The following example shows the output in Mplus, as well as how to reproduce
it using Stata. For this example we will use the same dataset we used for our
logit regression data analysis example. You can download the dataset for Mplus here:
logit.dat. The model we specify for this
example includes four variables, three predictors and one outcome. We use
Graduate Record Exam scores (gre), undergraduate grade point average (gpa),
and prestige of the undergraduate program (topnotch) to predict that whether an
applicant is admitted to graduate school. The Mplus input for this
model is:

data: file is logit.dat;

variable: names are admit gre topnotch gpa;
categorical = admit;

analysis: 
type = general;
estimator = ml;
! need to use estimator = ml to make this a logistic model;

model: admit on gre topnotch gpa;

output: stand;

Below are the results from the model described above. Note that Mplus produces
two types of standardized coefficients “Std” which are in the fifth column of
the output shown below,
and “StdXY” which are in the sixth column. The Std column contains coefficients standardized using the variance of continuous latent variables.
Because all of the variables in this model are manifest (i.e. observed) the
coefficients in this column are identical to those in the column of regular
coefficients (i.e. the “Estimates” column). The StdXY column contains the
coefficients standardized using the variance of the background and/or outcome
variables, in addition to the variance of continuous latent variables.

MODEL RESULTS

                   Estimates     S.E.  Est./S.E.    Std     StdYX

 ADMIT      ON
    GRE                0.002    0.001      2.314    0.002    0.152
    TOPNOTCH           0.437    0.292      1.498    0.437    0.086
    GPA                0.668    0.325      2.052    0.668    0.135

 Thresholds
    ADMIT$1            4.601    1.096      4.196    4.601    2.439

Now, from the latent variable point of view, there is a latent variable
behind the observed dichotomous variable and this latent variable is the
true outcome variable. In other word, the logistic regression is simply
modeling the latent variable using the linear relationship:

$$
y^{*} = beta_0 + beta_1* GRE + beta_2*TOPNOTCH + beta_3*GPA
$$

Notice that there is no random residual term here. Instead, we assume
that

$$
y^{*} – (beta_0 + beta_1* GRE + beta_2*TOPNOTCH + beta_3*GPA)
$$
obeys the standard logistic distribution. Therefore, the variance of (y^{*}) is the sum
of variance of the linear prediction plus the variance of standard logistic
distribution, which is (frac{pi^2}{3}), that is (Var(y^{*}) = Var(Xbeta) +frac{pi^2}{3}). This is
the formula that Mplus uses to calculate the variance for the outcome variable.

Now we are ready to replicate the results from Mplus in Stata. The first bold line below opens
the dataset, and the second runs the logistic regression model in Stata. Note
that the raw coefficients from Stata and Mplus are within rounding
error of each other, this should be the case, since we are running the same
model. We have also run fitstat to display many fit indices including the
variance for (y^{*}).

use https://stats.idre.ucla.edu/stat/stata/dae/logit.dta, clear
logit admit gre topnotch gpa, nolog

Logistic regression                               Number of obs   =        400
                                                  LR chi2(3)      =      21.85
                                                  Prob > chi2     =     0.0001
Log likelihood = -239.06481                       Pseudo R2       =     0.0437

------------------------------------------------------------------------------
       admit |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         gre |   .0024768   .0010702     2.31   0.021     .0003792    .0045744
    topnotch |   .4372236   .2918532     1.50   0.134    -.1347983    1.009245
         gpa |   .6675556   .3252593     2.05   0.040     .0300592    1.305052
       _cons |  -4.600814   1.096379    -4.20   0.000    -6.749678   -2.451949
------------------------------------------------------------------------------

fitstat

Measures of Fit for logit of admit

Log-Lik Intercept Only:       -249.988   Log-Lik Full Model:           -239.065
D(396):                        478.130   LR(3):                          21.847
                                         Prob > LR:                       0.000
McFadden's R2:                   0.044   McFadden's Adj R2:               0.028
ML (Cox-Snell) R2:               0.053   Cragg-Uhler(Nagelkerke) R2:      0.074
McKelvey & Zavoina's R2:         0.075   Efron's R2:                      0.052
Variance of y*:                  3.558   Variance of error:               3.290
Count R2:                        0.683   Adj Count R2:                    0.000
AIC:                             1.215   AIC*n:                         486.130
BIC:                         -1894.490   BIC':                           -3.873
BIC used by Stata:             502.095   AIC used by Stata:             486.130

How does fitstat compute the variance of (y^{*})? We have explained earlier
that (Var(y^{*}) = Var(Xbeta) +frac{pi^2}{3}) and now let’s check if this is the case.

predict xb, xb
sum xb

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
          xb |       400   -.8111861    .5180669  -2.166729   .4880949

return list

scalars:
                  r(N) =  400
              r(sum_w) =  400
               r(mean) =  -.8111860970774433
                r(Var) =  .2683933174379701
                 r(sd) =  .5180669044032538
                r(min) =  -2.166728973388672
                r(max) =  .4880948960781097
                r(sum) =  -324.4744388309773

display r(Var) + (_pi^2)/3
3.5582615

As you can see, they match very nicely. Now we are ready to calculate a standardized coefficient.
This is also called “full-standardization” since it requires both the
outcome variable and the predictor variable to be standardized. As always, we will need three pieces of
information, the standard deviation of (y^{*}), the standard
deviation of the predictor variable for which we want to create a standardized
coefficient, and the raw coefficient for that predictor variable.

To
obtain the standard deviation for the linear predictor, we will create a local
macro variable based on what have calculated above, this is the first line
of code below. Next we
summarize the predictor variable for which we want to create a standardized coefficient,
in this case gre, and save the standard deviation to a local macro
variable called “xstd.” Since Stata
automatically stores the coefficients from the last regression we ran, we can
access the coefficient for gre by typing _b[gre]. Now we are
ready to actually calculate the standardized coefficients. The second to
last command below creates a new local macro called “gre_std” and sets it equal
to the standardized coefficient for gre (i.e. _b[gre]*`xstd’/`ystd’).
The last command shown below tells Stata to display the contents of “gre_std”
which is the standardized coefficient for the relationship between gre
and the log odds of y. This value is approximately
0.1516, looking at the Mplus output above, we see that the standardized
coefficient (StdYX) for male is also estimated to be 0.152 by Mplus.

local ystd=sqrt(r(Var)+(_pi^2)/3)
sum gre

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
         gre |       400       587.7    115.5165        220        800

local xstd = r(sd)
local gre_std = _b[gre]*`xstd'/`ystd'
display "`gre_std'"
.1516774659729085

The commands and output below show the same process for the other two predictor variables
in the model.

sum topnotch

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
    topnotch |       400       .1625    .3693709          0          1

local xstd = r(sd)
local topnotch_std = _b[topnotch]*`xstd'/`ystd'
display "`topnotch_std'"
.0856144885799177
 
sum gpa

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
         gpa |       400      3.3899    .3805668       2.26          4

local xstd = r(sd)
local gpa_std = _b[gpa]*`xstd'/`ystd'
display "`gpa_std'"
.1346788501438455

Cautions, Flies in the Ointment

See Also