Principal Axis Factoring #2 rawpar.sps [1/2]

[krisscot]

[Art Kendall]

You sent 2 messages that appear to have attachments.  But the attachments do not appear to be any common formats.

Is this the syntax that can be downloaded from

https://people.ok.ubc.ca/brioconn/nfactors/nfactors.html

If not what filetype are the attachments?

Art Kendall
Social Research Consultants

On 2/11/2011 8:41 AM, krisscot wrote: ===================== To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD

[Mike]

Okay, it's been while since I've done this kind of review but if I make any

mistakes I hope that more knowledgeable people will correct my errors.

Here goes:

 

First, one way of thinking about factor analysis is that it takes a square

matrix and tries to determine if there is a smaller matrix that contains the

information of the larger matrix.

 

Consider a correlation matrix that has n variables.  This means we have

a n x n matrix.  But, by definition, if the correlation matrix has non-zero

off diagonal elements (i.e., non-zero correlations between variables),

then information in one variable is "redundant" with information in other

variables.  The question now is can this matrix be reduced to a smaller

number of rows/columns, say, an m x m (where m < n) where the rows

and columns are independent and the off-diagonal elements are all zero.

This number of row/columns in this smaller square matrix is referred

to as the "rank" of the matrix.  Old school factor analysis can be thought

of as being concerned with finding the rank of a correlation matrix;

the rank represents that number of "factors" needed to explain the

original correlation matrix.

 

Principal factor analysis is one way to reduce a correlation matrix to a

smaller matrix represented by independent factors.  This is done by

using a series of equations that relate the original matrix to the reduced

matrix.  A loading matrix relating the factors to original data is estimated

(the Lambda or loadings of variables on factors) and the eigenvalues are

part of the solution.  If m=5, then the value of the eigenvalues will be

one set of values because you are trying to calculating the loading

for five factors.  If you limit the number of factor to m=3, the solutions

or eigenvalues will be different because you are now estimate the

loading of variables on 3 factors.

 

In mathematical terms, eigenvalues greater than 1.00 should represent

solutions to solving the set of simultaneous equations needed to reduce

the correlation matrix to the factor matrix.  Eigenvalues less than 1.00

technically are really equal to zero.  But correlation matrices often will

behave badly (the pattern of interrelationships are far more complex than

represented by the simple pairwise correlations, e.g, errors in one variable

are correlated with errors in another variable), and the estimation procedure

might go awry, which is why negative eigenvalues (mathematically,

imaginary numbers are obtained as a solution) might occur.

 

So, when engaged in exploratory factor analysis, you are asking the program

what is the "rank" a correlation matrix can be reduced to, that is, a smaller

square matrix where rows/columns are independent.  The quality of the

data in important.  If you think you know what the rank should be or, in other

words, the number of factors that underlie the variables you have, then

you can specify the rank/number of factors.  You will get different estimates

depending upon how far from the "true" solution you are.  Unfortunately,

Principal factor can't be used to test factor models, all it can do is try to

provide a solution under certain assumptions (e.g., pairwise correlations

are the only relationships, errors are uncorrelated, the factors are independent

or orthogonal, etc.). 

 

Real life data and correlation matrices based on them is likely to be more

complex which is why Structural Equation Modeling (SEM) might be used

because you can explicitly identify how many factors there should be,

whether they are independent or correlated, whether the errors/unique

variances are constant and uncorrelated and so on.  But one would have

to take a course on SEM to fully understand and appreciate these issues.

 

I hope I didn't much of the above wrong.

 

-Mike Palij

New York University

[hidden email]

 

----- Original Message -----

From: [hidden email]

To: [hidden email]

Sent: Friday, February 11, 2011 8:41 AM

Subject: Principal Axis Factoring #2 rawpar.sps [1/2]

Thanks to everyone who responded, but I fear that I may not have explained myself well in the original email. Let me try again with a more basic question. I first have run a PAF not specifying any number of factors. When I consult the Total Variance Explained table below the Eigenvalues for the first three factors based on the PAF are 7.249, 2.324, and 1.041. When I run exactly the same code but specify 3 factors the Eigenvalues for these three factors based on the PAF are 7.197, 2.262, .986. I have placed these tables within the message below. This is my fundamental question: why are the eigenvalues different? They are both based on PAF extraction prerotation.

 

PAF solution without specifying the number of factors.

 

Total Variance Explained
Factor	Initial Eigenvalues			Extraction Sums of Squared Loadings			Rotation Sums of Squared Loadingsa
	Total	% of Variance	Cumulative %	Total	% of Variance	Cumulative %	Total
1	7.705	32.105	32.105	7.249	30.205	30.205	5.319
2	2.864	11.934	44.039	2.324	9.684	39.889	2.713
3	1.515	6.314	50.353	1.041	4.337	44.226	2.901
4	1.159	4.831	55.184	.629	2.620	46.846	4.688
5	1.067	4.448	59.632	.590	2.456	49.302	3.854
6	.800	3.333	62.965
7	.767	3.197	66.162
8	.685	2.855	69.017
9	.680	2.832	71.849
10	.646	2.691	74.540
11	.619	2.578	77.118
12	.569	2.369	79.487
13	.553	2.302	81.789
14	.534	2.224	84.013
15	.512	2.132	86.145
16	.476	1.984	88.129
17	.457	1.904	90.033
18	.455	1.895	91.927
19	.428	1.784	93.711
20	.390	1.625	95.336
21	.354	1.476	96.812
22	.288	1.198	98.010
23	.253	1.054	99.065
24	.224	.935	100.000
Extraction Method: Principal Axis Factoring.
a. When factors are correlated, sums of squared loadings cannot be added to obtain a total variance.

 

 

PAF solution specifying 3 factors:

 

Total Variance Explained
Factor	Initial Eigenvalues			Extraction Sums of Squared Loadings			Rotation Sums of Squared Loadingsa
	Total	% of Variance	Cumulative %	Total	% of Variance	Cumulative %	Total
1	7.705	32.105	32.105	7.197	29.989	29.989	5.720
2	2.864	11.934	44.039	2.262	9.424	39.413	3.975
3	1.515	6.314	50.353	.986	4.108	43.521	4.700
4	1.159	4.831	55.184
5	1.067	4.448	59.632
6	.800	3.333	62.965
7	.767	3.197	66.162
8	.685	2.855	69.017
9	.680	2.832	71.849
10	.646	2.691	74.540
11	.619	2.578	77.118
12	.569	2.369	79.487
13	.553	2.302	81.789
14	.534	2.224	84.013
15	.512	2.132	86.145
16	.476	1.984	88.129
17	.457	1.904	90.033
18	.455	1.895	91.927
19	.428	1.784	93.711
20	.390	1.625	95.336
21	.354	1.476	96.812
22	.288	1.198	98.010
23	.253	1.054	99.065
24	.224	.935	100.000
Extraction Method: Principal Axis Factoring.
a. When factors are correlated, sums of squared loadings cannot be added to obtain a total variance.

 

 

Thanks!

 

Kris