Standard methods of performing factor analysis ( i.e., those based on a matrix of
Pearson’s correlations) assume that the variables are continuous and follow a multivariate normal
distribution. If the model includes variables that are dichotomous or ordinal a factor analysis can be performed using a polychoric
correlation matrix. In Stata we can generate a matrix of polychoric
correlations using the user-written command polychoric. You can find and
install the polychoric command by typing search polychoric in the Stata
command window and following the directions the screen. For more information on
locating and installing user-written commands see our FAQ:
How do I use search to search for programs and
additional help?. Note that variables used with polychoric may be
binary (0/1), ordinal, or continuous, but cannot be nominal (unordered
categories). Also note that the correlations in the matrix produced by the
polychoric command are not all polychoric correlations. When both
variables have 10 or fewer observed values, a polychoric correlation is
calculated, when only one of the variables takes on 10 or fewer values ( i.e., one
variable is continuous and the other categorical) a polyserial correlation is
calculated, and if both variables take on more than 10 values a Pearson’s
correlation is calculated. Once we have a polychoric correlation matrix, we can use
the factormat command to perform an exploratory factor analysis using the
matrix as input, rather than raw variables.

The dataset for this example includes data on 1428 college students
and their instructors. The example analysis includes dichotomous variables, including
faculty sex (facsex) and faculty nationality (US citizen or foreign citizen,
facnat); ordered categorical variables, including faculty rank (facrank), student
rank (studrank) and grade (A, B, C, etc., grade);
and the continuous variables faculty salary (salary), years teaching at
the University of Texas (yrsut), and number of students in the class (nstud) in this analysis. These
variables were selected to represent a range of types of variables ( i.e., dichotomous, ordered categorical, and continuous),
and do not necessarily form substantively meaningful factors. 

Below we open the dataset and generate the polychoric correlation matrix for the
eight variables in our analysis. You may notice that the polychoric command
runs somewhat more slowly than Stata’s correlate and pwcorr
commands, this is normal.

use https://stats.idre.ucla.edu/stat/stata/output/m255, clearpolychoric facsex facnat facrank studrank grade salary yrsut nstud

Polychoric correlation matrix

              facsex      facnat     facrank    studrank       grade      salary
  facsex           1
  facnat  -.08153951           1
 facrank  -.33496545  -.54985327           1
studrank   .14701719  -.04503906   -.0006882           1
   grade  -.05250522  -.07768724   .03336171   .21606134           1
  salary  -.24422069  -.31687704   .75225252   .04830565   -.0073763           1
   yrsut  -.09789967  -.49838303   .68902129   .00459421   .01994406   .53046614
   nstud  -.46151997    .2795961  -.17809723  -.33304524  -.13713578  -.08439606

               yrsut       nstud
   yrsut           1
   nstud  -.31031949           1

The polychoric command does not does not display the number of cases (with listwise deletion)
used to generate the matrix,
but it does store the n in r(sum_w) so we can use the display command to view it. Then we use the
matrix command to store the polychoric correlation matrix (saved in r(R)
by the polychoric command) as r, so that we can use it with the
factormat command. The factormat command is followed by the name
of the matrix we wish to use for the analysis ( i.e., r). The n(…)
“option” gives the sample size, and is required. We have used the factors(…)
option to indicate that we wish to retain three factors.

display r(sum_w)
1338

global N = r(sum_w)

matrix r = r(R)
factormat r, n($N) factors(3)
(obs=1338)

Factor analysis/correlation                        Number of obs    =     1338
    Method: principal factors                      Retained factors =        3
    Rotation: (unrotated)                          Number of params =       21

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      2.42705      1.26666            0.6995       0.6995
        Factor2  |      1.16039      0.84359            0.3344       1.0340
        Factor3  |      0.31680      0.18808            0.0913       1.1253
        Factor4  |      0.12871      0.16060            0.0371       1.1624
        Factor5  |     -0.03189      0.08326           -0.0092       1.1532
        Factor6  |     -0.11515      0.05212           -0.0332       1.1200
        Factor7  |     -0.16727      0.08181           -0.0482       1.0718
        Factor8  |     -0.24908            .           -0.0718       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated: chi2(28) = 3824.64 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    -----------------------------------------------------------
        Variable |  Factor1   Factor2   Factor3 |   Uniqueness 
    -------------+------------------------------+--------------
          facsex |  -0.1902   -0.6651   -0.2171 |      0.4744  
          facnat |  -0.5913    0.2174    0.1465 |      0.5816  
         facrank |   0.9183    0.1642    0.0173 |      0.1295  
        studrank |   0.0645   -0.3558    0.3430 |      0.7516  
           grade |   0.0636   -0.1380    0.3316 |      0.8670  
          salary |   0.7365    0.1822    0.0751 |      0.4187  
           yrsut |   0.7520   -0.0762   -0.1107 |      0.4165  
           nstud |  -0.2861    0.6777   -0.0493 |      0.4565  
    -----------------------------------------------------------

The above factor analysis output can be interpreted in a manner similar to a standard factor analysis
model, including the use of rotation methods to increase interpretability.

See also

Stata Annotated Output: Factor Analysis