**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: <$15,000, $15,000-$25,000, $25,000-$50,000, $50,000-$75,000, $75,000-$100,000, and >$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, cleanlgpa 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 ratingVariable | 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 – This method is appropriate when you know into what interval each observation of the outcome variable falls, but you do not know the exact value of the observation.
- Ordered probit
*–*It is possible to conceptualize this model as an ordered probit regression with six ordered categories: 0 (0.0-2.0), 1 (2.0-2.5), 2 (2.5-3.0), 3 (3.0-3.4), 4 (3.4-3.8), and 5 (3.8-4.0). - Ordinal logistic regression – The results would be very similar in terms of which predictors are significant; however, the predicted values would be in terms of probabilities of membership in each of the categories. It would be necessary that the data meet the proportional odds assumption which, in fact, these data do not meet when converted into ordinal categories.
- OLS regression – You could analyze these data using OLS regression on the midpoints of the intervals. However, that analysis would not reflect our uncertainty concerning the nature of the exact values within each interval, nor would it deal adequately with the left- and right-censoring issues in the tails.

## 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.typeFitting 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

**write**and

**3.type**are statistically significant; the coefficient for

**rating**and

**2.type**are not (at the .05 level of significance).

**write**is statistically significant. A unit increase in writing score leads to a .005 increase in predicted GPA. One of the indicator variables for

**type**is also statistically significant. Compared to level 1 of

**type**, the predicted achievement for level 3 of

**type**increases by about .71. To determine if

**type**itself is statistically significant, we can use the

**constrast**command to obtain the two degree-of-freedom test of this variable. This is shown below.

**lgpa**and

**ugpa**of 0.78 and 0.57, respectively. This shows a substantial reduction. The output also contains an estimate of the standard error of sigma, as well as a 95% confidence interval. Stata does not compute sigma directly but actually computes the log of sigma (/lnsigma in the output).

contrast typeContrasts 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 typePredictive 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 R^{2} or pseudo-R^{2}. You can compute an approximate measure of fit by calculating the R^{2} 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 1.0000display .7494^2.56160036display .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-R^{2}s in addition to other measures of fit. The
Cox-Snell pseudo-R^{2}, 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.

fitstatMeasures 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

- Stata online manual
- Related Stata commands
- Annotated output for the intreg command

## References

- Long, J. S. and Freese, J. (2006).
*Regression Models for Categorical and Limited Dependent Variables Using Stata, Second Edition*. College Station, TX: Stata Press. - Long, J. S. (1997).
*Regression Models for Categorical and Limited Dependent Variables.*Thousand Oaks, CA: Sage Publications. - Stewart, M. B. (1983). On least squares estimation when the dependent variable is grouped.
*Review of Economic Studies*50: 737-753. - Tobin, J. (1958). Estimation of relationships for limited dependent variables.
*Econometrica*26: 24-36.