Lecture 3 (Chapter 4)

Linear Models for Longitudinal Data Linear Regression Model (Review) Ordinary Least Squares (OLS) Maximum Likelihood Estimation Distribution Theory

Linear Regression Model (A Review) Linear Regression Model (Review) Ordinary Least Squares (OLS) Maximum Likelihood Estimation Distribution Theory Will need to remember: –Basic matrix algebra –Likelihood function –Normal distribution

Regression Analysis Q: What are the goals of regression analysis? A: Estimation, Testing, and Prediction Estimation of the effect of one variable (exposure), called the predictor of interest, after adjusting, or controlling for, other measured variables –Remove confounding variables –Remove bias Testing whether variables are associated with the response Prediction of a response variable given a collection of covariates

Regression Analysis (cont’d) Classification of Variables Response Variable –Dependent Variable –Outcome Variable Predictor of interest (POI) –Exposure variable –Treatment assignment Confounding variables –Associated with response and POI –Not intermediate Precision variables –Associated with response and not with POI –Reduces response uncertainty

Ex: Impact of maternal smoking on low birth Measured variables –Birth weight (g) –Maternal smoking (yes/no) –Maternal age (yrs) maternal weight at last menses (kg) –Race –History of premature labor (yes/no) history of hypertension Response: birth weight Predictor of interest: maternal smoking Confounder: maternal age, race, history of premature labor Precision: maternal weight at last menses

Linear Regression Model

Review of linear model

Review of linear model (cont’d)

We have only one measurement per subject; and measurements on difference subjects are independent

Linear Model includes as special cases: –The analysis of variance (ANOVA), x ij are dummy variables –Multiple regression, x ij are continuous –The analysis of covariance, x ij are dummy and continuous variables is the expected value of the response variable Y, per unit change in its corresponding explanatory variable x, all other variables held fixed. Review of linear model (cont’d)

Principle of maximum likelihood (R.A. Fisher, 1925) –Estimate by the values that make the observations maximally likely –Likelihood P(Data| ), as a function of Estimation of

What is a Likelihood Function?

Likelihood Inference

Linear Model – I.I.D. Normal Errors

Distribution Theory for Linear Model mx1 (mxp) (px1)px1mxm Ordinary Least Squares (OLS) Estimate = Maximum Likelihood Estimate (MLE)

Distribution Theory for Linear Model Properties of “A”

Distribution Theory for Linear Model Properties of H Exercise: Check that HH’ =H

Distribution Theory for Linear Model Properties of