03GLS - PubH8452 Longitudinal Data Analysis Fall 2011 General Linear Models General Linear Models for Longitudinal Data Outline • Review of

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Unformatted text preview: PubH8452 Longitudinal Data Analysis - Fall 2011 General Linear Models General Linear Models for Longitudinal Data Outline • Review of multivariate normal distribution • Correlation specifications • Review of likelihood inference • Ordinary least squares (OLS) • Weighted least squares (WLS) • Maximum likelihood (ML) • Restricted maximum likelihood (REML) • Robust estimation 1 PubH8452 Longitudinal Data Analysis - Fall 2011 General Linear Models Review of Multivariate Normal Distribution The density function for a multivariate normal random vector Y is: f ( y ; μ , Σ) = 1 ( √ 2 π ) n | Σ | 1 / 2 exp ‰- 1 2 ( y- μ ) T Σ- 1 ( y- μ ) , where-∞ < y j < ∞ , j = 1 , . . . , n . • This distribution is completely specified by its first two moments, μ = E( Y ) and Σ = Var( Y ). • Each Y j has a marginal univariate normal distribution with mean μ j and variance σ 2 jj . • If we partition Σ as: Σ = Σ 11 Σ 12 Σ 21 Σ 22 ¶ , where Σ 11 is a n 1 × n 1 matrix, Σ 22 is a n 2 × n 2 matrix and n 1 + n 2 = n , then a subset of the Y j ’s Z 1 = ( Y 1 , . . . , Y n 1 ) also has a multivariate normal distribution with mean μ 1 = ( μ 1 , . . . , μ n 1 ) and variance Σ 11 . • Let Z 2 = ( Y n 1 +1 , . . . , Y n ), then the conditional distribution of Z 1 given Z 2 is also normal with mean μ 1- Σ 12 Σ- 1 22 ( z 2- μ 2 ) , and variance Σ 11- Σ 12 Σ- 1 22 Σ 21 . 2 PubH8452 Longitudinal Data Analysis - Fall 2011 General Linear Models • If B is a m × n matrix, then B Y (a linear transformation) is also multivariate normal, with mean B μ and variance B Σ B T . • Consider the MLE of the coefficients for a linear regression model, ˆ β = ( X T X )- 1 X T Y . If Y ∼ N ( X β , σ 2 I ), then ˆ β also has a multivariate normal distribution with mean E( ˆ β ) = ( X T X )- 1 X T X β = β , and variance: Var( ˆ β ) = ( X T X )- 1 X T σ 2 I £ ( X T X )- 1 X T / T = σ 2 ( X T X )- 1 X T X ( X T X )- 1 = σ 2 ( X T X )- 1 . • The random variable U ≡ ( Y- μ ) T Σ- 1 ( Y- μ ) ∼ χ 2 n . • Correlations and independence – For a multivariate normal vector, being uncorrelated implies independent. – It is not true for two marginally normally distributed variables because their joint distribution may fail to be normal. 3 PubH8452 Longitudinal Data Analysis - Fall 2011 General Linear Models General Linear Models for Longitudinal Data We aim to develop a general linear model framework for longitudinal data, in which the inference we make about the parameters of interest recognize the likely correlation structure in the data. There are two ways of achieving this: 1. To build explicit parametric models of the covariance structure....
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This note was uploaded on 11/21/2011 for the course PUBH 8452 taught by Professor Xianghualuo during the Fall '11 term at Minnesota.

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03GLS - PubH8452 Longitudinal Data Analysis Fall 2011 General Linear Models General Linear Models for Longitudinal Data Outline • Review of

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