1 introduction what does it mean when there is a strong positive correlation between x and y ?...
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1
Introduction
What does it mean when there is a strong positive correlation between x and y ?
Regression analysis aims to find a precise formula to relate the movements of y to those of x
The use of regression requires a good deal of thought and a good dose of skepticism
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Example: Sales vs. Advertising
Advertising (mil.$) Sales (mil.$)1.2 1201.6 1901.8 2602.2 2602.6 3003.1 2903.4 3303.6 3304.0 3404.2 310
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Example: Sales vs. Advertising
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Example: Sales vs. Advertising
It is believed that sales, S, are tied to advertising, A, by a simple linear equation:
What do 0 and 1 represent?
How can we find 0 and 1 ?
AS 10
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Sales vs. Advertising
The linear relation is usually not exact.
A more realistic model:
Where 0 and 1 are regression coefficients.
ii10i AS
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Measurement Error
Properties of distribution A mean of zero Symmetry around zero An assignment of greater probability to
small errors than to larger ones Errors are assumed to be:
Independent Have same variance
(homoscedasticity)
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Method of Least Squares
Consider the simple formula:
Where the measurement errors are independent samples from N(0, )
How to find the estimators of 0 and 1?
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Choosing the best line
y
x
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Least Squares Estimates
Suggest an index to measure discrepancy between points and line
Focus on vertical disparities between points and line
Sum of the square of the deviations:
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n
1i
2i10i
2i10 )xy(),(L
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Least Squares Estimates
n1i i
n1i i
n
1i
2n
1ii
2i
n
1ii
n
1iin
1iii
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x)n/1(x and y)n/1(y where
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xy
xyˆ
xˆyˆ