Let’s say that you have a dataset with a bunch of binary variables. Further, you believe that
these binary variables reflect underlying and unobserved continuous variables. You don’t
want to compute your confirmatory factor analysis (CFA) directly on the binary variables. You will want to compute the CFA
on tetrachoric correlations that reflect the associations among these underlying continuous
variables. We will demonstrate this by using data with five continuous variables and
creating binary variables from them by dichotomizing them at a point a little above their
mean values.

Let’s begin by loading the hsbdemo.dta dataset and creating binary variables for
read, write, math, science and socst.

use https://stats.idre.ucla.edu/stat/data/hsbdemo, clear

gen r=read

Now that we have the binary variables, let’s checkout the correlations among the continuous
version of the variables and the binary version.

corr read write math science socst

(obs=200)

             |     read    write     math  science    socst
-------------+---------------------------------------------
        read |   1.0000
       write |   0.5968   1.0000
        math |   0.6623   0.6174   1.0000
     science |   0.6302   0.5704   0.6307   1.0000
       socst |   0.6215   0.6048   0.5445   0.4651   1.0000

corr r w m s o
 
(obs=200)

             |        r        w        m        s        o
-------------+---------------------------------------------
           r |   1.0000
           w |   0.4109   1.0000
           m |   0.4750   0.5029   1.0000
           s |   0.3846   0.4320   0.4750   1.0000
           o |   0.4057   0.3910   0.3676   0.3009   1.0000

As you can see, the correlations among the binary version of the variables are much lower than
among the continuous version. The Pearson correlations tend to underestimate the relationship
between the underlying continuous variables that give rise to the binary variables. What we
need are the tetrachoric correlations which we can obtain using the tetrachoric
command.

tetrachoric r w m s o

(obs=200)

             |        r        w        m        s        o
-------------+---------------------------------------------
           r |   1.0000 
           w |   0.6145   1.0000 
           m |   0.6874   0.7176   1.0000 
           s |   0.5790   0.6411   0.6874   1.0000 
           o |   0.6148   0.5780   0.5556   0.4690   1.0000 

The tetrachoric correlations are much closer to the original correlations among the continuous
variables than the correlations among the binary values.

For comparison purposes we will compute a CFA on the original continuous data.

sem (FC->read write math science socst)

Endogenous variables

Measurement:  read write math science socst

Exogenous variables

Latent:       FC

Fitting target model:

Iteration 0:   log likelihood = -3469.2622  
[output omitted] 
Iteration 3:   log likelihood = -3468.8093  

Structural equation model                       Number of obs      =       200
Estimation method  = ml
Log likelihood     = -3468.8093

 ( 1)  [read]FC = 1
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement  |
  read  chi2 = 0.0163sem (FB->r w m s o)

Next, we will create the SSD dataset and compute the CFA on the tetrachoric correlations.

clear

ssd init r w m s o

Summary statistics data initialized.  Next use, in any order,

    ssd set observations (required)
        It is best to do this first.

    ssd set means (optional)
        Default setting is 0.

    ssd set variances or ssd set sd (optional)
        Use this only if you have set or will set correlations and, even then,
        this is optional but highly recommended.  Default setting is 1.

    ssd set covariances or ssd set correlations (required)

ssd set obs 200
  (value set)

    Status:
                       observations:    set
                              means:  unset
                    variances or sd:  unset
        covariances or correlations:  unset (required to be set)

ssd set cor  1.0000  ///
             0.6145    1.0000  ///
             0.6874   0.7176   1.0000  ///
             0.5790   0.6411   0.6874    1.0000   ///
             0.6148   0.5780   0.5556    0.4690    1.0000 
  (values set)

    Status:
                       observations:    set
                              means:  unset
                    variances or sd:  unset
        covariances or correlations:    set

sem (FT->r w m s o)

Endogenous variables

Measurement:  r w m s o

Exogenous variables

Latent:       FT

Fitting target model:

Iteration 0:   log likelihood = -1148.1182  
Iteration 1:   log likelihood = -1147.4763  
Iteration 2:   log likelihood = -1147.4673  
Iteration 3:   log likelihood = -1147.4673  

Structural equation model                       Number of obs      =       200
Estimation method  = ml
Log likelihood     = -1147.4673

 ( 1)  [r]FT = 1
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement  |
  r  chi2 = 0.0124

You will note that the model fit versus a saturated model is very close to the value that
was obtained when ran the CFA on the continuous variables.