Stata has added a maximum likelihood tetrachoric command to Stata 9.2. The matrix
of tetrachoric correlations is saved in r(Rho) for use pic pcamat or factormat.

If you need polychoric or polyserial correlations in addition to tetrachoric
then the polychoric command by Stas Kolenikov is meant for you. The correlation
matrix is displayed using the matrix list r(R) command.

You can download
polychoric over the internet by typing search
polychoric (see How can I used the search command to search for
programs and get additional
help? for more information about using search).

Now, let’s read in an example dataset,
tetra, which contains data several dichotomous variables
derived from the High School and Beyond study.

use https://stats.idre.ucla.edu/stat/stata/faq/tetra, clear

summarize female schtyp ses hon sci

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
      female |       200        .545    .4992205          0          1
      schtyp |       200         .84     .367526          0          1
         ses |       200        .765    .4250628          0          1
         hon |       200        .265    .4424407          0          1
         sci |       200        .335    .4731749          0          1

Let’s compute a standard correlation matrix followed by a tetrachoric
correlation matrix.

corr female schtyp ses hon sci
(obs=200)

             |   female   schtyp      ses      hon      sci
-------------+---------------------------------------------
      female |   1.0000
      schtyp |  -0.0153   1.0000
         ses |  -0.1512  -0.1776   1.0000
         hon |   0.1391   0.0148   0.0389   1.0000
         sci |  -0.1599  -0.0659   0.1435   0.3179   1.0000

tetrachoric female schtyp ses hon sci

(obs=200)

             |   female   schtyp      ses      hon      sci
-------------+---------------------------------------------
      female |   1.0000 
      schtyp |  -0.0291   1.0000 
         ses |  -0.2636  -0.4475   1.0000 
         hon |   0.2362   0.0304   0.0735   1.0000 
         sci |  -0.2562  -0.1253   0.2668   0.5037   1.0000 
         
mat list r(Rho)   /* note the upper-case R */

symmetric r(Rho)[5,5]
            female      schtyp         ses         hon         sci
female           1
schtyp  -.02907834           1
   ses  -.26359952  -.44745088           1
   hon   .23621208   .03040347   .07348643           1
   sci  -.25621193  -.12525624   .26676862   .50370596           1

And now for the polychoric version.

polychoric female schtyp ses hon sci
  
matrix list r(R)

symmetric r(R)[5,5]
            female      schtyp         ses         hon         sci
female           1
schtyp  -.02893308           1
   ses  -.26297321  -.44527627           1
   hon   .23643947   .03044205   .07352325           1
   sci  -.25634248  -.12494757    .2666472     .504248           1