Thanks for the explanation. My appreciation to Gottgried Helms and
Donald Burrill.
Just for a record, I found a detail example from the Edward Jackson's
book "A user's guide to principal components", pp271.
In summary, once I regressed the "Ys" on the four factor scores
("fs1", "fs2", "fs3", "fs4"), replace the four factor scores with
Factor Score Coefficients (FSC). See the Minitab output below.
Therfore, the regression equation is
Ys = 0.000 + 0.472*fs1 + 0.352*fs2 + 0.343*fs3 - 0.419*fs4
Ys = 0.000 + 0.472*(-0.105*X1s+1.146*X2s+0.089*X3s-0.303*X4s)
+ 0.352*(-0.195*X1s+0.097*X2s-0.081*X3s-0.278*X4s)
+ 0.343*(-0.084*X1s-0.087*X2s-1.104*X3s-0.237*X4s)
- 0.419*(-0.365*X1s+0.371*X2s-0.294*X3s-1.367*X4s)
= -0.311*X1s+0.389*X2s-0.242*X3s+0.248*X4s
Minitab output:
## Factor Analysis: X1s, X2s, X3s, X4s
>> ........
Principal Component Factor Analysis of the Correlation Matrix
>>......
Rotated Factor Loadings and Communalities
Varimax Rotation
Variable Factor1 Factor2 Factor3 Factor4 Communality
X1s 0.023 -0.981 0.023 0.191 1.000
X2s 0.962 -0.025 0.142 -0.234 1.000
X3s -0.139 0.023 -0.971 0.195 1.000
X4s 0.286 0.251 0.243 -0.892 1.000
Variance 1.0265 1.0265 1.0216 0.9255 4.0000
% Var 0.257 0.257 0.255 0.231 1.000
Factor Score Coefficients (FSC)
Variable Factor1 Factor2 Factor3 Factor4
X1s -0.105 -1.095 -0.084 -0.365
X2s 1.146 0.097 -0.087 0.371
X3s 0.089 -0.081 -1.104 -0.294
X4s -0.303 -0.278 -0.237 -1.361
## Regression Analysis: Ys versus fs1, fs2, fs3, fs4
The regression equation is
Ys = 0.000 + 0.472 fs1 + 0.352 fs2 + 0.343 fs3 - 0.419 fs4
Predictor Coef SE Coef T P VIF
Constant 0.0000 0.1097 0.00 1.000
fs1 0.4724 0.1114 4.24 0.000 1.0
fs2 0.3523 0.1114 3.16 0.004 1.0
fs3 0.3426 0.1114 3.08 0.005 1.0
fs4 -0.4190 0.1114 -3.76 0.001 1.0
S = 0.6398 R-Sq = 64.0% R-Sq(adj) = 59.1%
PRESS = 15.7834 R-Sq(pred) = 52.17%
.
.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
. http://jse.stat.ncsu.edu/ .
=================================================================
