Lecture 3 chapter 4
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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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Lecture 3 chapter 4

Lecture 3(Chapter 4)


Linear models for longitudinal data

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 (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

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

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

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

Linear Regression Model


Review of linear model

Review of linear model


Lecture 3 chapter 4

Review of linear model (cont’d)


Lecture 3 chapter 4

Review of linear model (cont’d)

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


Lecture 3 chapter 4

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.


Lecture 3 chapter 4

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


Lecture 3 chapter 4

What is a Likelihood Function?


Lecture 3 chapter 4

Likelihood Inference


Lecture 3 chapter 4

Linear Model – I.I.D. Normal Errors


Lecture 3 chapter 4

Distribution Theory for Linear Model

mx1 (mxp) (px1)px1mxm

Ordinary Least Squares (OLS) Estimate = Maximum Likelihood Estimate (MLE)


Lecture 3 chapter 4

Distribution Theory for Linear Model Properties of

“A”


Lecture 3 chapter 4

Distribution Theory for Linear Model Properties of

H

Exercise: Check that HH’ =H


Lecture 3 chapter 4

Distribution Theory for Linear ModelProperties of


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