Download Lecture 3

Lecture 3
(Chapter 4)
Linear Models for Longitudinal Data
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Linear Regression Model (Review)
Ordinary Least Squares (OLS)
Maximum Likelihood Estimation
Distribution Theory
Linear Regression Model (A Review)
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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)
Review of linear model (cont’d)
• We have only one measurement per
subject; and measurements on difference
subjects are independent
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
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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) px1
mxm
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