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Lecture 3 (Chapter 4)

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)

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Lecture 3 (Chapter 4)

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  1. Lecture 3(Chapter 4)

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

  3. 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

  4. 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

  5. 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

  6. 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

  7. Linear Regression Model

  8. Review of linear model

  9. Review of linear model (cont’d)

  10. Review of linear model (cont’d) • We have onlyone measurement per subject; and measurements on difference subjects areindependent

  11. Review of linear model (cont’d) • Linear Model includes as special cases: • The analysis of variance (ANOVA), xij are dummy variables • Multiple regression, xij are continuous • The analysis of covariance, xij 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.

  12. Estimation of • 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

  13. What is a Likelihood Function?

  14. Likelihood Inference

  15. Linear Model – I.I.D. Normal Errors

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

  17. Distribution Theory for Linear Model Properties of “A”

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

  19. Distribution Theory for Linear ModelProperties of

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