{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–5. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern