**Note:** This example was done using Mplus version 6.12. The syntax may not work, or may function differently, with other versions of Mplus.

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 analysis

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****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). In addition to the three-category variable **prog**, the dataset
contains a dummy variable for each level of **prog** (**prog1**, **prog2**,
and **prog3**), for example, **prog1** is equal to 1 when **prog**=1
(general), and 0 otherwise. The dataset does not contain any missing values. (Note that the names of
variables should NOT be included at the top of the data file. Instead, the
variables are named as part of the **variable** command.) You may want to run
the descriptive statistics in a general use statistics package, such as SAS,
Stata or SPSS, because the options for obtaining descriptive statistics are
limited in Mplus. Even if you chose to run descriptive statistics in another
package, it is
a good idea to run a model with **type=basic** before you do anything else,
just to make sure the dataset is being read correctly.

Lets start by looking at some descriptive statistics generated in another
package. The first table gives the descriptive statistics for the three
continuous variables, and the second table tabulates the categorical
variable **prog**. As expected the highest value of **apt** is 800. In this dataset, the lowest value of **apt** is 352
indicating that no students received a score of 200 (i.e., the lowest
score possible), thus even though censoring from below was possible, it
does not occur in this dataset.

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 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

As we mentioned above, even if you’ve already run descriptive statistics in
another package, you probably want to run an Mplus model with **type=basic**
to make sure your data has been read in properly. The input file for such a
model is shown below. We have also used the **type = plot1**
option of the **plot** command, so that we can use Mplus to generate
histograms and scatterplots.

Data: file is http://stats.idre.ucla.edu/wp-content/uploads/2016/02/tobit.dat; Variable: names are id read math prog apt prog1 prog2 prog3; usevariables are read math apt prog1 prog2 prog3; Analysis: type = basic; Plot: type = plot1;

As we mentioned above, you will want to look at this output carefully to be sure that
the dataset was read into Mplus correctly. For example, checking to make sure that
you have the correct number of observations, and that the variables all have
means that are close to those from the descriptive statistics generated in a
general purpose statistical package. If there are missing values for some or all
of the variables, the descriptive statistics generated by Mplus may not match
those from a general purpose statistical package exactly, because by default, Mplus versions
5.0 and later use maximum likelihood based procedures for handling missing
values. Looking at the output shown below we can confirm that the number of
observations is correct and that the means of the variables are consistent with
those from a general purpose statistical package. Later on we will use the variance of
**apt** as a point of comparison, so we
will make note of this variance (9795.194) shown on the diagonal of the covariance
matrix below.

SUMMARY OF ANALYSIS Number of groups 1 Number of observations 200 <output omitted> ESTIMATED SAMPLE STATISTICS Means READ MATH APT PROG1 PROG2 ________ ________ ________ ________ ________ 1 52.230 52.645 640.035 0.225 0.525 Means PROG3 ________ 1 0.250 Covariances READ MATH APT PROG2 PROG3 ________ ________ ________ ________ ________ READ 105.123 MATH 63.615 87.768 APT 656.273 681.595 9844.416 PROG2 2.075 2.157 19.906 0.251 PROG3 -1.515 -1.564 -19.677 -0.132 0.188 Correlations READ MATH APT PROG2 PROG3 ________ ________ ________ ________ ________ READ 1.000 MATH 0.662 1.000 APT 0.645 0.733 1.000 PROG2 0.404 0.460 0.401 1.000 PROG3 -0.340 -0.385 -0.457 -0.607 1.000

The **plot** command included in the input file above allows us to view histograms of our variables. We can view
the histogram by clicking on the "Graph" menu, and then moving down to
click on "View graphs." In the window that appears select
"Histograms" and click "view." A
second window will appear, where we can select the variable we wish to plot. Below is a
histogram of **apt**.

Looking at the above histogram showing the distribution of **apt**, we can
see the censoring in the data, that is, there are more cases with scores of
750 to 800 (i.e., the bin labeled 777.5) 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. To produce
this graph we proceeded as before, but after we selected **apt** as the
variable to be plotted, we moved to the "Display properties" tab (in the same
window), here we set the number of bins to be the range of **apt** plus one
(800-352+1=449), this produces a histogram with a bin for each integer value
from 352 to 800. 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.

Next we’ll explore the bivariate relationships in our dataset. We can view
the histogram by going to the "Graph" menu, and down to "View graphs," then
selecting "Scatterplots" in the window that appears. Clicking view will show a
second window, where we can select the variables we wish to plot. Below is a scatterplot showing **read** and **apt**. Note the collection of cases
near the top of the 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, the focus of this page.
- OLS Regression – You could analyze these data using OLS regression. OLS regression will treat the 800 as the actual values and not as the lower limit of the top academic aptitude. A limitation of this approach is that when the variable is censored, OLS provides inconsistent estimates of the parameters, meaning that the coefficients from the analysis will not necessarily approach the "true" population parameters as the sample size increases. See Long (1997, chapter 7) for a more detailed discussion of problems of using OLS regression with censored data.
- Truncated Regression – There is sometimes confusion about the difference between truncated data and censored data. With censored variables, all of the observations are in the dataset, but we don’t know the "true" values of some of them. With truncation some of the observations are not included in the analysis because of the value of the variable. When a variable is censored, regression models for truncated data provide inconsistent estimates of the parameters. See Long (1997, chapter 7) for a more detailed discussion of problems of using regression models for truncated data to analyze censored data.

## Tobit analysis

Below is the content of an Mplus input file for a tobit regression model.
Because we are not using all of the variables in the dataset in the model, we
use the **usevariables**
option of the **variables** command to indicate which variables should be
included in the model. The **censored** option
declares that the variable **apt** is censored. The **(a)**
following **apt** on
the **censored** option indicates that the variable is censored from above (i.e., right censoring). If we had
censoring from below (i.e., left-censoring), we would have used the **(b)** option instead.
By default in version 6.12, Mplus uses a robust weighted least squares
estimator. You can use maximum likelihood estimation with robust standard errors
by specifying **estimator = mlr** in the analysis command**.** If you
would like maximum likelihood estimation without robust standard errors, use **
estimator = ml **in the **analysis** command.

data: file is http://stats.idre.ucla.edu/wp-content/uploads/2016/02/tobit.dat ; variable: names are id read math prog apt prog1 prog2 prog3; usevariables are read math apt prog2 prog3; censored are apt (a); model: apt on read math prog2 prog3; output: stdyx;

SUMMARY OF ANALYSIS Number of groups 1 Number of observations 200 Number of dependent variables 1 Number of independent variables 4 Number of continuous latent variables 0 Observed dependent variables Censored APT Observed independent variables READ MATH PROG2 PROG3 Estimator WLSMV Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Parameterization DELTA

- At the top of the output we see that 200 observations were used.
- From the output we see that the model includes one censored dependent (i.e., outcome) variable and 4 independent (predictor) variables.
- The analysis summary is followed by a block of technical information about the model, which we will not discuss except to note that the estimator is listed as WLSMV (the default).

SUMMARY OF CENSORED LIMITS APT 800.000 THE MODEL ESTIMATION TERMINATED NORMALLY

- The section headed SUMMARY OF CENSORED LIMITS describes the censoring of values in the dataset, Mplus has correctly
concluded that the maximum score for
**apt**is 800.

MODEL FIT INFORMATION Number of Free Parameters 6 Chi-Square Test of Model Fit Value 0.000* Degrees of Freedom 0 P-Value 0.0000 * The chi-square value for MLM, MLMV, MLR, ULSMV, WLSM and WLSMV cannot be used for chi-square difference testing in the regular way. MLM, MLR and WLSM chi-square difference testing is described on the Mplus website. MLMV, WLSMV, and ULSMV difference testing is done using the DIFFTEST option. RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 90 Percent C.I. 0.000 0.000 Probability RMSEA <= .05 0.000 CFI/TLI CFI 1.000 TLI 1.000 Chi-Square Test of Model Fit for the Baseline Model Value 930585.250 Degrees of Freedom 5 P-Value 0.0000 WRMR (Weighted Root Mean Square Residual) Value 0.000

MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value APT ON READ 2.698 0.637 4.234 0.000 MATH 5.914 0.774 7.640 0.000 PROG2 -12.715 13.525 -0.940 0.347 PROG3 -46.144 14.233 -3.242 0.001 Intercepts APT 209.568 33.189 6.314 0.000 Residual Variances APT 4313.421 504.344 8.553 0.000

- The section titled MODEL RESULTS includes the coefficient estimates (labeled
Estimate), their standard errors, the ratio of each estimate to its standard
error (i.e., the z-score, labeled Est./S.E.), and the associated p-values.
The coefficients for
**read**and**math**are statistically significant, as is the coefficient for**prog**=3. Tobit regression coefficients are interpreted in a similar manner to OLS regression coefficients; however, the linear effect is on the uncensored latent variable, not the observed outcome. See McDonald and Moffitt (1980) for more details.- For a one unit increase in
**read**, there is a 2.7 point increase in the predicted value of**apt**. - A one unit increase in
**math**is associated with a 5.91 unit increase in the predicted value of**apt**.

- For a one unit increase in
- The residual variance of apt is 4313.42. Some packages give the square
root of this parameter denoted sigma. This value can be compared to the
variance of
**apt**(9795.194 in the output from type=basic), a substantial reduction. Mplus also gives the standard error, est./s.e. and an associated p-value for the residual variance of**apt**. That the residual variance is statistically significant means that the estimated value (9795.194) is statistically significantly different from 0. The validity of this test is a matter of debate among statisticians, and some programs will produce the estimate and standard error, but not the test of statistical significance.

Because we used the **stdyx** option of the **output** command, the output
includes standardized coefficients. We did this primarily to obtain the
R-square values for the output variables, so we have omitted the standardized
output to save space. Based on this output, the model explains about 62% of the
variance in **apt**.

<output omitted> R-SQUARE Observed Residual Variable Estimate Variance APT 0.616

We may also want to test that the coefficients for **prog2**, and **prog3**,
all equal to zero. This type of test can also be
described as an overall test for the effect of **prog**. There are
multiple ways to test this type of hypothesis, the **model test** command
requests one of them, a Wald test. The Mplus input file shown
below is similar to the first regression model, except that the coefficients for **
prog2**, and **prog3**
are assigned the names **p2**, and **p3**, respectively. Note that
each variables to be tested must be alone on a line followed by its label in
parentheses. In the
**model test** command,
these coefficient names (i.e., p**2**, and **p3**) are used to test that each of the coefficients is equal to 0.

Data: File is http://stats.idre.ucla.edu/wp-content/uploads/2016/02/tobit.dat; Variable: Names are id read math prog apt prog1 prog2 prog3; usevariables are read math apt prog2 prog3; censored are apt (a); Model: apt on read math prog2 (p1) prog3 (p2); Model test: p1 = 0; p2 = 0;

The majority of the output from this model is the same as the first model, so we will
only show part of the output generated by the **model test** command.

Wald Test of Parameter Constraints Value 11.906 Degrees of Freedom 2 P-Value 0.0026

The test statistic of 11.906, with 2 degrees of freedom and an associated p-value of
0.0026 indicates that the overall effect of **prog** is statistically significant.

We can also test additional hypotheses about the differences in the coefficients for
different levels of **prog**. Below we test that the coefficient for **prog2** is equal to the
coefficient for **prog3**. In the output below we see that the two coefficient
are significantly different.

Data: File is http://stats.idre.ucla.edu/wp-content/uploads/2016/02/tobit.dat; Variable: Names are id read math prog apt prog1 prog2 prog3; Missing are all (-9999) ; usevariables are read math apt prog2 prog3; censored are apt (a); Model: apt on read math prog2 (p1) prog3 (p2); Model test: p1 = p2;Wald Test of Parameter Constraints Value 6.979 Degrees of Freedom 1 P-Value 0.0082

The test statistic of 6.979, with 1 degree of freedom and an associated p-value of 0.0082
indicates that the coefficient for **prog**=2 is significantly different from
the coefficient for **prog**=3.

## See also

## References

Long, J. S. 1997. *Regression Models for Categorical and Limited Dependent Variables.*
Thousand Oaks, CA: Sage Publications.

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

Tobin, J. 1958. Estimation of relationships for limited dependent variables.
*Econometrica* 26: 24-36.