regression analysis -...
TRANSCRIPT
-
Regression Analysis
Basic concept
-
Regression Analysis
1805(Legendre)1809(Gauss)
Francis Galton regression to the mean
-
Types of Relationships
Y
X
Y
X
Y
Y
X
X
Linear relationships Curvilinear relationships
DCOVA
Copyright 2014 Statistics for Managers Using Microsoft Excel 7th Edition, Global Edition
-
Types of Relationships(continued)
Y
X
Y
X
Strong relationships
Y
Y
X
X
Weak relationships
DCOVA
Copyright 2014 Statistics for Managers Using Microsoft Excel 7th Edition, Global Edition
Y
X
Y
X
No relationship
-
independent variableX
dependent variable Y
explanatory variable
response variable
-
Y
(X)
simple regression analysisy=fx
multiple regression analysisy=fx1, x2, x3, ., xn
multivariate regression analysisy1, y2, .., ym=fx1, x2, ., xn
linear regression
nonlinear regression
-
simple regression analysis
-
xi
yi
xiyi
= 0 + 1
fy xi
= 0 + 1
-
xix1
= = 0 + 1
fy xi
yxiyx1
xi
x1
50 kg
50kg170 cm
-
1.y
2.homocedasticityy
Var(y)=12 = 2
2 = = 2 =
3.
= 0 + 1 01
4.XY
5.0
N0, 2 Cov , j 0
-
()
xix1
= 0 + 1
( xi , yi )
(xi , i )
( xi , yi ) 00 11
-
Y
XXi
i
( xi , yi )
(xi , i )
=
= 0 + 1
()i Xi
-
i
=
min2 = min
2 = min 0 12 = min (SSE)
-
min 0 12= min (SSE) 0 1
0
SSE
0= 0
SSE
1= 0
2 0 1(1) = 0
2 0 1() = 0
= 0 + 1 (1)
= 0 + 12
1 =( )( )
( )2
0 =
2
2 ()
2
=
2 2
=
2
Sxy XYSx
2 X
1
(1) 0 = 1 x
y 0
-
1. 0 1 0 1 E 0 0
E 1 1
2. 0 1 2
Var 0
2
2
2 Var 12
2
3. 0 1
0~ 0,
2
2
2 1~ 1,2
2
4. 0 1
-
1
()
(F)
-
(Coefficient of Determination)
Y
XXi
i
( xi , yi )
(xi , i )
ei
y = y +
=
= 0 + 1
Coefficient of Determinationr
-
y2 = y +
2
y2 = y
2 + 2
SST
SSR
SSE
R y2 = 0 + 1
2= 1 + 1
2
12
2 = 12xi
2 nx2
T y2 =yi
2 ny2 = n 1 y2 = n y
2
SSESST - SSR
= 12 n 1
1
1
2 = 12 n 1
2 = 12 n
2
-
SSESSR
2 =
= 1
R2
2 =
2
2 =
0 + 1 2
2 =
1 + 1 2
2 =
12
2
2
=12
2 2
2 2
=12 1 1
2
1 1
2
= 1
2
= 122
2 =
2
2
Coefficient of Determinationr
-
Examples of Approximate R2 Values
R2 = 1
Y
X
Y
X
R2 = 1
R2 = 1
Perfect linear relationship
between X and Y:
100% of the variation in Y is
explained by variation in X
DCOVA
Coefficient of Determinationr
-
Examples of Approximate R2 Values
Y
X
Y
X
0 < R2 < 1
Weaker linear relationships
between X and Y:
Some but not all of the
variation in Y is explained
by variation in X
DCOVA
Coefficient of Determinationr
-
Examples of Approximate R2 Values
R2 = 0
No linear relationship
between X and Y:
The value of Y does not
depend on X. (None of the
variation in Y is explained
by variation in X)
Y
XR2 = 0
DCOVA
Coefficient of Determinationr
-
F
H01 = 0 ()
H11 0 ()
xy
SSTSSR()SSE()
-
F
F
SSR 1 MSR=SSR/1
F*=SR
SE SSE n-2 MSE=SSE/(n-2)
SST n-1
F* F, 1, n-2 H0
MSRSSRH0
-
R2F
MSRSSRH0R
2
2 =
F*=SR
SE=
SSR 1
SSE n2
F*=SR
SE=
SSR 1
SSE n2=
n2SSE
SSTSSE
SST
=(n2)R2
SSTSSR
SST
=(n2)R2
1R2
=R2
1
(1 R2)
n 2
R2H0R2FH0 R
2F
-
2
-
Y
XXi
i
( xi , yi )
(xi , i )
1 0
Slope = 1 =
(0, 0)
Intercept
b
a
= 0 + 1
-
1
H01 = 0 ()
H11 0 ()
1~ 1,2
2 2
t
t =1 1S1
=1
MSE xi x
2
t*t/2, n-2 H0XY
-
1
1~ 1,2
2 2
t1-
t =1 1S1
=1
MSE xi x
2
t1-
1 t2,n2
MSE
xi x2 1 1 + t
2,n2
MSE
xi x2
-
1
H01 0 ()
H11 > 0 ()
1~ 1,2
2 2
t
t =1 1S1
=1
MSE xi x
2
t*t, n-2 H0XY
-
1
H01 0 ()
H11 < 0 ()
1~ 1,2
2 2
t
t =1 1S1
=1
MSE xi x
2
t*- t, n-2 H0XY
-
0
H00 = 0 ()
H10 0 ()
2
t
t =0 0S0
=0
2
2
t*t/2, n-2 H0
0~ 0,
2
22
-
0
2
t1-
t1-
0 t2,n2
2
2 0 0 + t
2,n2
2
2
0~ 0,
2
22
t =0 0S0
=0
2
2
-
Y
X
= 0 + 1
50kg
50 kg
50kg170 cm
50kg
-
= 0 + 1
fy xi
xi
(xi , i )
-
xy
Eyxixi
t2,df
= t2, 2
1
+
2
2
= 0 + 1 xi
~ ,1
+
2
2
2
(MSE2)
-
xy
yixi
= t2, 2
1 +1
+
2
2
= 0 + 1 xi
(y)~ 0 , 1 +1
+
2
2
2
(y)~ E(y), Var()
-
1.
= t2, 2
1
+
2
2
2.MSE
3. xi
4. n
5. xi
http://faculty.cas.usf.edu/mbrannick/regression/Prediction.html
-
(Correlation Analysis)
3
-
(Correlation Analysis)
=
=
2
2
= ,
=
1 1
-
(correlation coefficient)
1.xyxy = 0
2. xy = 0 xy
xy
3.xy = 1
4.xy = 1
-
=
2
2=
1. rxy xy
2. xy = 0 xy
3. 2 = 2
4. rxy = R2 1
-
1 =
2
=
1 =
2 =
=
= 1 xy
2 = 122
2
2 = 1
2
= 2
= 2 1
-
()
0 = 0
1 0
=
1 2
2
t > t2,n2 H0
t =1
MSE xi x
2
-
The End