Lecture 2 Linear Regression: A Model for the Mean

[Pages:56]Lecture 2 Linear Regression: A Model for the Mean

Sharyn O'Halloran

Closer Look at:

Linear Regression Model

Least squares procedure Inferential tools Confidence and Prediction Intervals

Assumptions Robustness Model checking Log transformation (of Y, X, or

both)

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Linear Regression: Introduction

Data: (Yi, Xi) for i = 1,...,n

Interest is in the probability distribution of Y as a function of X

Linear Regression model:

Mean of Y is a straight line function of X, plus an error term or residual

Goal is to find the best fit line that minimizes the sum of the error terms

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Estimated regression line

Steer example (see Display 7.3, p. 177)

Equation for estimated regression line:

7

Intercept=6.98

.73

6.5

Fitted line

1

Y^ = 6.98-.73X

PH

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Error term

1

2

ltime

Fitted v alues

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Create a new variable ltime=log(time)

Regression analysis

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Regression Terminology

Regression: the mean of a response variable as a function of one or more explanatory variables:

?{Y | X}

Regression model: an ideal formula to approximate the regression

Simple linear regression model:

?{Y | X } = 0 + 1X

"mean of Y given X" or "regression of Y on X"

Intercept

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Slope

Unknown parameter

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Regression Terminology

Y

X

Dependent variable

Independent variable

Explained variable

Explanatory variable

Response variable

Control variable

Y's probability distribution is to be explained by X

b0 and b1 are the regression coefficients

(See Display 7.5, p. 180)

Note: Y = b0 + b1 X is NOT simple regression

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Regression Terminology: Estimated coefficients

0 + 1X ^ 0 + ^ 1 X ^ 0 ^ 1

0 + 1X

^ 0 + ^ 1 X

0+ 1

^ 0 + ^ 1

Choose ^ 0 and ^1 to make the residuals small

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