Stata Data Analysis Examples: Interval Regression



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Stata Data Analysis Examples
Interval Regression

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, 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 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 and from reading and writing test scores. The measure of GPA is a self-report response to 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 which there is only interval censoring.

Description of the Data

Let's pursue Example 3 from above.

We have a hypothetical data file, intregex.dta with 30 observations. The GPA score is represented by two values, the lower interval score (lgpa) and the upper interval score (ugpa). The reading, writing test scores and the teacher rating are read, write and rating respectively.

Let's look at the data.

use http://www.ats.ucla.edu/stat/stata/dae/intregex, clear

list 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

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
          id |        30        15.5    8.803408          1         30
        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
        read |        30    171.6667    94.39767         50        350

Graphing these data can be rather tricky. So just to get an idea of what the distribution of GPA is we will do separate histograms for lgpa and ugpa.

histogram lgpa, normal



histogram ugpa, normal



correlate lgpa ugpa write rating read
(obs=30)

             |     lgpa     ugpa    write   rating     read
-------------+---------------------------------------------
        lgpa |   1.0000
        ugpa |   0.9488   1.0000
       write |   0.6206   0.6572   1.0000
      rating |   0.5355   0.5904   0.4763   1.0000
        read |   0.5747   0.5997   0.3182   0.4489   1.0000
 
         
graph matrix write rating read lgpa ugpa, half jitter(2)

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a interval regression analysis, let's consider some other methods that you might use.

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.
It is possible to conceptualize this model as an ordered logistic 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). 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.

Stata Interval Regression

intreg lgpa ugpa write rating read

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 = -38.212102  
Iteration 1:   log likelihood = -36.680551  
Iteration 2:   log likelihood = -36.662189  
Iteration 3:   log likelihood = -36.662185  
Iteration 4:   log likelihood = -36.662185  

Interval regression                               Number of obs   =         30
                                                  LR chi2(3)      =      30.17
Log likelihood = -36.662185                       Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       write |   .0052829   .0015363     3.44   0.001     .0022718    .0082939
      rating |    .016789    .009751     1.72   0.085    -.0023226    .0359005
        read |    .002329   .0008046     2.89   0.004      .000752     .003906
       _cons |   .9133711   .4794007     1.91   0.057     -.026237    1.852979
-------------+----------------------------------------------------------------
    /lnsigma |  -1.090882   .1516747    -7.19   0.000    -1.388159   -.7936051
-------------+----------------------------------------------------------------
       sigma |   .3359201   .0509506                      .2495343    .4522116
------------------------------------------------------------------------------

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

Just to be a bit more conservative we will run the model again using the robust option to obtain the Huber-White robust standard errors.

intreg lgpa ugpa write rating read, robust

Fitting constant-only model:

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

Fitting full model:

Iteration 0:   log pseudolikelihood = -38.212102  
Iteration 1:   log pseudolikelihood = -36.680551  
Iteration 2:   log pseudolikelihood = -36.662189  
Iteration 3:   log pseudolikelihood = -36.662185  
Iteration 4:   log pseudolikelihood = -36.662185  

Interval regression                               Number of obs   =         30
                                                  Wald chi2(3)    =      54.85
Log pseudolikelihood = -36.662185                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       write |   .0052829   .0014818     3.57   0.000     .0023786    .0081872
      rating |    .016789   .0106434     1.58   0.115    -.0040717    .0376497
        read |    .002329   .0010766     2.16   0.031     .0002189     .004439
       _cons |   .9133711   .4883437     1.87   0.061     -.043765    1.870507
-------------+----------------------------------------------------------------
    /lnsigma |  -1.090882   .1267837    -8.60   0.000    -1.339373   -.8423906
-------------+----------------------------------------------------------------
       sigma |   .3359201   .0425892                      .2620098    .4306797
------------------------------------------------------------------------------

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

The output looks very much like the output from an OLS regression. At the top, the number of observations used in the analysis is given along a likelihood-ratio chi-squared. The likelihood-ratio chi-squared tests the difference between the full model (with predictors) and the constant only model. Below that is the p-value for the chi-squared with three degrees of freedom. Obviously this model, as a whole, is statistically significant. Next comes the log-likelihood for the full model.

The intreg command does not compute an R² or pseudo-R². You can compute a rough-and-ready 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   0.7963   1.0000 

display .7494^2

.56160036

display .7963^2

.63409369

The calculated values of approximately .60 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 (findit spostado), which provides a number of pseudo-R²'s in addition to other measures of fit. The McKelvey-Zavoina pseudo-R²'s, which computes the R² using the variances of the latent variable and the latent predicted variable [ Var(predicted-y*)/Var(y*) ], is close to our rough-and-ready estimates above.

fitstat

Measures of Fit for intreg of lgpa ugpa

Log-Lik Intercept Only:        -51.747   Log-Lik Full Model:            -36.662
D(25):                          73.324   LR(3):                          30.170
                                         Prob > LR:                       0.000
McFadden's R2:                   0.292   McFadden's Adj R2:               0.195
ML (Cox-Snell) R2:               0.634   Cragg-Uhler(Nagelkerke) R2:      0.655
McKelvey & Zavoina's R2:         0.677                              
Variance of y*:                  0.350   Variance of error:               0.113
AIC:                             2.777   AIC*n:                          83.324
BIC:                           -11.706   BIC':                          -19.967
BIC used by Stata:              90.330   AIC used by Stata:              83.324

In the main body of the table we have the interval regression coefficients, the standard error of the coefficients, a Wald t-test (coefficient/se) and the p-value associated with each t-test. By default, we also get a 95% confidence interval for the coefficients. With the level() option you can request a different confidence interval.

The ancillary statistic /sigma is equivalent to the standard error of estimate in OLS regression. The value of 0.34 can be compared to the standard deviations for lgpa and ugpa of 0.78 and 0.57. 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).

Finally, the output provides a summary of the number of left-censored, uncensored, right-censored and interval-censored values.

Sample Write-Up of the Analysis

Before we begin the sample write-up we need to get the output into a form more acceptable for publication. The estout command (findit estout by Ben Jann of ETH Zurich), will get us closer to what we want.

estout, cells(b(star fmt(%8.2f)) se(par fmt(%8.2f))) stats(sigma ll chi2, fmt(%8.2f))

        b/se
model   
write   0.0053***
        (0.0015)
rating  0.0168
        (0.0106)
read    0.0023*
        (0.0011)
_cons   0.9134
        (0.4883)
lnsigma 
_cons   -1.0909***
        (0.1268)
sigma   0.34
ll      -36.66
chi2    54.85

With a little bit of manual editing and remembering to change to add the McKelvey-Zavoina pseudo-R² we can produce an acceptable table of the output.

model   
write             0.0053***
                 (0.0015)
rating            0.0168
                 (0.0106)
read              0.0023*
                 (0.0011)
constant          0.9134
                 (0.4883)

standard error   
  of estimate     0.34
log-likelihood  -36.66
chi-squared      54.85
pseudo R-squared  0.68 

legend: coefficient/(standard error)  *** p<0.001

The interval regression model predicting GPA from reading, writing and teacher ratings was statistically significant (chi-squared = 54.85, df = 3, p < 0.001). Both the reading and writing test scores were statically significant at the 0.001 and 0.05 levels, respectively. The teacher ratings were not significant (p = 0.12). The McKelvey and Zavoina pseudo-R² was 0.68 indicating that these three predictors accounted for approximately 68% of the variability in the latent outcome variable. A unit change in reading and writing lead to a 0.0053 and 0.0023 increase in the predicted GPA, respectively.

Additional Example

The example we just covered only had interval-censored data, it is also possible to use intreg with data that that have both interval-censored data along with left-censored and/or right-censored data as in Example 2.

Stata Data Analysis Examples Interval Regression