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

Interval regression is used to model outcomes that have interval censoring. 
In other words, you know the ordered category into which each observation falls,
but you do not know the exact value of the observation.  Interval
regression is a generalization of censored regression.

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 interval regression

Example 1.  We wish to model annual income using years of education and marital status.  However,
we do not have access to the precise values for income.  Rather, we only have data on the income ranges:
$100,000.  Note
that the extreme values of the categories on either end of the range are either left-censored
or right-censored.  The other categories are interval censored, that is, each interval is both
left- and right-censored.  Analyses of this type require a generalization of censored regression
known as interval regression.

Example 2.  We wish to predict GPA from teacher ratings of effort and from reading and writing test
scores.  The measure of GPA is a self-report response to the following item:

Select the category that best represents your overall GPA.
  less than 2.0
  2.0 to 2.5
  2.5 to 3.0
  3.0 to 3.4
  3.4 to 3.8
  3.8 to 3.9
  4.0 or greater

Again, we have a situation with both interval censoring and left- and right-censoring.  We do not know the exact value of GPA for each student; we only know the interval in which their
GPA falls.

Example 3. We wish to predict GPA from teacher ratings of effort, writing
test scores and the type of program in which the student was enrolled
(vocational, general or academic).  The measure of GPA is a self-report response to the following item:

Select the category that best represents your overall GPA.
  0.0 to 2.0
  2.0 to 2.5
  2.5 to 3.0
  3.0 to 3.4
  3.4 to 3.8
  3.8 to 4.0

This is a slight variation of Example 2.  In this example, there is only interval censoring.

Description of the data

Let’s pursue Example 3 from above.

We have a hypothetical data file, intreg_data.dta with 30 observations.  The GPA score is represented by two values, the lower interval score (lgpa) and the upper
interval score (ugpa).  The writing test scores, the teacher rating
and the type of program (a nominal variable which has three levels) are write, rating and type, respectively.

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/intreg_data, clearlist lgpa ugpa, clean

       lgpa   ugpa  
  1.    2.5      3  
  2.    3.4    3.8  
  3.    2.5      3  
  4.      0      2  
  5.      3    3.4  
  6.    3.4    3.8  
  7.    3.8      4  
  8.      2    2.5  
  9.      3    3.4  
 10.    3.4    3.8  
 11.      2    2.5  
 12.      2    2.5  
 13.      2    2.5  
 14.    2.5      3  
 15.    2.5      3  
 16.    2.5      3  
 17.    3.4    3.8  
 18.    2.5      3  
 19.      2    2.5  
 20.      3    3.4  
 21.    3.4    3.8  
 22.    3.8      4  
 23.      2    2.5  
 24.      3    3.4  
 25.    3.4    3.8  
 26.      2    2.5  
 27.      2    2.5  
 28.      2    2.5  
 29.    2.5      3  
 30.    2.5      3  

Note that there are two GPA responses for each observation, lgpa for the lower end of the
interval and ugpa for the upper end.

summarize lgpa ugpa write rating

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
        lgpa |        30         2.6    .7754865          0        3.8
        ugpa |        30    3.096667    .5708332          2          4
       write |        30    113.8333    49.94278         50        205
      rating |        30    57.53333    8.303441         48         72

tabstat lgpa ugpa, by(type) stats(n mean sd)

Summary statistics: N, mean, sd
  by categories of: type 

      type |      lgpa      ugpa
-----------+--------------------
vocational |         8         8
           |      1.75    2.4375
           |  .7071068  .1767767
-----------+--------------------
   general |        10        10
           |      2.78      3.24
           |  .3852849  .3373096
-----------+--------------------
  academic |        12        12
           |  3.016667  3.416667
           |  .6336522  .5474458
-----------+--------------------
     Total |        30        30
           |       2.6  3.096667
           |  .7754865  .5708332
--------------------------------

Graphing these data can be rather tricky.  Just to get an idea of what the distribution of
GPA is, we will do separate histograms for lgpa and ugpa.  We
will also correlate the variables in the dataset.

histogram ugpa, normal xlabel(0(1)4) name(hugpa)
histogram lgpa, normal xlabel(0(1)4) name(hlgpa)
graph combine hlgpa hugpa, ycommon xsize(7)

correlate lgpa ugpa write rating
(obs=30)

             |     lgpa     ugpa    write   rating
-------------+------------------------------------
        lgpa |   1.0000
        ugpa |   0.9488   1.0000
       write |   0.6206   0.6572   1.0000
      rating |   0.5355   0.5904   0.4763   1.0000

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. 

Interval regression

We will use the intreg command to run the interval regression
analysis.  The intreg command requires two outcome variables, the
lower limit of the interval and the upper limit of the interval.  The i. before
type 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.  Note that this syntax was introduced
in Stata 11.

intreg lgpa ugpa write rating i.type

Fitting constant-only model:

Iteration 0:   log likelihood = -52.129849  
Iteration 1:   log likelihood =  -51.74803  
Iteration 2:   log likelihood = -51.747288  
Iteration 3:   log likelihood = -51.747288  

Fitting full model:

Iteration 0:   log likelihood = -35.224403  
Iteration 1:   log likelihood = -33.142851  
Iteration 2:   log likelihood = -33.128906  
Iteration 3:   log likelihood = -33.128905  

Interval regression                               Number of obs   =         30
                                                  LR chi2(4)      =      37.24
Log likelihood = -33.128905                       Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       write |   .0052847   .0016921     3.12   0.002     .0019683    .0086012
      rating |   .0133143   .0091197     1.46   0.144    -.0045601    .0311886
             |
        type |
          2  |    .374853     .19275     1.94   0.052      -.00293    .7526359
          3  |   .7097467   .1668399     4.25   0.000     .3827466    1.036747
             |
       _cons |   1.103863   .4452887     2.48   0.013     .2311137    1.976613
-------------+----------------------------------------------------------------
    /lnsigma |  -1.237263   .1596419    -7.75   0.000    -1.550155   -.9243703
-------------+----------------------------------------------------------------
       sigma |   .2901775   .0463245                      .2122151    .3967812
------------------------------------------------------------------------------

  Observation summary:         0  left-censored observations
                               0     uncensored observations
                               0 right-censored observations
                              30       interval observations

contrast type


Contrasts of marginal linear predictions

Margins      : asbalanced

------------------------------------------------
             |         df        chi2     P>chi2
-------------+----------------------------------
model        |
        type |          2       18.71     0.0001
------------------------------------------------

The two degree-of-freedom chi-square test indicates that type is a
statistically significant predictor of lgpa and ugpa.

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, because they
are adjusted for write and rating also.

margins type

Predictive margins                                Number of obs   =         30
Model VCE    : OIM

Expression   : Linear prediction, predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        type |
          1  |   2.471456     .13236    18.67   0.000     2.212035    2.730877
          2  |   2.846309    .118957    23.93   0.000     2.613157     3.07946
          3  |   3.181203   .0969802    32.80   0.000     2.991125     3.37128
------------------------------------------------------------------------------

The expected mean GPA for students in program type 1 (vocational) is 2.47; the
expected mean GPA for students in program type 3 (academic) is 3.18.

If you would like to
compare interval 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
-------------+---------------------------------------------------------------
           . |     30   -51.74729   -33.12891      6     78.25781    86.66499
-----------------------------------------------------------------------------
               Note:  N=Obs used in calculating BIC; see [R] BIC note

The intreg command does not compute an R² or pseudo-R².  You can compute an approximate measure of fit by calculating the R² between the
predicted and observed values.

predict p
correlate lgpa ugpa p
(obs=30)

             |     lgpa     ugpa        p
-------------+---------------------------
        lgpa |   1.0000
        ugpa |   0.9488   1.0000
           p |   0.7494   0.8430   1.0000 

display .7494^2

.56160036

display .8430^2

.710649

The calculated values of approximately .56 and .71 are probably close to what you would find in an OLS
regression if you had actual GPA scores.  You can also make use of the Long and Freese utility command fitstat (search spostado)
(see How can I use
the search command to search for programs and get additional help? for more
information about using search),
which provides a number of pseudo-R²s in addition to other measures of fit.  The
Cox-Snell pseudo-R², in which the ratio of the likelihoods reflects
the improvement of the full model over the intercept-only model,
is close to our approximate estimates above.

fitstat

Measures of Fit for intreg of lgpa ugpa

Log-Lik Intercept Only:        -51.747   Log-Lik Full Model:            -33.129
D(23):                          66.258   LR(4):                          37.237
                                         Prob > LR:                       0.000
McFadden's R2:                   0.360   McFadden's Adj R2:               0.225
ML (Cox-Snell) R2:               0.711   Cragg-Uhler(Nagelkerke) R2:      0.734
McKelvey & Zavoina's R2:         0.760                              
Variance of y*:                  0.351   Variance of error:               0.084
AIC:                             2.675   AIC*n:                          80.258
BIC:                           -11.970   BIC':                          -23.632
BIC used by Stata:              86.665   AIC used by Stata:              78.258

Things to consider

See also

References