Version info: Code for this page was tested in Stata 12.

Truncated regression is used to model dependent variables for which some of the
observations are not included in the analysis because of the value of the
dependent variable. 

Please note: The purpose of this page is to show how to use various data
analysis commands.  It does not cover all aspects of the research process which
researchers are expected to do.  In particular, it does not cover data
cleaning and checking, verification of assumptions, model diagnostics or
potential follow-up analyses.

Examples of truncated regression

Example 1.

A study of students in a special GATE (gifted and talented education) program
wishes to model achievement as a function of language skills and the type of
program in which the student is currently enrolled.  A major concern is
that students are required to have a minimum achievement score of 40 to enter
the special program.  Thus, the sample is truncated at an achievement score
of 40.

Example 2. A researcher has data for a sample of Americans whose income is
above the poverty line.  Hence, the lower part of the distribution of
income is truncated.  If the researcher had a sample of Americans whose
income was at or below the poverty line, then the upper part of the income
distribution would be truncated.  In other words, truncation is a result of
sampling only part of the distribution of the outcome variable.

Description of the data

Let’s pursue Example 1 from above.

We have a hypothetical data file, truncreg.dta, with 178 observations.  The
outcome variable is called achiv, and the language test score
variable is called langscore.  The variable prog is a categorical predictor variable with
three levels indicating the type of program in which the students were
enrolled. 

Let’s look at the data.  It is always a good idea to start with descriptive
statistics.

use https://stats.idre.ucla.edu/stat/stata/dae/truncreg, clear

summarize achiv langscore

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
       achiv |       178    54.23596     8.96323         41         76
   langscore |       178    54.01124    8.944896         31         67

tabstat achiv, by(prog) stats(n mean sd)

Summary for variables: achiv
     by categories of: prog (type of program)

    prog |         N      mean        sd
---------+------------------------------
 general |        40    51.575   7.97074
academic |       101  56.89109  9.018759
vocation |        37  49.86486  7.276912
---------+------------------------------
   Total |       178  54.23596   8.96323
----------------------------------------

histogram achiv, bin(6) start(40) freq normaltabulate prog

    type of |
    program |      Freq.     Percent        Cum.
------------+-----------------------------------
    general |         40       22.47       22.47
   academic |        101       56.74       79.21
   vocation |         37       20.79      100.00
------------+-----------------------------------
      Total |        178      100.00

Analysis methods you might consider

Below is a list of some analysis methods you may have
encountered.  Some of the methods listed are quite reasonable, while others have
either fallen out of favor or have limitations.    

Truncated regression

Below we use the truncreg command to estimate a truncated regression
model.  The i. before prog indicates that it is a factor
variable (i.e., categorical variable), and that it should be included in the
model as a series of indicator variables.  The ll() option in the truncreg command indicates the value at which the left truncation
take place.  There is also a ul() option to indicate the value of
the right truncation, which was not needed in this example. 

truncreg achiv langscore i.prog, ll(40)

(note: 0 obs. truncated)

Fitting full model:

Iteration 0:   log likelihood = -598.11669  
Iteration 1:   log likelihood = -591.68374  
Iteration 2:   log likelihood = -591.31208  
Iteration 3:   log likelihood = -591.30981  
Iteration 4:   log likelihood = -591.30981  

Truncated regression
Limit:   lower =         40                             Number of obs =    178
         upper =       +inf                             Wald chi2(3)  =  54.76
Log likelihood = -591.30981                             Prob > chi2   = 0.0000

------------------------------------------------------------------------------
       achiv |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   langscore |   .7125775   .1144719     6.22   0.000     .4882168    .9369383
             |
        prog |
          2  |   4.065219   2.054938     1.98   0.048     .0376131    8.092824
          3  |  -1.135863   2.669961    -0.43   0.671    -6.368891    4.097165
             |
       _cons |   11.30152   6.772731     1.67   0.095     -1.97279    24.57583
-------------+----------------------------------------------------------------
      /sigma |   8.755315    .666803    13.13   0.000     7.448405    10.06222
------------------------------------------------------------------------------

test 2.prog 3.prog

 ( 1)  [eq1]2.prog = 0
 ( 2)  [eq1]3.prog = 0

           chi2(  2) =    7.19
         Prob > chi2 =    0.0274

The two degree-of-freedom chi-square test indicates that prog is a
statistically significant predictor of achiv.

We can use the margins
command to obtain the expected cell means.  Note that these are different
from the means we obtained with the tabstat command above.

margins prog

Predictive margins                                Number of obs   =        178
Model VCE    : OIM

Expression   : Linear prediction, predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        prog |
          1  |   49.78871   1.897166    26.24   0.000     46.07034    53.50709
          2  |   53.85393   1.150041    46.83   0.000     51.59989    56.10797
          3  |   48.65285   2.140489    22.73   0.000     44.45757    52.84813
------------------------------------------------------------------------------

In the table above, we can see that the expected mean of avchiv for
the first level of prog is approximately 49.79; the expected mean for
level 2 of prog is 53.85; the expected mean for the third level of
prog is 48.65.

marginsplot

If you would like to
compare truncated regression models, you can issue the estat ic command to get the log
likelihood, AIC and BIC values.

estat ic

-----------------------------------------------------------------------------
       Model |    Obs    ll(null)   ll(model)     df          AIC         BIC
-------------+---------------------------------------------------------------
           . |    178           .   -591.3098      5      1192.62    1208.529
-----------------------------------------------------------------------------
               Note:  N=Obs used in calculating BIC; see [R] BIC note

The truncreg output includes neither an R² nor a pseudo-R².  You can compute
a rough estimate of the degree of association by correlating achiv with the predicted
value and squaring the result.

predict p
correlate p achiv
(obs=178)

             |        p    achiv
-------------+------------------
           p |   1.0000
       achiv |   0.5524   1.0000


display r(rho)^2
.30519203

The calculated value of .31 is rough estimate of the R² you would find in an OLS
regression.  The squared correlation between the observed and predicted
academic aptitude values is about 0.31, indicating that these predictors
accounted for over 30% of the variability in the outcome variable. 

Things to consider

See also

References