How to Calculate the Component Score Coefficient Matrix

[Alex Cao]

Hi, I'm trying to calculate the "Component Score Coefficient Matrix"
when using the extraction method of Principal Component Analysis and a
covariance matrix (for a program I am writing).  I want the
Standardized coefficients but can't seem to get the right numbers.
I'm thinking it is related somehow to the "Component Matrix" but am
not sure.  Any help would be appreciated.

Alex

[Dan Zetu]

I hope I am understanding your question correctly. The component scores are
already standardized, therefore the score coefficients are standardized
coefficients. I am not sure this was your question though.

Dan

>From: Alex Cao <[hidden email]>
>Reply-To: Alex Cao <[hidden email]>
>To: [hidden email]
>Subject: How to Calculate the Component Score Coefficient Matrix
>Date: Thu, 24 Aug 2006 13:34:18 -0400
>
>Hi, I'm trying to calculate the "Component Score Coefficient Matrix"
>when using the extraction method of Principal Component Analysis and a
>covariance matrix (for a program I am writing).  I want the
>Standardized coefficients but can't seem to get the right numbers.
>I'm thinking it is related somehow to the "Component Matrix" but am
>not sure.  Any help would be appreciated.
>
>Alex

[Alex Cao]

Sorry, let me be more clear about my previous email.  I want to
reproduce the "component score coefficient matrix" in a non-SPSS
program and am having trouble figuring out how it is calculated.

Alex


On 8/24/06, Dan Zetu <[hidden email]> wrote:

> I hope I am understanding your question correctly. The component scores are
> already standardized, therefore the score coefficients are standardized
> coefficients. I am not sure this was your question though.
>
> Dan
>
>
> >From: Alex Cao <[hidden email]>
> >Reply-To: Alex Cao <[hidden email]>
> >To: [hidden email]
> >Subject: How to Calculate the Component Score Coefficient Matrix
> >Date: Thu, 24 Aug 2006 13:34:18 -0400
> >
> >Hi, I'm trying to calculate the "Component Score Coefficient Matrix"
> >when using the extraction method of Principal Component Analysis and a
> >covariance matrix (for a program I am writing).  I want the
> >Standardized coefficients but can't seem to get the right numbers.
> >I'm thinking it is related somehow to the "Component Matrix" but am
> >not sure.  Any help would be appreciated.
> >
> >Alex
>

[Hector Maletta]

Alex,
As Dan explains, one of the matrices you get in the output produced by the
FACTOR procedure in SPSS is the matrix of factor score coefficients. The
rows represent observed variables, and the columns represent extracted
factors. The cells contain the coefficients. The score of factor Fi is the
sum of biZi, where Zi is the standardized version of observed variable Xi,
and Fi is a standardized factor score. Both Zi and Fi have zero means and
unit standard deviations. My symbol b stands for the Greek letter beta
usually reserved for the standardized regression coefficients in the
standardized regression equation which lacks a constant. The direct or
non-standardized regression equation would be Y=b0+sum(biXi) where b is the
old and reliable b of our own Latin alphabet.
If you browse the SPSS algorithm for the FACTOR procedure (available in the
installation CD and also at the SPSS website) you may find the
matrix-algebra formula of the coefficients, enabling you to put them into
your program. It is a standard regression, only without the error term.
Hector

-----Mensaje original-----
De: SPSSX(r) Discussion [mailto:[hidden email]] En nombre de Dan
Zetu
Enviado el: Thursday, August 24, 2006 2:40 PM
Para: [hidden email]
Asunto: Re: How to Calculate the Component Score Coefficient Matrix

I hope I am understanding your question correctly. The component scores are
already standardized, therefore the score coefficients are standardized
coefficients. I am not sure this was your question though.

Dan

>From: Alex Cao <[hidden email]>
>Reply-To: Alex Cao <[hidden email]>
>To: [hidden email]
>Subject: How to Calculate the Component Score Coefficient Matrix
>Date: Thu, 24 Aug 2006 13:34:18 -0400
>
>Hi, I'm trying to calculate the "Component Score Coefficient Matrix"
>when using the extraction method of Principal Component Analysis and a
>covariance matrix (for a program I am writing).  I want the
>Standardized coefficients but can't seem to get the right numbers.
>I'm thinking it is related somehow to the "Component Matrix" but am
>not sure.  Any help would be appreciated.
>
>Alex

[Hector Maletta]

Alex,
I have just sent you (privately) the SPSS FACTOR algorithm. I hope you find
the answer there.
In the most common applications, factor scores are obtained by regression,
where the score of the kth factor for the jth case is Fkj=sum(h=1 to m) of
(bkhj Zhj), summed over the m variables.
For all factors and variables this can be expressed more compactly as
F=S'inv(R)Z where S' is the transpose of the factor structure matrix, inv(R)
is the inverse of the correlation coefficients matrix (with ones in the
diagonal in the case of Principal Component method), and Z is the matrix of
values for N cases and M standardized variables.
Regression estimation of factor scores sometimes (but rather rarely) happens
to generate scores that are somewhat inter-correlated, even when the factors
are supposed to be orthogonal. For this reason other methods of estimation
such as Bartlet's method are occasionally used.
Hope this helps. I have not checked, however, whether the above description
exactly fits the algorithm followed by SPSS programming.
Hector

-----Mensaje original-----
De: SPSSX(r) Discussion [mailto:[hidden email]] En nombre de Alex
Cao
Enviado el: Thursday, August 24, 2006 3:03 PM
Para: [hidden email]
Asunto: Re: How to Calculate the Component Score Coefficient Matrix

Sorry, let me be more clear about my previous email.  I want to
reproduce the "component score coefficient matrix" in a non-SPSS
program and am having trouble figuring out how it is calculated.

Alex


On 8/24/06, Dan Zetu <[hidden email]> wrote:
> I hope I am understanding your question correctly. The component scores
are

> already standardized, therefore the score coefficients are standardized
> coefficients. I am not sure this was your question though.
>
> Dan
>
>
> >From: Alex Cao <[hidden email]>
> >Reply-To: Alex Cao <[hidden email]>
> >To: [hidden email]
> >Subject: How to Calculate the Component Score Coefficient Matrix
> >Date: Thu, 24 Aug 2006 13:34:18 -0400
> >
> >Hi, I'm trying to calculate the "Component Score Coefficient Matrix"
> >when using the extraction method of Principal Component Analysis and a
> >covariance matrix (for a program I am writing).  I want the
> >Standardized coefficients but can't seem to get the right numbers.
> >I'm thinking it is related somehow to the "Component Matrix" but am
> >not sure.  Any help would be appreciated.
> >
> >Alex
>