ABSTRACT

A very general linear mixed model for longitudinal data has been proposed by Laird and Ware (1982) based on the work of Harville (1974, 1976, 1977), https://www.w3.org/1998/Math/MathML"> y i    =   X i β   +   Z i γ i   + ϵ i ,   https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203748640/4fa687a6-cf95-4be5-a509-4d5b776421ec/content/eq111.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where Y i is an n i × 1 column vector of the response variable for subject i, X i , is an n i , × b design matrix, β is a b × 1 vector of regression coefficients assumed to be fixed, Z i is an n i × g design matrix for the random effects, γ i which are assumed to be independently distributed across subjects with distribution, https://www.w3.org/1998/Math/MathML"> γ i   ∼   N ( 0 , σ 2 B ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203748640/4fa687a6-cf95-4be5-a509-4d5b776421ec/content/eq112.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where B, for between, is an arbitrary covariance matrix. The within subject errors, ϵi, are assumed to be distributed https://www.w3.org/1998/Math/MathML"> ϵ i   ∼   N ( 0 , σ 2 W i ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203748640/4fa687a6-cf95-4be5-a509-4d5b776421ec/content/eq113.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where W i , for within, is a covariance matrix which may be parameterized using a few parameters with a scale factor, σ2, factored out. Often it is assumed that W i = I, the identity matrix. Various possibilities for parameterizing the W i matrices will be discussed later. For example, W i may have the structure of a first order autoregression. The ϵ i are also independently distributed from subject to subject and independent of the γi . Model (2.1) is very general since different subjects can have different numbers of observations as well as different observation times. The subscript i in the vector Y i , and the matrices X i , Z i , and W i ; indicates that these vectors and matrices are subject specific. The matrices X i and Z i are not necessarily of full rank.