Download Summary formula sheet for simple linear regression Slope b = (Y

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Summary formula sheet for simple linear
regression
_
_
_
Slope b = ! (Yi -Y)(Xi -X) / ! (Xi -X)2
_
Variance 5 2 / ! (Xi -X)2
_
_
Intercept a= Y - b X _
2
1
X _
2
Variance of a [ n + !
]
5
2
(Xi -X)
Estimated mean at X_ 0 a + b X0
(X0 -X)_2
1
2
Variance [ n + !
]
5
2
(Xi -X)
Estimated individual at _X0 a + b X0
(X0 -X)_2
1
2
Variance [1 + n + !
]
5
(Xi -X)2
_
Total SS = ! (Yi -Y)2
Regression
_ SS =_
_
2
[ ! (Yi -Y)(Xi -X)] / ! (Xi -X)2
Error SS = Total SS - Regression SS
R2 = Regression SS/ Total SS = "proportion
explained"
MSE = error mean square = estimate of 5 2
= Error SS/ df
df= degrees of freedom = n-2 for simple
linear.
Example
Data points (x1 ,y1 ), (x2 ,y2 ), ...., (xn ,yn )
_ (1,5), (2,7),_ (3,9), (4,6), (5,8)
x = 15/5=3, y = 7
Corrected sum of squares for x:
n
_
!(xi - x)2 = Sxx = (1-3)2 +...+(5-3)2 = 10
i=1
Corrected sum of squares for y:
n
_
!(yi -y)2 = Syy = (5-7)2 +...+(8-7)2 = 10
i=1
Corrected sum of cross products = Sxy =
n
_
_
!(xi - x)(yi -y) =
i=1
(-2)(-2)+(-1)(0)+...+(2)(1) = 5 =
__
!xi yi -n x y = 110-5(3)(7)
n
i=1
Slope: b = Sxy /Sxx = 5/10 = 0.5
Intercept:
_
_
y - b x = 7-0.5(3) = 5.5
^
y=5.5
+ 0.5x
y
y^
r=y- y^
5
6
-1
7
6.5
0.5
9
7
2
6
7.5
-1.5
!ri2 = "Error sum of squares" =
8
8
0
n
i=1
SSE = 1+0.25+4+2.25=7.5
SSE is also Syy - S2xy /Sxx = Syy - b2 Sxx =
10-52 /10
Variance of b:
MSE/Sxx œ 2.5/10 = 0.25.
ÈMSE/Sxx is called "standard error" of b.
Task: test H0 : true slope is 0
t = b/È0.25 = 1 which is not an
unusual t.
data a; input x y @@; cards;
15
27
39 46 58
;
proc reg; model Y =X / p;
run;
Dependent Variable: y
Analysis of Variance
Source
DF
Model
1
Error
3
Corr Total 4
Root MSE
Dependent Mean
Coeff Var
Sum of
Squares
2.50000
7.50000
10.00000
1.58114
7.00000
22.58770
Mean
Square F Value
2.50000
1.00
2.50000
R-Square
Adj R-Sq
Pr > F
0.3910
0.2500
0.0000
Parameter Estimates
Variable
Intercept
x
DF
1
1
Parameter Standard
Estimate
Error t Value
5.50000 1.65831
3.32
0.50000 0.50000
1.00
Output Statistics
Pr > |t|
0.0452
0.3910
Obs
Dep Var
y
Predicted
Value
Residual
1
2
3
4
5
5.0000
7.0000
9.0000
6.0000
8.0000
6.0000
6.5000
7.0000
7.5000
8.0000
-1.0000
0.5000
2.0000
-1.5000
0
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