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Tobit Analysis

**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.

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.

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 http://www.ats.ucla.edu/stat/stata/dae/tobit, clearsummarize apt read mathVariable | 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 75tabulate progtype of | program | Freq. Percent Cum. ------------+----------------------------------- academic | 45 22.50 22.50 general | 105 52.50 75.00 vocational | 50 25.00 100.00 ------------+----------------------------------- Total | 200 100.00histogram 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 1.0000graph 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**.

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 upper 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.

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

- The final log likelihood (-1041.0629) is shown at the top of the output, it can be used in comparisons of nested models, but we won't show an example of that here.
- Also at the top of the output we see that all 200 observations in our data set were used in the analysis (fewer observations would have been used if any of our variables had missing values).
- The likelihood ratio chi-square of 188.97 (df=4) with a p-value of 0.0001 tells us that our model as a whole fits significantly better than an empty model (i.e., a model with no predictors).
- In the table we see the coefficients, their standard errors, the t-statistic,
associated p-values, and the 95% confidence interval of the coefficients.
The coefficients for
**read**and**math**are statistically significant, as is the coefficient for**prog**=3. Tobit regression coefficients are interpreted in the similiar 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**. - The terms for
**prog**have a slightly different interpretation. The predicted value of**apt**is 46.14 points lower for students in a vocational program (**prog**=3) than for students in an academic program (**prog**=1).

- For a one unit increase in
- The ancillary statistic /sigma is analogous to the square root of the residual variance in OLS regression. The value of 65.67 can be compared to the standard deviation of academic aptitude which was 99.21, a substantial reduction. The output also contains an estimate of the standard error of /sigma as well as the 95% confidence interval.
- Finally, the output provides a summary of the number of left-censored, uncensored and right-censored values.

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 **findit fitstat** (see
How can I use the
findit command to search for programs and get additional help? for more
information about using **findit**).

fitstatMeasures 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

- Stata Online Manual
- Related Stata Commands
**cnreg**-- censored normal regression, in which the censoring values may change from observation to observation.**intreg**-- interval regression, in which observations may be point data, interval data, left-censored data or right-censored data.

- Long, J. S. (1997).
*Regression Models for Categorical and Limited Dependent Variables.*Thousand Oaks, CA: Sage Publications. - Tobin, J. (1958). Estimation of relationships for limited dependent variables.
*Econometrica*26: 24-36.

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

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