Factor Command

Joe Prendergast

Join Date: Apr 2020
Posts: 25

14 May 2020, 12:01

Hello,

I have an observation about the Factor command. I'd be interested in anyone's thoughts.

My understanding about factor models, including what is see in the STATA Factor Methods and Formulas, is that when I change the number of factors used, all of the loadings should change because the Unique Effects matrix will change. But I'm not seeing that. For example, the addition of a third factor doesn't change the loadings for the first 2 factors:

Code:

    -------------------------------------------------

        Variable |  Factor1   Factor2 |   Uniqueness

    -------------+--------------------+--------------

           sp_sq |   0.3567    0.1766 |      0.8416  

         baml_sq |   0.5766   -0.1127 |      0.6548  

     tbill_ds_sq |   0.2186   -0.1298 |      0.9354  

          wti_sq |   0.3099    0.0112 |      0.9038  

         gold_sq |   0.2100    0.2460 |      0.8954  

         usuk_sq |   0.2276    0.1811 |      0.9154  

        ust10_sq |   0.5281   -0.1248 |      0.7055  

    -------------------------------------------------

Code:

    -----------------------------------------------------------

        Variable |  Factor1   Factor2   Factor3 |   Uniqueness

    -------------+------------------------------+--------------

           sp_sq |   0.3567    0.1766    0.0240 |      0.8410  

         baml_sq |   0.5766   -0.1127   -0.0066 |      0.6547  

     tbill_ds_sq |   0.2186   -0.1298    0.0919 |      0.9269  

          wti_sq |   0.3099    0.0112   -0.0648 |      0.8996  

         gold_sq |   0.2100    0.2460    0.0468 |      0.8932  

         usuk_sq |   0.2276    0.1811   -0.0310 |      0.9144  

        ust10_sq |   0.5281   -0.1248   -0.0142 |      0.7053  

    -----------------------------------------------------------

The other issue I'm having is that the factors(#) option has no effect on the output - I always get 3 factors reported (for my dataset). To get the results above I went into the factor.ado and set the factors variable directly. That may be related to the observation above if I didn't change the .ado correctly.

Joe

Tags: factor

William Lisowski

Join Date: Dec 2014
Posts: 10150

14 May 2020, 13:16

To begin with, I really hope you kept a copy of the unedited version of factor.ado and can replace your edited version with the original. Which you should do before doing anything else.

You wrote

My understanding about factor models, including what is see in the STATA Factor Methods and Formulas, is that when I change the number of factors used, all of the loadings should change because the Unique Effects matrix will change.

From the the Methods and formulas section of the documentation for factor we have

Suppose that there are p variables and q factors. Let Ψ represent the p × p diagonal matrix of uniquenesses, ...

and from that I infer that the matrix of uniquenesses is a function of the number of variables and not of the number of factors.

From the same documentation we can read that the factors() option selects the maximum number of factors to be retained, but also, factors correspondting to eigenvalues below that specficied by the mineigen() option will not be retained. You need to take both into account.

Code:

. webuse bg2, clear
(Physician-cost data)

. factor bg2cost1-bg2cost6, pcf factors(3)
(obs=568)

Factor analysis/correlation                      Number of obs    =        568
    Method: principal-component factors          Retained factors =          2
    Rotation: (unrotated)                        Number of params =         11

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      1.70622      0.30334            0.2844       0.2844
        Factor2  |      1.40288      0.49422            0.2338       0.5182
        Factor3  |      0.90865      0.18567            0.1514       0.6696
        Factor4  |      0.72298      0.05606            0.1205       0.7901
        Factor5  |      0.66692      0.07456            0.1112       0.9013
        Factor6  |      0.59236            .            0.0987       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(15) =  269.07 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    -------------------------------------------------
        Variable |  Factor1   Factor2 |   Uniqueness
    -------------+--------------------+--------------
        bg2cost1 |   0.3581    0.6279 |      0.4775  
        bg2cost2 |  -0.4850    0.5244 |      0.4898  
        bg2cost3 |  -0.5326    0.5725 |      0.3886  
        bg2cost4 |  -0.4919    0.3254 |      0.6521  
        bg2cost5 |   0.6238    0.3962 |      0.4539  
        bg2cost6 |   0.6543    0.3780 |      0.4290  
    -------------------------------------------------

. factor bg2cost1-bg2cost6, pcf mineigen(.8)
(obs=568)

Factor analysis/correlation                      Number of obs    =        568
    Method: principal-component factors          Retained factors =          3
    Rotation: (unrotated)                        Number of params =         15

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      1.70622      0.30334            0.2844       0.2844
        Factor2  |      1.40288      0.49422            0.2338       0.5182
        Factor3  |      0.90865      0.18567            0.1514       0.6696
        Factor4  |      0.72298      0.05606            0.1205       0.7901
        Factor5  |      0.66692      0.07456            0.1112       0.9013
        Factor6  |      0.59236            .            0.0987       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(15) =  269.07 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    -----------------------------------------------------------
        Variable |  Factor1   Factor2   Factor3 |   Uniqueness
    -------------+------------------------------+--------------
        bg2cost1 |   0.3581    0.6279   -0.2586 |      0.4106  
        bg2cost2 |  -0.4850    0.5244   -0.4741 |      0.2650  
        bg2cost3 |  -0.5326    0.5725    0.0626 |      0.3847  
        bg2cost4 |  -0.4919    0.3254    0.6926 |      0.1725  
        bg2cost5 |   0.6238    0.3962    0.3650 |      0.3207  
        bg2cost6 |   0.6543    0.3780    0.0137 |      0.4288  
    -----------------------------------------------------------

.

Last edited by William Lisowski; 14 May 2020, 13:20.

Comment

Sven-Kristjan Bormann

Join Date: Jul 2018

Posts: 310
#3

14 May 2020, 13:36

You should probably post the correlation matrix, so that we can run the factormat-command to check your results.
As far as I understand the factor-option, only the display of the output should change, but there will always as many factors as there are variables. Only the number of retained factors can change by setting the factor-option
Comment
Joe Prendergast

Join Date: Apr 2020

Posts: 25
#4

15 May 2020, 09:30

Thank guys, Yes, William, of course I made my own copy of the ado file and put it in my personal directory. I find delving into Stata's code is a good way to improve my own Stata programming skills. Stats Tech Services gave my some good guidance on my question:

Dear Joe,

Let me start by describing how the -factor- command works,
so I can also explain why this makes sense.

Let's see the description for the -factors()- option:

"factors(#) sets the maximum number of factors to be retained
for later use by the postestimation commands"

It doesn't refer to how many factors are computed; it refers to
how many factors will be retained for further computations.

When you call -factor- with -factors()- option, the command
will compute all the factors (i.e., as many factors as variables
you have). This will explain 100% of the variance.

also, from the documentation,
"factor always prints the full set of eigenvalues but prints
the corresponding eigenvectors only for retained factors"
(which answers your last question).

Now, let's look at "Methods and Formulas". My guess is that you
are looking at the equation

x = f*Delta' + e
and wondering why the solution with q<p would be equivalent to
obtaining just the first q factors of the solution computed with
p factors. I'll try to write an intuitive explanation.

The key is in the orthogonality. If I want to find a subspace
of dimension q=3 that minimizes the error, this subspace will
certainly contain the subspace of dimension 2 (2 equations) that
minimizes the error. But because my factors are orthogonal, I
can express the subspace for q=3 as f1, f2, f3, when f1 and f2 are
the solution from q=2, and f3 is a new vector, orthogonal to f1
and f2.

Of course, all this can be rotated, but this particular solution
allows us to have a clear sort order for the factors.

Sincerely,
Isabel

Isabel Canette, Ph.D.
Principal Mathematician & Statistician
StataCorp LLC

One strange thing though. I wrote a Python program to try to replicate Stata's methodology. I included estimates of the diagonal unique variance matrix. I matched their PCA results exactly but didn't come anywhere close to their FACTOR (unrotated) results.
Comment

William Lisowski

Join Date: Dec 2014
Posts: 10150

15 May 2020, 09:50

You do not tell us the exact commands whose results you are comparing, so let me say that if you are comparing your Python program to the results of pca and to the results of factor, then you should use

Code:

factor ..., pcf

to get results comparable to pca.

Code:

. webuse bg2
(Physician-cost data)

. 
. * Principal Components Analysis
. quietly pca bg2cost1-bg2cost6

. matrix list e(Ev)

e(Ev)[1,6]
     Eigenvalues:  Eigenvalues:  Eigenvalues:  Eigenvalues:  Eigenvalues:  Eigenvalues:
           Comp1         Comp2         Comp3         Comp4         Comp5         Comp6
r1     1.7062152     1.4028764     .90865173     .72297905      .6669203     .59235727

. 
. * Principal factors
. quietly factor bg2cost1-bg2cost6

. matrix list e(Ev)

e(Ev)[1,6]
                Factor1     Factor2     Factor3     Factor4     Factor5     Factor6
Eigenvalues    .8538927   .54107173   .02321134  -.14966742  -.18917716   -.2511521

. 
. * Principal component factors
. quietly factor bg2cost1-bg2cost6, pcf

. matrix list e(Ev)

e(Ev)[1,6]
               Factor1    Factor2    Factor3    Factor4    Factor5    Factor6
Eigenvalues  1.7062152  1.4028764  .90865173  .72297905   .6669203  .59235727

Comment

Joe Prendergast

Join Date: Apr 2020

Posts: 25
#6

15 May 2020, 16:56

For PCA I'm using:

Code:

correlate *sq matrix m_corr = r(C) pcamat m_corr, n(8881)

And for FACTOR:

Code:

correlate *sq matrix m_corr = r(C) factormat m_corr, n(8881)

I realize if I use the pcf option I can get FACTORMAT to do PCA. What's bothering me is I set up my Python code to do FACTOR (without pcf) and it's instead giving me PCA results instead of FACTOR (without pcf) results.

I realize the question can't be answered here, without delving into my Python code. So more of an observation. I did follow the approach in the Stata documentation, which is the same as that in Rencher, A. C. 1998. Multivariate Statistical Inference and Applications, New York: Wiley. I always wonder if I really know what I'm getting when I use a Stats package instead of doing it myself.
Comment

Joe Prendergast

Join Date: Apr 2020
Posts: 25

20 May 2020, 14:45

I'm still struggling with the STATA FACTORMAT command output. I wrote python code following their approach and got very different results. Independently, the popular FactorAnalyzer python module results agree exactly with mine. The three sets of output are below. I also attached my data.

Stata technical services hasn't responded to my request to refer this to their programmers.

STATA:

Code:

. correlate dm*
(obs=8,881)

             | dm_sq_sp dm_sq_~l dm_sq_~s dm_sq_~i dm_sq_~d dm_sq_~k dm_sq~10
-------------+---------------------------------------------------------------
    dm_sq_sp |   1.0000
  dm_sq_baml |   0.1450   1.0000
dm_sq_tbil~s |   0.0238   0.0766   1.0000
   dm_sq_wti |   0.0881   0.1008  -0.0038   1.0000
  dm_sq_gold |   0.1398   0.0893   0.0448   0.1002   1.0000
  dm_sq_usuk |   0.0938   0.0875  -0.0123   0.0779   0.1037   1.0000
 dm_sq_ust10 |   0.1402   0.5072   0.0393   0.0898   0.0831   0.1878   1.0000


. matrix m_corr = r(C)

. noisily factormat m_corr, n(8881)
(obs=8,881)

Factor analysis/correlation                      Number of obs    =      8,881
    Method: principal factors                    Retained factors =          3
    Rotation: (unrotated)                        Number of params =         18

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      0.96859      0.79038            1.3361       1.3361
        Factor2  |      0.17821      0.13988            0.2458       1.5819
        Factor3  |      0.03833      0.07148            0.0529       1.6348
        Factor4  |     -0.03315      0.02620           -0.0457       1.5891
        Factor5  |     -0.05935      0.05372           -0.0819       1.5072
        Factor6  |     -0.11308      0.14154           -0.1560       1.3512
        Factor7  |     -0.25462            .           -0.3512       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(21) = 3848.50 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    -----------------------------------------------------------
        Variable |  Factor1   Factor2   Factor3 |   Uniqueness
    -------------+------------------------------+--------------
        dm_sq_sp |   0.2670    0.1801    0.0339 |      0.8952  
      dm_sq_baml |   0.6014   -0.1474    0.0226 |      0.6161  
    dm_sq_tbil~s |   0.0857   -0.0211    0.1561 |      0.9679  
       dm_sq_wti |   0.1909    0.1694   -0.0161 |      0.9346  
      dm_sq_gold |   0.2063    0.2398    0.0523 |      0.8972  
      dm_sq_usuk |   0.2500    0.1491   -0.0904 |      0.9071  
     dm_sq_ust10 |   0.6220   -0.1233   -0.0340 |      0.5968  
    -----------------------------------------------------------

FactorAnalyzer:

Code:

common eigenvalues
[1.2275 0.3476 0.168 ]
factor loadings
[[ 0.2438  0.7677  0.0796  0.1689  0.2011  0.2444  0.6661]
 [ 0.2089 -0.2384 -0.0102  0.1661  0.366   0.2865 -0.0583]
 [ 0.0824  0.1176  0.1144  0.0387  0.1985 -0.2652 -0.1517]]
communalities
[0.1099 0.66   0.0195 0.0576 0.2138 0.2122 0.4701]

My Code:

Code:

common eigenvalues
[[1.2275 0.3477 0.168 ]]
factor loadings
[[-0.2438 -0.7678 -0.0796 -0.1689 -0.2011 -0.2444 -0.666 ]
 [-0.2089  0.2384  0.0102 -0.1661 -0.3661 -0.2865  0.0583]
 [ 0.0823  0.1175  0.1144  0.0387  0.1986 -0.2652 -0.1518]]
communalities
[0.1098 0.6601 0.0195 0.0576 0.2139 0.2122 0.47  ]

Attached Files

Data.csv (696.9 KB, 1 view)

Last edited by Joe Prendergast; 20 May 2020, 14:51.

Comment

William Lisowski

Join Date: Dec 2014
Posts: 10150

20 May 2020, 15:45

The following four examples suggest to me that the factorization technique you need in Stata is ml (maximum likelihood factor) or ipf (iterated principal factor).

Code:

preserve
collapse (mean) dm*
mkmat dm*, matrix(fa_m)
restore, preserve
collapse (sd) dm*
mkmat dm*, matrix(fa_s)
restore
correlate dm*
matrix m_corr = r(C)

factormat m_corr, n(8881) means(fa_m) sds(fa_s)
factormat m_corr, n(8881) means(fa_m) sds(fa_s) pcf
factormat m_corr, n(8881) means(fa_m) sds(fa_s) ipf
factormat m_corr, n(8881) means(fa_m) sds(fa_s) ml

Code:

. factormat m_corr, n(8881) means(fa_m) sds(fa_s)
(obs=8,881)

Factor analysis/correlation                      Number of obs    =      8,881
    Method: principal factors                    Retained factors =          3
    Rotation: (unrotated)                        Number of params =         18

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      0.96859      0.79039            1.3361       1.3361
        Factor2  |      0.17821      0.13988            0.2458       1.5819
        Factor3  |      0.03833      0.07148            0.0529       1.6348
        Factor4  |     -0.03315      0.02620           -0.0457       1.5891
        Factor5  |     -0.05935      0.05372           -0.0819       1.5072
        Factor6  |     -0.11308      0.14154           -0.1560       1.3512
        Factor7  |     -0.25462            .           -0.3512       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(21) = 3848.52 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    -----------------------------------------------------------
        Variable |  Factor1   Factor2   Factor3 |   Uniqueness 
    -------------+------------------------------+--------------
        dm_sq_sp |   0.2670    0.1801    0.0339 |      0.8952  
      dm_sq_baml |   0.6014   -0.1474    0.0226 |      0.6161  
    dm_sq_tbil~s |   0.0857   -0.0211    0.1561 |      0.9679  
       dm_sq_wti |   0.1909    0.1694   -0.0161 |      0.9346  
      dm_sq_gold |   0.2063    0.2398    0.0523 |      0.8972  
      dm_sq_usuk |   0.2500    0.1491   -0.0904 |      0.9071  
     dm_sq_ust10 |   0.6220   -0.1233   -0.0340 |      0.5968  
    -----------------------------------------------------------

. factormat m_corr, n(8881) means(fa_m) sds(fa_s) pcf
(obs=8,881)

Factor analysis/correlation                      Number of obs    =      8,881
    Method: principal-component factors          Retained factors =          3
    Rotation: (unrotated)                        Number of params =         18

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      1.75948      0.67933            0.2514       0.2514
        Factor2  |      1.08015      0.06657            0.1543       0.4057
        Factor3  |      1.01359      0.08889            0.1448       0.5505
        Factor4  |      0.92470      0.02416            0.1321       0.6826
        Factor5  |      0.90054      0.05948            0.1286       0.8112
        Factor6  |      0.84106      0.36059            0.1202       0.9314
        Factor7  |      0.48047            .            0.0686       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(21) = 3848.52 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    -----------------------------------------------------------
        Variable |  Factor1   Factor2   Factor3 |   Uniqueness 
    -------------+------------------------------+--------------
        dm_sq_sp |   0.4426    0.3484    0.1472 |      0.6611  
      dm_sq_baml |   0.7430   -0.4152   -0.0430 |      0.2738  
    dm_sq_tbil~s |   0.1413   -0.2238    0.8753 |      0.1638  
       dm_sq_wti |   0.3293    0.4447   -0.0669 |      0.6894  
      dm_sq_gold |   0.3604    0.5654    0.3199 |      0.4481  
      dm_sq_usuk |   0.4101    0.2877   -0.2987 |      0.6598  
     dm_sq_ust10 |   0.7650   -0.3690   -0.1669 |      0.2508  
    -----------------------------------------------------------

. factormat m_corr, n(8881) means(fa_m) sds(fa_s) ipf
(obs=8,881)

Factor analysis/correlation                      Number of obs    =      8,881
    Method: iterated principal factors           Retained factors =          6
    Rotation: (unrotated)                        Number of params =         21

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      1.22107      0.89575            0.6764       0.6764
        Factor2  |      0.32532      0.16265            0.1802       0.8565
        Factor3  |      0.16267      0.09531            0.0901       0.9466
        Factor4  |      0.06736      0.04208            0.0373       0.9840
        Factor5  |      0.02529      0.02137            0.0140       0.9980
        Factor6  |      0.00392      0.00416            0.0022       1.0001
        Factor7  |     -0.00024            .           -0.0001       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(21) = 3848.52 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    -----------------------------------------------------------------------------------------
        Variable |  Factor1   Factor2   Factor3   Factor4   Factor5   Factor6 |   Uniqueness 
    -------------+------------------------------------------------------------+--------------
        dm_sq_sp |   0.2512    0.2372    0.0828   -0.0219   -0.1112    0.0233 |      0.8604  
      dm_sq_baml |   0.6976   -0.1820    0.1531   -0.0574    0.0128    0.0166 |      0.4530  
    dm_sq_tbil~s |   0.0812   -0.0062    0.1804    0.1731    0.0600    0.0171 |      0.9270  
       dm_sq_wti |   0.1762    0.2015    0.0283   -0.1574    0.0878    0.0013 |      0.8951  
      dm_sq_gold |   0.1967    0.3474    0.1238    0.0522    0.0031   -0.0368 |      0.8212  
      dm_sq_usuk |   0.2461    0.2401   -0.2462    0.0639    0.0350    0.0282 |      0.8150  
     dm_sq_ust10 |   0.7310   -0.1300   -0.1519    0.0455   -0.0145   -0.0257 |      0.4227  
    -----------------------------------------------------------------------------------------

. factormat m_corr, n(8881) means(fa_m) sds(fa_s) ml
(obs=8,881)
number of factors adjusted to 3
Iteration 0:   log likelihood = -390.03776
Iteration 1:   log likelihood =  -25.96514
Iteration 2:   log likelihood = -16.395233
Iteration 3:   log likelihood = -6.0827653
Iteration 4:   log likelihood = -4.1684206
Iteration 5:   log likelihood = -4.1449844
Iteration 6:   log likelihood = -4.1449523

Factor analysis/correlation                      Number of obs    =      8,881
    Method: maximum likelihood                   Retained factors =          3
    Rotation: (unrotated)                        Number of params =         18
                                                 Schwarz's BIC    =     171.94
    Log likelihood = -4.144952                   (Akaike's) AIC   =    44.2899

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      1.20624      0.83971            0.6898       0.6898
        Factor2  |      0.36653      0.19050            0.2096       0.8993
        Factor3  |      0.17603            .            0.1007       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(21) = 3848.52 Prob>chi2 = 0.0000
    LR test:   3 factors vs. saturated:  chi2(3)  =    8.29 Prob>chi2 = 0.0405

Factor loadings (pattern matrix) and unique variances

    -----------------------------------------------------------
        Variable |  Factor1   Factor2   Factor3 |   Uniqueness 
    -------------+------------------------------+--------------
        dm_sq_sp |   0.2128    0.2258    0.1131 |      0.8909  
      dm_sq_baml |   0.7978   -0.1365    0.0672 |      0.3404  
    dm_sq_tbil~s |   0.0845   -0.0134    0.1085 |      0.9809  
       dm_sq_wti |   0.1461    0.1759    0.0726 |      0.9425  
      dm_sq_gold |   0.1518    0.3547    0.2531 |      0.7870  
      dm_sq_usuk |   0.1921    0.3685   -0.2283 |      0.7752  
     dm_sq_ust10 |   0.6604    0.0648   -0.1597 |      0.5343  
    -----------------------------------------------------------

Comment

Joe Prendergast

Join Date: Apr 2020

Posts: 25
#9

21 May 2020, 05:31

Thank you. I am iterating in my code and evidently so evidently is the FactorAnalyzer python module.
Comment
William Lisowski

Join Date: Dec 2014

Posts: 10150
#10

21 May 2020, 06:12

A quick search of the web led me to

https://factor-analyzer.readthedocs.io/en/latest/

where we see

The factor_analyzer package allows users to perfrom EFA using either (1) a minimum residual (MINRES) solution, (2) a maximum likelihood (ML) solution, or (3) a principal factor solution.

and further reading shows

method ({'minres', 'ml', 'principal'}, optional) – The fitting method to use, either MINRES or Maximum Likelihood. Defaults to ‘minres’.

It's often useful to read the fine material provided by the authors of software packages.
Comment
Joe Prendergast

Join Date: Apr 2020

Posts: 25
#11

21 May 2020, 13:46

I knew that factor_analyzer was doing the iterative minimization because that's what I did with my own python code and I matched factor_analyzer exactly. What I missed was that I needed to use the ipf option in STATA to match my results and the factor_analyzer results.
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