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Lecture 3 (Chapter 4)

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Lecture 3(Chapter 4)

- Linear Regression Model (Review)
- Ordinary Least Squares (OLS)
- Maximum Likelihood Estimation
- Distribution Theory

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

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

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

- 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

Review of linear model (cont’d)

Review of linear model (cont’d)

- We have onlyone measurement per subject; and measurements on difference subjects areindependent

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.

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

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 ModelProperties of