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Linear Regression Models: Basics and Applications

Understand linear regression modeling, from OLS to Maximum Likelihood Estimation, using distribution theory in longitudinal data analysis. Learn goals of regression analysis: Estimation, Testing, and Prediction.

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Linear Regression Models: Basics and Applications

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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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