## 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, ϵ_{i}, 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, σ^{2}, 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 γ_{i}
. 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.