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

The tobit model, also called a censored regression model, is designed to estimate
linear relationships between variables when there is either left- or right-censoring
in the dependent variable (also known as censoring from below and above,
respectively). Censoring from above takes place when cases with a value at or
above some threshold, all take on the value of that threshold, so
that the true value might be equal to the threshold, but it might also be higher.
In the case of censoring from below, values those that fall at or below some
threshold are censored.

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 and potential follow-up analyses.

Examples of tobit regression

Example 1.

In the 1980s there was a federal law restricting speedometer readings to no more than 85 mph. So if
you wanted to try and predict a vehicle’s top-speed from a combination of horse-power and engine size,
you would get a reading no higher than 85, regardless of how fast the vehicle was really traveling.
This is a classic case of right-censoring (censoring from above) of the data. The only thing we are certain of is that
those vehicles were traveling at least 85 mph.

Example 2. A research project
is studying the level of lead in home drinking water as a function of the age of
a house and family income. The water testing kit cannot detect lead
concentrations below 5 parts per billion (ppb). The EPA considers levels above
15 ppb to be dangerous. These data are an example of left-censoring (censoring
from below).

Example 3. Consider the situation in which we have a measure of academic
aptitude (scaled 200-800) which we want to model using reading and math test
scores, as well as, the type
of program the student is enrolled in (academic, general, or vocational). The problem here is that students who answer all questions on
the academic aptitude test correctly receive a score of 800, even though it is likely that these
students are not “truly” equal in aptitude. The same is true of students who
answer all of the questions incorrectly. All such students would have a score of
200, although they may not all be of equal aptitude.

Description of the data

Let’s pursue Example 3 from above.

We have a hypothetical data file, tobit.dta with 200 observations.
The academic aptitude variable is apt, the reading and math test scores are read
and math respectively. The variable prog is the type of program
the student is in, it is a categorical (nominal) variable that takes on three
values, academic (prog = 1), general (prog = 2), and vocational (prog
= 3).

Let’s look at the data.
Note that in this dataset, the lowest
value of apt is 352. No students received a score of 200 (i.e. the lowest
score possible), meaning that even though censoring from below was possible, it
does not occur in the dataset.

use https://stats.idre.ucla.edu/stat/stata/dae/tobit, clearsummarize apt read math

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
         apt |       200     640.035    99.21903        352        800
        read |       200       52.23    10.25294         28         76
        math |       200      52.645    9.368448         33         75


tabulate prog

    type of |
    program |      Freq.     Percent        Cum.
------------+-----------------------------------
   academic |         45       22.50       22.50
    general |        105       52.50       75.00
 vocational |         50       25.00      100.00
------------+-----------------------------------
      Total |        200      100.00


histogram apt, normal bin(10) xline(800)

Looking at the above histogram showing the distribution of apt, we can
see the censoring in the data, that is, there are far more cases with scores of
750 to 800 than
one would expect looking at the rest of the distribution. Below is an alternative histogram
that further highlights the excess of cases where apt=800. In the
histogram below, the discrete option produces a histogram where each
unique value of apt has its own bar. The freq option causes the y-axis to
be labeled with the frequency for each value, rather than the density. Because
apt is continuous, most values of apt are unique in the dataset,
although close to the center of the distribution there are a few values of
apt that have two or three cases. The spike on the far right of the
histogram is the bar for cases where apt=800, the height of this bar
relative to all the others clearly shows the excess number of cases with this value.

histogram apt, discrete freq

Next we’ll explore the bivariate relationships in our dataset.

correlate read math apt
(obs=200)

             |     read     math      apt
-------------+---------------------------
        read |   1.0000
        math |   0.6623   1.0000
         apt |   0.6451   0.7333   1.0000


graph matrix read math apt, half jitter(2)

In the last row of the scatterplot matrix shown above, we see the
scatterplots showing read and apt, as well as math
and apt. Note the collection of cases at the top of each scatterplot
due to the censoring in the distribution of apt.

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.

Tobit regression

Below we run the tobit model, using read, math, and
prog to predict apt. The ul( ) option in the tobit command indicates the value at which the right-censoring
begins (i.e., the upper limit). There is also a ll( ) option to indicate the value of the left-censoring
(the lower limit) which was not needed in this example. The i. before
prog indicates that prog is a factor
variable (i.e.,
categorical variable), and that it should be included in the model as a series
of dummy variables. Note that this syntax was introduced in Stata 11.

tobit apt read math i.prog, ul(800)

Tobit regression                                  Number of obs   =        200
                                                  LR chi2(4)      =     188.97
                                                  Prob > chi2     =     0.0000
Log likelihood = -1041.0629                       Pseudo R2       =     0.0832

------------------------------------------------------------------------------
         apt |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        read |   2.697939    .618798     4.36   0.000     1.477582    3.918296
        math |   5.914485   .7098063     8.33   0.000     4.514647    7.314323
             |
        prog |
          2  |  -12.71476   12.40629    -1.02   0.307    -37.18173     11.7522
          3  |   -46.1439   13.72401    -3.36   0.001     -73.2096   -19.07821
             |
       _cons |    209.566   32.77154     6.39   0.000     144.9359    274.1961
-------------+----------------------------------------------------------------
      /sigma |   65.67672   3.481272                      58.81116    72.54228
------------------------------------------------------------------------------
  Obs. summary:          0  left-censored observations
                       183     uncensored observations
                        17 right-censored observations at apt>=800

We can test for an overall effect of prog using the test command. Below we see that the overall effect of
prog is statistically significant.

test 2.prog 3.prog

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

       F(  2,   196) =    5.98
            Prob > F =    0.0030

We can also test additional hypotheses about the differences in the
coefficients for different levels of prog. Below we
test that the coefficient for prog=2 is equal to the coefficient for
prog=3. In the output below we see that the coefficient for prog=2 is
significantly different than the coefficient for prog=3.

test 2.prog = 3.prog

 ( 1)  [model]2.prog - [model]3.prog = 0

       F(  1,   196) =    6.66
            Prob > F =    0.0106

We may also wish to see measures of how well our model fits. This can be
particularly useful when comparing competing models. One method of doing this is
to compare the predicted values based on the tobit model to the observed values in
the dataset. Below we use predict to generate predicted values of apt
based on the model. Next we correlate the observed values of apt with the
predicted values (yhat).

predict yhat
(option xb assumed; fitted values)

correlate apt yhat
(obs=200)

             |      apt     yhat
-------------+------------------
         apt |   1.0000
        yhat |   0.7825   1.0000

The correlation between the predicted and observed values of apt is 0.7825. If we square this
value, we get the multiple squared correlation, this indicates predicted values
share about 61% (0.7825^2 = 0.6123) of their variance with apt. Additionally,
we can use the user-written command
fitstat to produce a variety of fit statistics. You can find more information
on fitstat by typing search fitstat (see
How can I use the
search command to search for programs and get additional help? for more
information about using search).

fitstat

Measures of Fit for tobit of apt

Log-Lik Intercept Only:      -1135.545   Log-Lik Full Model:          -1041.063
D(193):                       2082.126   LR(4):                         188.965
                                         Prob > LR:                       0.000
McFadden's R2:                   0.083   McFadden's Adj R2:               0.077
ML (Cox-Snell) R2:               0.611   Cragg-Uhler(Nagelkerke) R2:      0.611
McKelvey & Zavoina's R2:         0.616                              
Variance of y*:              11230.171   Variance of error:            4313.432
AIC:                            10.481   AIC*n:                        2096.126
BIC:                          1059.550   BIC':                         -167.772
BIC used by Stata:            2113.916   AIC used by Stata:            2094.126

See Also

References

McDonald, J. F. and Moffitt, R. A. 1980. The Uses of Tobit Analysis. The Review of Economics and Statistics
Vol 62(2): 318-321.