Matrix-End Matrix question

[Salbod]

I'm trying to use Matrix-End Matrix to work out a small factor analytic problem (4 variables and 5 cases) from Tabachnick and Fidell's Using Multivariate Statistics (5th Ed.), p 615-628.  I'm able to compute the four eigenvalues of 2.016305104, 1.941513814, .037812306, and .004368776 and the associated eigenvector matrix (see Matrix-End Matrix code below). However, Tabachnick et al., in the section I'm working from,  reanalyses the correlation matrix restricting the solution to two values and vectors: "In this example, when no limit is placed on the number of factors, eigenvalues of 2.02, 1.94, 04, and .00 are computed for each of the four possible factors. Only the first two factors, with values over 1.00, are large enough to be retained in subsequent analyses. FA us rerun specifying extraction of just the first two factors; they have eignevalues of 2.00 and 1.91, respectively, as indicated in Table 13.3. [The new Eigenvectors are included too; p 617-618].

How do I only extract a two factor solution in Matrix-End Matrix?

TIA

Stephen Salbod, Pace University, NYC



MATRIX.
***** This section creates data matrix-A.
COMPUTE A = {32,64,65,67;
             61,37,62,65;
             59,40,45,43;
             36,62,34,35;
             62,46,43,40}.

***** This section creates a deviation matrix-X (A-M).

COMPUTE N = NROW(A).
COMPUTE ONE=MAKE(N,N,1).
COMPUTE X = A - (ONE*A)*(1/N).

***** This section computes the a variance-covariance matrix-S .

COMPUTE tranX = TRANSPOS(X).
COMPUTE prodX = tranX*X.
COMPUTE S = (1/(N-1))*prodX.
COMPUTE invS = INV(S).
PRINT S /TITLE = "Covariance Matrix S".
PRINT invS /TITLE = "Inverse of S".

***** This section computes the correlation matrix-R.

COMPUTE rtdiagS = SQRT(DIAG(S)).
COMPUTE rtD = MDIAG(rtdiagS).
COMPUTE invrtD = inv(rtD).
COMPUTE R = invrtD*S*invrtD.

PRINT R /TITLE = "R, the correlation matrix"
        /FORMAT = "F6.3".

CALL EIGEN(R, EIGENVEC, EIGENVAL).

PRINT EIGENVEC.
PRINT EIGENVAL.


END MATRIX.

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[Maguin, Eugene]

Stephen,

I'm not familiar with T&F nor am I algebraically familiar with different
eigenvalue/vector extraction routines. However, my limited knowledge of
eigenvalues makes me suspicious that the eigenvalues/vectors depend on the
number of them extracted. Since T&F report different values for the first
two factors I wonder whether they may have indulged in some type of
adjustment of the main diagonal, i.e., to adjust the communalities, between
the first and second extractions. What do they say about the model
(principle components, common factor, etc) and the communalities?

Gene Maguin

-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Salbod, Mr. Stephen
Sent: Wednesday, January 18, 2012 10:19 AM
To: [hidden email]
Subject: Matrix-End Matrix question

I'm trying to use Matrix-End Matrix to work out a small factor analytic
problem (4 variables and 5 cases) from Tabachnick and Fidell's Using
Multivariate Statistics (5th Ed.), p 615-628.  I'm able to compute the four
eigenvalues of 2.016305104, 1.941513814, .037812306, and .004368776 and the
associated eigenvector matrix (see Matrix-End Matrix code below). However,
Tabachnick et al., in the section I'm working from,  reanalyses the
correlation matrix restricting the solution to two values and vectors: "In
this example, when no limit is placed on the number of factors, eigenvalues
of 2.02, 1.94, 04, and .00 are computed for each of the four possible
factors. Only the first two factors, with values over 1.00, are large enough
to be retained in subsequent analyses. FA us rerun specifying extraction of
just the first two factors; they have eignevalues of 2.00 and 1.91,
respectively, as indicated in Table 13.3. [The new Eigenvectors are included
too; p 617-618].

How do I only extract a two factor solution in Matrix-End Matrix?

TIA

Stephen Salbod, Pace University, NYC



MATRIX.
***** This section creates data matrix-A.
COMPUTE A = {32,64,65,67;
             61,37,62,65;
             59,40,45,43;
             36,62,34,35;
             62,46,43,40}.

***** This section creates a deviation matrix-X (A-M).

COMPUTE N = NROW(A).
COMPUTE ONE=MAKE(N,N,1).
COMPUTE X = A - (ONE*A)*(1/N).

***** This section computes the a variance-covariance matrix-S .

COMPUTE tranX = TRANSPOS(X).
COMPUTE prodX = tranX*X.
COMPUTE S = (1/(N-1))*prodX.
COMPUTE invS = INV(S).
PRINT S /TITLE = "Covariance Matrix S".
PRINT invS /TITLE = "Inverse of S".

***** This section computes the correlation matrix-R.

COMPUTE rtdiagS = SQRT(DIAG(S)).
COMPUTE rtD = MDIAG(rtdiagS).
COMPUTE invrtD = inv(rtD).
COMPUTE R = invrtD*S*invrtD.

PRINT R /TITLE = "R, the correlation matrix"
        /FORMAT = "F6.3".

CALL EIGEN(R, EIGENVEC, EIGENVAL).

PRINT EIGENVEC.
PRINT EIGENVAL.


END MATRIX.

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[David Marso]

I no longer have my copy of T&F (BTW: I was a student of both authors in the 80's).
AFAIK:  The Eigenvalues are *NOT* going to change as a function of the number of retained components.
The number of retained components serves only as a base for which vectors are subjected to rotation!
I believe we need more detail re what is going on between pg 615-618.

Gene Maguin wrote

Stephen,

I'm not familiar with T&F nor am I algebraically familiar with different
eigenvalue/vector extraction routines. However, my limited knowledge of
eigenvalues makes me suspicious that the eigenvalues/vectors depend on the
number of them extracted. Since T&F report different values for the first
two factors I wonder whether they may have indulged in some type of
adjustment of the main diagonal, i.e., to adjust the communalities, between
the first and second extractions. What do they say about the model
(principle components, common factor, etc) and the communalities?

Gene Maguin

-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Salbod, Mr. Stephen
Sent: Wednesday, January 18, 2012 10:19 AM
To: [hidden email]
Subject: Matrix-End Matrix question

I'm trying to use Matrix-End Matrix to work out a small factor analytic
problem (4 variables and 5 cases) from Tabachnick and Fidell's Using
Multivariate Statistics (5th Ed.), p 615-628.  I'm able to compute the four
eigenvalues of 2.016305104, 1.941513814, .037812306, and .004368776 and the
associated eigenvector matrix (see Matrix-End Matrix code below). However,
Tabachnick et al., in the section I'm working from,  reanalyses the
correlation matrix restricting the solution to two values and vectors: "In
this example, when no limit is placed on the number of factors, eigenvalues
of 2.02, 1.94, 04, and .00 are computed for each of the four possible
factors. Only the first two factors, with values over 1.00, are large enough
to be retained in subsequent analyses. FA us rerun specifying extraction of
just the first two factors; they have eignevalues of 2.00 and 1.91,
respectively, as indicated in Table 13.3. [The new Eigenvectors are included
too; p 617-618].

How do I only extract a two factor solution in Matrix-End Matrix?

TIA

Stephen Salbod, Pace University, NYC



MATRIX.
***** This section creates data matrix-A.
COMPUTE A = {32,64,65,67;
             61,37,62,65;
             59,40,45,43;
             36,62,34,35;
             62,46,43,40}.

***** This section creates a deviation matrix-X (A-M).

COMPUTE N = NROW(A).
COMPUTE ONE=MAKE(N,N,1).
COMPUTE X = A - (ONE*A)*(1/N).

***** This section computes the a variance-covariance matrix-S .

COMPUTE tranX = TRANSPOS(X).
COMPUTE prodX = tranX*X.
COMPUTE S = (1/(N-1))*prodX.
COMPUTE invS = INV(S).
PRINT S /TITLE = "Covariance Matrix S".
PRINT invS /TITLE = "Inverse of S".

***** This section computes the correlation matrix-R.

COMPUTE rtdiagS = SQRT(DIAG(S)).
COMPUTE rtD = MDIAG(rtdiagS).
COMPUTE invrtD = inv(rtD).
COMPUTE R = invrtD*S*invrtD.

PRINT R /TITLE = "R, the correlation matrix"
        /FORMAT = "F6.3".

CALL EIGEN(R, EIGENVEC, EIGENVAL).

PRINT EIGENVEC.
PRINT EIGENVAL.


END MATRIX.

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[Salbod]

TabachnickFidell.pdf

Here is the section I'm working from. --Steve

[David Marso]

It is *NOT* obvious where they are obtaining the alleged eigenvectors.  
AFAICT they are *INCORRECT*.
They certainly don't match anything in SPSS output.
They would not vary based upon the number of components extracted (likewise for the eigenvalues).
My mods of your program with additional calcs.
DATA LIST FREE /V1 TO V4.
BEGIN DATA
32,64,65,67  61,37,62,65  59,40,45,43  36,62,34,35  62,46,43,40
end data.
FACTOR  /VARIABLES v1 v2 v3 v4 /CRITERIA FACTORS(4).
FACTOR  /VARIABLES v1 v2 v3 v4 /CRITERIA FACTORS(2).

MATRIX.
***** This section creates data matrix-A.
COMPUTE A = {32,64,65,67;  61,37,62,65;  59,40,45,43;  36,62,34,35;  62,46,43,40}.
***** This section creates a deviation matrix-X (A-M).
COMPUTE X = A - (MAKE(NROW(A),NROW(A),1)*A)*(1/NROW(A)).

***** This section computes the a variance-covariance matrix-S .

COMPUTE S = (1/(NROW(A)-1))*TRANSPOS(X)*X.
PRINT S /TITLE = "Covariance Matrix S".
PRINT (INV(S))/TITLE = "Inverse of S".
***** This section computes the correlation matrix-R.

COMPUTE rtD = MDIAG(SQRT(DIAG(S))).
COMPUTE invrtD = Ginv(rtD).
COMPUTE R = invrtD*S*invrtD.

*PRINT R /TITLE = "R, the correlation matrix" /FORMAT = "F6.3".
CALL EIGEN(R, EIGENVEC, EIGENVAL).
COMPUTE COMPMat=EIGENVEC *MDIAG(SQRT(EIGENVAL)).
PRINT EIGENVAL /TITLE "EigenValues".
PRINT EIGENVEC / TITLE "Eigenvectors"
PRINT COMPMAT / TITLE "Components Matrix".
*-----------------------------------------------
** VARIOUS FUBAR ATTEMPTS AT FIGURING OUT WTF T&F are doing *.
COMPUTE REDVEC=EIGENVEC(:,1:2).
COMPUTE EVAL2=T(REDVEC)*R*REDVEC.
PRINT EVAL2.
COMPUTE CRED=COMPMAT(:,1:2).
COMPUTE NEWR=CRED*T(CRED).
PRINT NEWR.
CALL EIGEN(NewR, EIGVEC2, EIGVAL2).
PRINT EIGVEC2.
PRINT EIGVAL2.
END MATRIX.

Salbod wrote

TabachnickFidell.pdf

Here is the section I'm working from. --Steve

[David Marso]

I was able to reproduce the eigenvalues from the T&F 2 factor solution by using principal axis factoring (PAF) rather than principle components (PC) . Consult the algorithms for specifics on the difference (primarily, the substitution of the diagonals of R with the estimated communalities at each iteration of the solution).  I have *NO* idea where they are getting the published eigenvectors.

[Salbod]

Hi David,
   I appreciate all the time and thought you've put into this problem. Your modifications to the code are enlightening (makes it readable). And, once again (thank you for pointing it out) I missed the obvious: the communality estimates in the major diagonal. I can now see T&F are unclear in their presentation--that is a major step for me. Thank you!

 
--Steve    

[David Marso]

Hi Steve,
  Moving a little further: I tried the following *QAD* (PAF) implementation with MATRIX.
Results are pretty close but not exact so wondering where I might be straying (see Factor loadings are not as close as one would expect and one of the final communalities exceeds 1).
Might work better with a data set with larger N?
Anyhow, definitely just illustrative and not intended for production use.
--
DATA LIST FREE /V1 TO V4.
BEGIN DATA
32,64,65,67  61,37,62,65  59,40,45,43  36,62,34,35  62,46,43,40
end data.
FACTOR  /VARIABLES v1 v2 v3 v4 /CRITERIA FACTORS(2)/CRIT ITER(50)/extraction=paf .
CORR V1 TO V4 / MATRIX OUT (*).
MATRIX.
MGET FILE */TYPE=COR .
PRINT CR /TITLE = "R, the correlation matrix" /FORMAT = "F6.3".
***------------------------------------------------------------***.
CALL EIGEN(CR, EVEC, EVALue).
COMPUTE COMPMat=EVEC *MDIAG(SQRT(EVALue)).
PRINT EVALue /TITLE "Initial EigenValues".
PRINT EVEC / TITLE "Initial Eigenvectors".
PRINT COMPMAT / TITLE "Initial Components Matrix".
COMPUTE NFAC=NCOL(CR).
LOOP #=1 TO NCOL(CR).
+  DO IF EVALue(#) <1.
+    COMPUTE EVALue(#)=0.
+    COMPUTE NFAC=NFAC-1.
+  END IF.
END LOOP.
LOOP #0=1 TO 25.
* CR-(I-MDiag(Communalities)).
+  CALL EIGEN(CR-MDIAG(MAKE(NCOL(CR),1,1)-RSUM(EVEC&*EVEC *MDIAG(EVALue))) ,EVEC, EVALue).
+  COMPUTE EVALue=EVALUE(1:NFAC).
+  COMPUTE EVEC=EVEC(:,1:NFAC).
END LOOP.
COMPUTE COMPMat=EVEC *MDIAG(SQRT(EVALue)).
COMPUTE C=MDIAG(RSUM(EVEC&*EVEC *MDIAG(EVALue)) ).
PRINT EVEC /TITLE "Eigenvectors from PAF".
PRINT EVALue / TITLE "Eigenvalues from PAF".
PRINT COMPMAT /TITLE "Factor Matrix".
PRINT (T(DIAG(C)))/TITLE "Communalities".
END MATRIX.

** OUTPUT from FACTOR.
Communalities
  Initial Extraction
V1 .961 .970
V2 .953 .960
V3 .990 .989
V4 .991 .996
Extraction Method: Principal Axis Factoring.

Total Variance Explained
Factor Initial Eigenvalues Extraction Sums of Squared Loadings
        Total % of Variance Cumulative % Total % of Variance Cumulative %
1 2.016 50.408 50.408 2.005 50.118 50.118
2 1.942 48.538 98.945 1.909 47.733 97.852
3 .038 .945 99.891
4 .004 .109 100.000
Extraction Method: Principal Axis Factoring.

Factor Matrix
  Factor
        1 2
V1 -.400 .900
V2 .251 -.947
V3 .932 .348
V4 .956 .286
Extraction Method: Principal Axis Factoring.
a 2 factors extracted. 4 iterations required.

** Output from MATRIX **.
R, the correlation matrix
  1.000  -.953  -.055  -.130
  -.953  1.000  -.091  -.036
  -.055  -.091  1.000   .990
  -.130  -.036   .990  1.000

Initial EigenValues
   2.016305104
   1.941513814
    .037812306
    .004368776

Initial Eigenvectors
   .3524130186   .6143298243   .6624913038   .2439450833
  -.2511248167  -.6637642199   .6758934421   .1988000044
  -.6273986982   .3222290586   .2754625014  -.6531919454
  -.6473888045   .2796146654  -.1685044144   .6887014135

Initial Components Matrix
   .5004147107   .8559962110   .1288239950   .0161239696
  -.3565888485  -.9248772154   .1314300926   .0131400280
  -.8908851872   .4489882182   .0535647483  -.0431738443
  -.9192704702   .3896100835  -.0327663348   .0455209036

Eigenvectors from PAF
  -.2880998600   .6499228582
   .1819297804  -.6830412264
   .6508185424   .2551003796
   .6784801033   .2144265791

Eigenvalues from PAF
   2.005761789
   1.909914895

Factor Matrix
  -.4080211955   .8981912491
   .2576579056  -.9439607249
   .9217212384   .3525478843
   .9608969018   .2963368260

Communalities
    .973228816    .957449446    .973860052   1.011138370

------ END MATRIX -----

Salbod wrote

Hi David,
   I appreciate all the time and thought you've put into this problem. Your modifications to the code are enlightening (makes it readable). And, once again (thank you for pointing it out) I missed the obvious: the communality estimates in the major diagonal. I can now see T&F are unclear in their presentation--that is a major step for me. Thank you!

 
--Steve