Stata FAQ: How can I do factor analysis with missing data in Stata?



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Stata FAQ
How can I do factor analysis with missing data in Stata?

Trying to run factor analysis with missing data can be problematic. One issue is that traditional multiple imputation methods, such as mi estimate or mim, don't work with Stata's factor command. Truxillo (2005) has suggested an approach using maximum likelihood with the expectation-maximization (EM) algorithm to estimate of the covariance matrix. Stata's mi command computes an EM covariance matrix as part of the imputation process. We will demonstrate how to use this EM covariance matrix to obtain a factor solution.

To begin, we will load a Stata 11 dataset fa_missing, get some descriptive statistics and compute the complete case covariance matrix.

use http://www.ats.ucla.edu/stat/data/fa_missing, clear  /* Stata 11 data file */

summarize

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
      item13 |      1419    4.450317    .7374944          1          5
      item14 |      1428    4.518207    .7086049          1          5
      item15 |      1424    4.434691    .7478835          1          5
      item16 |      1420    4.270423    .8387034          1          5
      item17 |      1423    4.158819    .8969815          1          5
-------------+--------------------------------------------------------
      item18 |      1424    3.924157    1.032095          1          5
      item19 |      1420    4.072535    .9665034          1          5
      item20 |      1396    3.770774    .9137137          1          5
      item21 |      1422    3.769339    .9863042          1          5
      item22 |      1414    3.592645    1.122807          1          5
-------------+--------------------------------------------------------
      item23 |      1423    3.800422    .9639492          1          5
      item24 |      1417    3.653493    .9308223          1          5
      item25 |      1398    2.285408    .9892487          1          5
      item26 |      1414    2.077086    1.058313          1          5
      item27 |      1420    1.496479    .7294192          1          5
-------------+--------------------------------------------------------
      item28 |      1419    2.273432    .9677116          1          5
      
count  /* count total number of observations */

 1428
 
corr, cov  /* complete case covariance matrix */

(obs=1331)

             |   item13   item14   item15   item16   item17   item18   item19   item20   item21   item22
-------------+------------------------------------------------------------------------------------------
      item13 |  .536077
      item14 |  .338379   .48707
      item15 |  .321189  .321527  .536102
      item16 |  .345401  .289728  .305775  .681157
      item17 |  .376869   .34434  .382685  .435569  .799719
      item18 |  .310446  .313639  .348313  .350631   .52091  1.07943
      item19 |  .202336  .211924  .250935  .262903  .387361    .6311  .935065
      item20 |  .202417  .199169  .234828  .235136   .33403  .490213   .39099   .82419
      item21 |  .340947  .302693  .364249  .369725  .525901  .565762  .472584  .377907  .966934
      item22 |  .271303  .257165   .30236  .328247  .450612  .625471  .521898  .389491  .553577  1.24822
      item23 |   .39691  .374938  .405833   .35907  .526654  .567222  .406462  .352182  .558713   .52722
      item24 |  .305477   .28148  .290924  .325099  .432752  .457068  .332405   .30078  .449396  .456496
      item25 |  .008449 -.011696 -.038745 -.030012 -.039108 -.042074 -.064693 -.026436 -.025697 -.047378
      item26 |  .014954 -.024045 -.002687 -.019264 -.021647 -.018859  .018107 -.026555  .002384 -.019735
      item27 | -.036163 -.045486 -.046055 -.065249 -.055178 -.070832 -.053228 -.036927 -.062904 -.099815
      item28 | -.000554 -.013315 -.033624 -.048267 -.028426 -.051824 -.016597 -.044399 -.031681  -.07906

             |   item23   item24   item25   item26   item27   item28
-------------+------------------------------------------------------
      item23 |  .913566
      item24 |  .618358  .848286
      item25 | -.031721 -.043576  .976103
      item26 |  .014638 -.025494   .10275  1.10263
      item27 | -.059988 -.063666  .123452  .170048   .51931
      item28 | -.004233 -.049099   .23827  .210952  .353081  .941695

From the output above, you can see that there are a total of 1,428 observations with 1,365 complete cases. All of the variables have missing cases except for item14. item20 has the most missing data with only 1,396 nonmissing cases.

We will use the mlong format for mi set but this approach will work with any of the mi data formats. When you register variables to be imputed (mi register imputed) you should also include the variables without missing values, such as item14, so that they will be included in the EM covariance matrix. Next, run the mi impute mvn command with the emonly option. Notice that there are no variables to the right of the equal sign. In fact, there is no equal sign at all. After running mi impute, the EM covariance matrix can be found in the saved results in r(Sigma_em) which we will then save to the matrix cov_em for use in factormat.

mi set mlong

mi register imputed item13-item28

(97 m=0 obs. now marked as incomplete)

mi impute mvn item13-item28, emonly

note: variable item14 contains no soft missing (.) values; imputing nothing

Iteration 0:   Observed log likelihood = -9021.7844
Iteration 1:   Observed log likelihood = -4116.7934
Iteration 2:   Observed log likelihood = -4113.8728
Iteration 3:   Observed log likelihood = -4113.8685
Iteration 4:   Observed log likelihood = -4113.8685
Iteration 5:   Observed log likelihood = -4113.8685

Expectation-maximization estimation      Number obs           =      1428
                                         Number missing       =       167
                                         Number patterns      =        34
Prior: uniform                           Obs per pattern: min =         1
                                                          avg =        42
                                                          max =      1331

Observed log likelihood = -4113.8685 at iteration 5

------------------------------------------------------------------------------------------------------
             |    item13     item14     item15     item16     item17     item18     item19     item20 
-------------+----------------------------------------------------------------------------------------
Coef         |                                                                                        
       _cons |  4.451285   4.518207   4.435308   4.268804   4.156375   3.922213   4.070152   3.767296 
-------------+----------------------------------------------------------------------------------------
Sigma        |                                                                                        
      item13 |  .5430297    .348556   .3350714   .3450292   .3822566   .3104201   .2059764   .2103791 
      item14 |   .348556   .5017693   .3455828   .3003632    .355927   .3212322   .2256632   .2100503 
      item15 |  .3350714   .3455828   .5584064   .3186633    .400073   .3582909   .2667201   .2515707 
      item16 |  .3450292   .3003632   .3186633    .705225   .4391928   .3453138   .2643872   .2404926 
      item17 |  .3822566    .355927    .400073   .4391928   .8085157   .5141493   .3895897   .3465985 
      item18 |  .3104201   .3212322   .3582909   .3453138   .5141493   1.068469   .6312669   .4935741 
      item19 |  .2059764   .2256632   .2667201   .2643872   .3895897   .6312669   .9371905   .4000026 
      item20 |  .2103791   .2100503   .2515707   .2404926   .3465985   .4935741   .4000026   .8360225 
      item21 |  .3460404   .3144694   .3764157   .3618431   .5214286   .5644802   .4791261   .3844396 
      item22 |  .2817349   .2724658   .3191765   .3361343   .4594059   .6214813   .5312196   .3984811 
      item23 |  .4075676   .3925719   .4292744   .3747293   .5356915    .567577   .4185394   .3681236 
      item24 |  .3165909   .3003985      .3143    .334972   .4410231   .4564384   .3410914   .3120366 
      item25 |  -.004998  -.0312026  -.0558893  -.0478347   -.049516  -.0489516  -.0731514  -.0307672 
      item26 |  .0131501  -.0212114   .0033069  -.0091686  -.0192239  -.0085362   .0180015  -.0241016 
      item27 | -.0416338   -.054581  -.0551466  -.0723354  -.0615237  -.0827423  -.0596916  -.0426282 
      item28 | -.0053221  -.0246628  -.0425885  -.0575029  -.0363375  -.0585845  -.0250774  -.0471011 
------------------------------------------------------------------------------------------------------

------------------------------------------------------------------------------------------------------
             |    item21     item22     item23     item24     item25     item26     item27     item28 
-------------+----------------------------------------------------------------------------------------
Coef         |                                                                                        
       _cons |  3.770048   3.593149    3.79814   3.655047   2.285293   2.077139    1.49686   2.273121 
-------------+----------------------------------------------------------------------------------------
Sigma        |                                                                                        
      item13 |  .3460404   .2817349   .4075676   .3165909   -.004998   .0131501  -.0416338  -.0053221 
      item14 |  .3144694   .2724658   .3925719   .3003985  -.0312026  -.0212114   -.054581  -.0246628 
      item15 |  .3764157   .3191765   .4292744      .3143  -.0558893   .0033069  -.0551466  -.0425885 
      item16 |  .3618431   .3361343   .3747293    .334972  -.0478347  -.0091686  -.0723354  -.0575029 
      item17 |  .5214286   .4594059   .5356915   .4410231   -.049516  -.0192239  -.0615237  -.0363375 
      item18 |  .5644802   .6214813    .567577   .4564384  -.0489516  -.0085362  -.0827423  -.0585845 
      item19 |  .4791261   .5312196   .4185394   .3410914  -.0731514   .0180015  -.0596916  -.0250774 
      item20 |  .3844396   .3984811   .3681236   .3120366  -.0307672  -.0241016  -.0426282  -.0471011 
      item21 |   .970728   .5586832   .5705437   .4616899  -.0378778   .0058816  -.0728921  -.0373168 
      item22 |  .5586832   1.261583   .5445269   .4737303  -.0510983  -.0252185  -.0972336  -.0784298 
      item23 |  .5705437   .5445269   .9342235   .6368121  -.0503701   .0146983   -.066706   -.009803 
      item24 |  .4616899   .4737303   .6368121   .8657582  -.0596746  -.0287266  -.0696632  -.0554101 
      item25 | -.0378778  -.0510983  -.0503701  -.0596746   .9778334   .0906248   .1282772   .2393721 
      item26 |  .0058816  -.0252185   .0146983  -.0287266   .0906248   1.118956   .1671461    .196613 
      item27 | -.0728921  -.0972336   -.066706  -.0696632   .1282772   .1671461   .5314816   .3568273 
      item28 | -.0373168  -.0784298   -.009803  -.0554101   .2393721    .196613   .3568273   .9353024 
------------------------------------------------------------------------------------------------------

Note: no imputation performed.

matrix cov_em = r(Sigma_em)

matrix list cov_em

symmetric cov_em[16,16]
            item13      item14      item15      item16      item17      item18      item19      item20
item13   .54302971
item14     .348556   .50176934
item15   .33507137   .34558277   .55840641
item16   .34502917    .3003632   .31866333     .705225
item17   .38225661   .35592696   .40007302   .43919281   .80851571
item18   .31042005   .32123222   .35829089   .34531382   .51414933   1.0684692
item19   .20597637   .22566319   .26672014   .26438724   .38958966   .63126692   .93719047
item20   .21037912   .21005025   .25157069   .24049261   .34659848   .49357406   .40000261   .83602251
item21   .34604038   .31446937   .37641575    .3618431   .52142858   .56448016   .47912613   .38443958
item22   .28173489   .27246583   .31917653   .33613426   .45940591   .62148129   .53121959   .39848113
item23   .40756762   .39257192   .42927439   .37472927   .53569152   .56757702   .41853944   .36812361
item24    .3165909   .30039846       .3143   .33497204    .4410231   .45643844   .34109143   .31203658
item25  -.00499796  -.03120263  -.05588928  -.04783472  -.04951603  -.04895159  -.07315143  -.03076718
item26   .01315008  -.02121138   .00330686   -.0091686   -.0192239  -.00853618   .01800149  -.02410163
item27  -.04163384    -.054581  -.05514657  -.07233544  -.06152372  -.08274233  -.05969162  -.04262815
item28  -.00532215  -.02466275   -.0425885  -.05750295  -.03633748  -.05858451  -.02507744  -.04710109

            item21      item22      item23      item24      item25      item26      item27      item28
item21   .97072797
item22   .55868325   1.2615831
item23   .57054369   .54452686   .93422349
item24   .46168988    .4737303   .63681207   .86575819
item25  -.03787779  -.05109826  -.05037005  -.05967465   .97783345
item26   .00588162  -.02521849   .01469832  -.02872665   .09062481   1.1189565
item27  -.07289205  -.09723358  -.06670603   -.0696632    .1282772   .16714608   .53148159
item28  -.03731678  -.07842976  -.00980302  -.05541012   .23937206   .19661302   .35682735   .93530244

We will use the factormat command with the EM estimate of the covariance matrix to obtain our factor solution. The factormat command requires that the sample size, n, be entered along with the name of the covariance matrix. In her paper, Truxillo discusses three methods for specifying nominal sample size, 1) column-wise minimum, 2) column-wise average and 3) pairwise minimum. Column-wise minimum is just the number of complete cases for the variables with the most missing values which is the value we will use for this example. If you will recall from above that value is 1,396.

factormat cov_em, n(1396) fact(4) ml

(obs=1396)
Iteration 0:   log likelihood = -236.78484
Iteration 1:   log likelihood = -85.766521
(...omitted...)
Iteration 90:  log likelihood = -85.345691
Iteration 91:  log likelihood = -85.345691

Factor analysis/correlation                        Number of obs    =     1396
    Method: maximum likelihood                     Retained factors =        4
    Rotation: (unrotated)                          Number of params =       58
                                                   Schwarz's BIC    =  590.691
    Log likelihood = -85.34569                     (Akaike's) AIC   =  286.691

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      5.83062      4.66732            0.7030       0.7030
        Factor2  |      1.16329      0.34933            0.1403       0.8432
        Factor3  |      0.81396      0.32778            0.0981       0.9414
        Factor4  |      0.48619            .            0.0586       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated: chi2(120) = 9652.01 Prob>chi2 = 0.0000
    LR test:   4 factors vs. saturated:  chi2(62) =  169.61 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    ---------------------------------------------------------------------
        Variable |  Factor1   Factor2   Factor3   Factor4 |   Uniqueness 
    -------------+----------------------------------------+--------------
          item13 |   0.7069    0.0779   -0.3640    0.1850 |      0.3274  
          item14 |   0.7087    0.0258   -0.3268    0.1776 |      0.3588  
          item15 |   0.7260   -0.0043   -0.2333    0.1634 |      0.3919  
          item16 |   0.6158   -0.0577   -0.2016    0.2151 |      0.5306  
          item17 |   0.7618   -0.0252   -0.0548    0.1732 |      0.3860  
          item18 |   0.7099   -0.1273    0.3400    0.1912 |      0.3276  
          item19 |   0.5856   -0.1310    0.4220    0.2273 |      0.4102  
          item20 |   0.5331   -0.0986    0.2382    0.1632 |      0.6227  
          item21 |   0.7143   -0.0444    0.1319    0.0987 |      0.4606  
          item22 |   0.6024   -0.1166    0.2541    0.0639 |      0.5549  
          item23 |   0.8857    0.1157    0.0221   -0.2968 |      0.1134  
          item24 |   0.7246    0.0169    0.0385   -0.2133 |      0.4276  
          item25 |  -0.0717    0.2901    0.0348    0.0744 |      0.9039  
          item26 |  -0.0061    0.2680    0.0632    0.0585 |      0.9207  
          item27 |  -0.1399    0.6198    0.1288    0.1428 |      0.5593  
          item28 |  -0.0586    0.7349    0.1431    0.1606 |      0.4102  
    ---------------------------------------------------------------------

rotate, varimax normalize blanks(.3)

Factor analysis/correlation                        Number of obs    =     1396
    Method: maximum likelihood                     Retained factors =        4
    Rotation: orthogonal varimax (Kaiser on)       Number of params =       58
                                                   Schwarz's BIC    =  590.691
    Log likelihood = -85.34569                     (Akaike's) AIC   =  286.691

    --------------------------------------------------------------------------
         Factor  |     Variance   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      3.25325      0.32888            0.3922       0.3922
        Factor2  |      2.92437      1.70143            0.3526       0.7448
        Factor3  |      1.22294      0.32944            0.1474       0.8923
        Factor4  |      0.89350            .            0.1077       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated: chi2(120) = 9652.01 Prob>chi2 = 0.0000
    LR test:   4 factors vs. saturated:  chi2(62) =  169.61 Prob>chi2 = 0.0000

Rotated factor loadings (pattern matrix) and unique variances

    ---------------------------------------------------------------------
        Variable |  Factor1   Factor2   Factor3   Factor4 |   Uniqueness 
    -------------+----------------------------------------+--------------
          item13 |   0.7843                               |      0.3274  
          item14 |   0.7534                               |      0.3588  
          item15 |   0.6961    0.3120                     |      0.3919  
          item16 |   0.6114                               |      0.5306  
          item17 |   0.6040    0.4695                     |      0.3860  
          item18 |   0.3043    0.7487                     |      0.3276  
          item19 |             0.7454                     |      0.4102  
          item20 |             0.5561                     |      0.6227  
          item21 |   0.4237    0.5581                     |      0.4606  
          item22 |             0.5829                     |      0.5549  
          item23 |   0.5049    0.4350              0.6651 |      0.1134  
          item24 |   0.4010    0.3935              0.5019 |      0.4276  
          item25 |                       0.3044           |      0.9039  
          item26 |                                        |      0.9207  
          item27 |                       0.6562           |      0.5593  
          item28 |                       0.7676           |      0.4102  
    ---------------------------------------------------------------------
    (blanks represent abs(loading)<.3)

Factor rotation matrix

    --------------------------------------------------
                 | Factor1  Factor2  Factor3  Factor4 
    -------------+------------------------------------
         Factor1 |  0.6778   0.5954  -0.0605   0.4270 
         Factor2 |  0.1006  -0.1816   0.9512   0.2282 
         Factor3 | -0.6581   0.7251   0.1934   0.0609 
         Factor4 |  0.3120   0.2945   0.2326  -0.8728 
    --------------------------------------------------

Almost identical results to these were obtain using SAS proc mi with proc factor and using Mplus with the missing data option.

Reference

Truxillo, C. 2005. Maximum likelihood parameter estimation with incomplete data. Proceedings of the Thirtieth Annual SAS(r) Users Group International Conference. <http://www2.sas.com/proceedings/sugi30/111-30.pdf >

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Stata FAQ How can I do factor analysis with missing data in Stata?

Reference

Stata FAQ
How can I do factor analysis with missing data in Stata?