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
6
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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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