Transcript Ch12.1 Simple Linear Regression The Simple Linear Regression Model There exists parameters 0 , 1 and  2 such that for any fixed.

Ch12.1 Simple Linear Regression
The Simple Linear Regression Model
There exists parameters 0 , 1 and  2 such that for
any fixed value of x, the dependent variable is related to
x through the model equation
y  0  1x  
ε is a random variable (called the random deviation) with
E(ε) = 0 and V(ε) = σ2
One can see how a dependent variable is related to an
independent variable with a scatter plot.
Ch12.1
Ch12.1 Estimating Model Parameters
Principle of Least Squares
The vertical deviation of the point (xi,yi) from the line
y = b0 + b1x is yi – (b0 + b1xi)
The sum of squared vertical deviations from the points
( x1 , y1 ), ( x2 , y2 ),..., ( xn , yn )to the line is:
n
f (b0 , b1 )    yi   b0  b1xi  
i 1
Ch12.2
2
The error sum of squares, denoted SSE, is
SSE=   yi  yˆi 
2


   yi  ˆ0  ˆ1 xi 


2
and the estimate of σ2 is
SSE
ˆ  s 

n2
2
2
  yi  yˆi 
2
n2
A computational formula for the SSE, is
SSE=  yi2  ˆ0  yi  ˆ1  xi yi
The total sum of squares, denoted SST, is
SST  S yy    yi  y   
2
2
yi
   yi  / n
2
The coefficient of determination, denoted by r2, is given by
SSE
r  1
SST
2
Ch12.2
The least-squares (regression) line for the data is given by
y  ˆ0  ˆ1x where
b1  ˆ1
x y    x   y  / n


 x    x  / n
i
i
i
2
2
i
i
and
b0  ˆ0 
i
ˆ x
y


 i 1i
n
 y  ˆ1 x
ˆ1,..., y
ˆ n are obtained by
The fitted (predicted) values y
substituting x1,..., xn into the equation of the
estimated regression line: yˆ1  ˆ0  ˆ1x1,..., yˆn  ˆ0  ˆ1xn .
The residuals are the vertical deviations y1  yˆ1,..., yn  yˆn
from the estimated line.
Ch12.2