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03GLS - PubH8452 Longitudinal Data Analysis Fall 2011...

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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
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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
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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
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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. 2. To use methods of inference which are robust to mis-specification of the covariance structure. For the moment, we assume the observation times are common for all subjects, that is t ij = t j , j = 1 , . . . , n for all i = 1 , . . . , m . The general linear model assumes: All subjects are independent, that is, if X i is stochastic, ( Y i , X i ) are independent; if X i is fixed by design, Y i are independent.
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