi 是 q 维的随机效应向量,Xi 和 Zi 是已知的设计矩阵,固定效应和随机效应的维数分别为 nixp 和 nixq。假定矩阵 Ψ 的相关结构具有复合对称性,可控制组内误差之间的依赖关系,从而推导出轮廓限制最大似然函数关于模型相关参数的前两个偏导数的分析公式。导数的分析表示有助于有效利用牛顿-拉斐尔森(Newton-Raphson)或莱文伯格-马夸特(Levenberg-Marquardt)等数值算法。受限极大似然(REML)估计是一种统计技术,用于估计混合效应模型中的参数,特别是线性混合模型中的参数。它是最大似然估计方法的延伸,旨在提供无偏、有效的参数估计,尤其是在涉及相关数据的情况下。在 REML 方法的框架内,似然函数经过调整,以去除与固定效应相关的干扰参数。这种修改有助于提高参数估计的效率,尤其是在主要重点是估计方差成分或模型包含固定效应和随机效应的情况下。
Likelihood function derivatives for a linear mixed model with compound symmetry assumption
The paper explores the properties of linear mixed models with simple random effects of the form: yi = Xiβ + ZiYi + εi, i = 1, . . . ,M, Yi ∼ N(0, Ψ), εi ∼ Т(0, σ2I), where M is the number of distinct groups, each consisting of ni observations. Random effects Yi and within-group errors εi are independent across different groups and within the same group. β is a p-dimensional vector of fixed effects, Yi is a q-dimensional vector of random effects, and Xi and Zi are known design matrices of dimensions nixp and nixq, 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 Ψ 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.