Transcript Trivariate Regression By Hand

Trivariate Regression
By Hand
Variable
ar
ideal
misanth
ar
ideal
N
54
154
154
Mean
2.37969
3.65024
2.32078
ar
ideal
1.00000
0.05312
0.5129
0.05312
1.00000
0.5129
Std Dev
0.53501
0.53278
0.67346
misanth
0.22064
0.0060
r
p
-0.13975
0.0839
r
p
Beta Weights
1 
2 
r y 1  r y 2 r12
1 r
2
12
r y 2  r y 1r 12
1 r
2
12

. 05312  . 22064 (  0 . 13975 )
1  . 13975

2
 . 0856
. 22064  . 05312 (  0 . 13975 )
1  . 13975
2
 . 2326
Zˆ AR  (. 0856 ) Z Idealism  (. 2326 ) Z Misanthrop
y
Unstandardize the Weights
bi 
b Idealism 
b Misanthrop
y
 isy
si
. 0856 (. 535 )
. 53278

 . 086
. 2326 (. 535 )
. 67346
 . 185
Obtain the Intercept
a Y 
 bX
i
a  2 . 38  . 086 ( 3 . 65 )  . 185 ( 2 . 32 )  1 . 637
AR  1 . 637  (. 086 )Idealism  (. 185 )Misanthrop y
Obtain R2
r y 1  r y 2  2 r y 1r y 2 r 12
2
R
. 05312
2
2
y 12
 . 22064

2
2
1 r

 2 (. 05312 )(. 22064 )(  0 . 13975 )
1  . 13975
R 
2
12
2
. 0559  . 236
 . 0559
R2 For 2 or More Predictors
R
R
2
y 12
2
y 12 ... i ... p


r
i yi
r
2
y yˆ
 . 0856 (. 05312 )  . 2326 (. 22064 )  . 0559
Test the R2
• SSy = (N-1)s2 = 153(.53501)2 = 43.794
• SSregr = R2SSY = .0559(43.794) = 2.447;
df = p = 2; MSregr = 2.447/2 = 1.2235
• SSerror = SSY – SSregr = 43.794 - 2.447 =
41.347; df = n-p-1 = 151;
MSerror = 41.347/151 = 0.2738
• F(2, 151) = 1.2235/0.2738 = 4.468, p = .013
Obtain the Semipartial Corrs2
sr i  R
2
sr
sr
2
Idealism
R
 . 0559  . 22064
2
Misanthrop y
sr Idealism 
2
y 12 ... i ... p
2
y 12 ...( i )... p
2
 . 0559  . 05312
. 0072  . 0849
sr Misanthrop
y

 . 0072
2
 . 0531
. 0531  . 2304
Obtain the Partial Corrs2
pr i 
2

pr
2
Idealism
pr
2
Misanthrop y
sr i
1 R
2
2
y 12 ...( i )... p
. 0072
1  . 22064

2
 . 0076
. 0531
1  . 05312
2
 . 0533
Test the Partials
t  sr
N  p 1
1 R
df = N - p – 1 = 151
2
t Ideal  . 085
t Misanth  . 230
154  2  1
1  . 0559
 1 . 07
154  2  1
1  . 0559
 2 . 91
Also See
• Obtaining Exact Significance Levels with
SAS
• Correlation and Regression Analysis:
SAS - bivariate and trivariate, with plots
– Annotated output