Sangdon Lee schrieb:
>
> Dear All,
>
> I have four independent variables X1s X2s X3s X4s and one dependent
> variable Ys. All of them are mean-centered and variance-scaled. I
> performed factor analysis using principal component extraction with
> Varimax rotation and 4 factors were extracted. (I'll reduce the number
> of factors later. For this question 4 factors were extracted.)
>
> The 4 factor scores (fs1 fs2 fs3 fs4) were computed and Ys was
> regressed on the 4 factor scores. I'm wondering how the principal
> component regression (PCR) can re-expressed with the four independent
> variables instead of the 4 factor scores.
>
> For example, let's say the PCR equation is Ys = 0.000 + 0.472*fs1 +
> 0.352*fs2 + 0.343*fs3 - 0.419*fs4. If I replace the fs1 with factor
> score coefficients (FSC) for factor 1, fs2 with FSC for factor 2, and
> so forth, would I get the coefficients for the four independent
> variables ?
>
> I appreciate any help.
>
> Sangdon Lee, Ph.D.,
> GM Tech. Center.
> [EMAIL PROTECTED]
With Ax the factor-loadingsmatrix of the Varimax-solution for X,
FSC the factor-scores and X and Y the data (horizontal vectors for each variable)
Ax * FSC = X
so
FSC = Ax^-1 * X
Maybe I don't understand your question right.
But for the case I do, it's just a matrix equation.
[ fs1-scores...........
[ fs2-scores......... FSC = Matrix of factor-scores
[ fs3-scores.........
[ fs4-scores..........
------------------------------------------------------------------------
[L11 L12 L13 L14] ![ x1-data ................
[L21 L22 L23 L24] ![ x2-data ................
[L31 L32 L33 L34] ![ x3-data ................ X
[L41 L42 L43 L44] ![ x4-data ................
[L51 L52 L53 L54] ![ y -data ................ Y
L = loadingsmatrix, where L1_4 (the submatrix of L from row 1 to 4)
is in varimax-position, L5 is the submatrix L from row 5 to 5, only a
single row). [X,Y] the combined matrix of X and Y-scores.
Then
FSC = inv( L1_4) * X
And
[X,Y] = [L1_4 , L5] * FSC
So
[X,Y] = [L1_4 , L5] * inv(L1_4) * X
and the B1_4 part of
B1_4 = L1_4 * inv(A1_4)
is a identity-matrix and the B5-part (the loadings of Y on the variables)
B5 = L5 * inv(A1_4)
contains the beta weights (are in terms of X)
Y = B5 * X
Gottfried Helms
.
.
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