### Stata Annotated Output Tobit Regression

This page shows an example of tobit regression analysis with footnotes explaining the output. The data in this example were gathered on undergraduates applying to graduate school and includes undergraduate GPAs, the reputation of the school of the undergraduate (a topnotch indicator), the students' GRE score, and whether or not the student was admitted to graduate school.

The range of possible GRE scores is 200 to 800. This means that our outcome variable is both left censored and right-censored.  In other words, if two students score an 800, they are equal according to our scale but might not truly be equal in aptitude.  (In other words, we have a ceiling effect.)  The same is true of two students scoring 200 (a floor effect).  Tobit regression generates a model that predicts the outcome variable to be within the specified range.

If we are interested in predicting a student's GRE score using their undergraduate GPA and the reputation of their undergraduate institution, we should first consider GRE as an outcome variable.

use http://www.ats.ucla.edu/stat/stata/dae/logit.dta, clear
summarize(gre)
    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
gre |       400       587.7    115.5165        220        800
histogram gre, bin(10) freq

To generate a tobit model in Stata, list the outcome variable followed by the predictors and then specify the lower limit and/or upper limit of the outcome variable.  The lower limit is specified in parentheses after ll and the upper limit is specified in parentheses after ul.   A tobit model can be used to predict an outcome that is censored from above, from below, or both.

tobit gre gpa topnotch, ll(200) ul(800)
Tobit regression                                  Number of obs   =        400
LR chi2(2)      =      70.93
Prob > chi2     =     0.0000
Log likelihood = -2331.4314                       Pseudo R2       =     0.0150

------------------------------------------------------------------------------
gre |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
gpa |   111.3085   15.19665     7.32   0.000     81.43273    141.1842
topnotch |   46.65774   15.75356     2.96   0.003     15.68716    77.62833
_cons |   205.8515   51.24073     4.02   0.000     105.1152    306.5879
-------------+----------------------------------------------------------------
/sigma |   111.4882   4.143727                      103.3419    119.6345
------------------------------------------------------------------------------
Obs. summary:          0  left-censored observations
375     uncensored observations
25 right-censored observations at gre>=800

#### Tobit Regression Output

Tobit regression                                  Number of obsb   =        400
LR chi2(2)c      =      70.93
Prob > chi2d     =     0.0000
Log likelihooda = -2331.4314                      Pseudo R2e       =     0.0150

------------------------------------------------------------------------------
gref|      Coef.g  Std. Err.h     ti   P>|t|j    [95% Conf. Interval]k
-------------+----------------------------------------------------------------
gpa |   111.3085   15.19665     7.32   0.000     81.43273    141.1842
topnotch |   46.65774   15.75356     2.96   0.003     15.68716    77.62833
_cons |   205.8515   51.24073     4.02   0.000     105.1152    306.5879
-------------+----------------------------------------------------------------
/sigmal|   111.4882   4.143727                      103.3419    119.6345
------------------------------------------------------------------------------
Obs. summary:m         0  left-censored observations
375     uncensored observations
25 right-censored observations at gre>=800

a. Log likelihood - This is the log likelihood of the fitted model. It is used in the Likelihood Ratio Chi-Square test of whether all predictors' regression coefficients in the model are simultaneously zero.

b. Number of obs - This is the number of observations in the dataset for which all of the response and predictor variables are non-missing.

c. LR chi2(2) - This is the Likelihood Ratio (LR) Chi-Square test that at least one of the predictors' regression coefficient is not equal to zero.  The number in the parentheses indicates the degrees of freedom of the Chi-Square distribution used to test the LR Chi-Square statistic and is defined by the number of predictors in the model (2).

d. Prob > chi2 - This is the probability of getting a LR test statistic as extreme as, or more so, than the observed statistic under the null hypothesis; the null hypothesis is that all of the regression coefficients are simultaneously equal to zero.  In other words, this is the probability of obtaining this chi-square statistic (70.93) or one more extreme if there is in fact no effect of the predictor variables.  This p-value is compared to a specified alpha level, our willingness to accept a type I error, which is typically set at 0.05 or 0.01.  The small p-value from the LR test,  <0.0001, would lead us to conclude that at least one of the regression coefficients in the model is not equal to zero. The parameter of the chi-square distribution used to test the null hypothesis is defined by the degrees of freedom in the prior line, chi2(2)

e. Pseudo R2 - This is McFadden's pseudo R-squared.  Tobit regression does not have an equivalent to the R-squared that is found in OLS regression; however, many people have tried to come up with one.  There are a wide variety of pseudo-R-square statistics.  Because this statistic does not mean what R-square means in OLS regression (the proportion of variance of the response variable explained by the predictors), we suggest interpreting this statistic with great caution. For more information on pseudo R-squareds, see What are Pseudo R-Squareds?.

f. gre - This is the response variable predicted by the model.  We are using a tobit model because this response variable is censored: the GRE scores are scaled from 200 to 800 and cannot fall outside of this range.

g. Coef. - These are the regression coefficients. 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. The expected GRE score changes by Coef. for each unit increase in the corresponding predictor.

gpa - If a subject were to increase his gpa by one point, his expected GRE score would increase by 111.3085 points while holding all other variables in the model constant. Thus, the higher a student's gpa, the higher the predicted GRE score.

topnotch - If a subject attended a topnotch institution for her undergraduate education, her expected GRE score would be 46.65774 points higher than a subject with the same grade point average who attended a non-topnotch institution.  Thus, subjects from topnotch undergraduate institutions have higher predicted GRE scores than subjects from non-topnotch undergraduate institutions if grade point averages are held constant.

_cons - If all of the predictor variables in the model are evaluated at zero, the predicted GRE score would be _cons = 205.8515.  For subjects from non-topnotch undergraduate institutions (topnotch evaluated at zero) with zero gpa, the predicted GRE score would be 205.8515.  This may seem very low, considering the mean GRE score is 587.7, but note that evaluating gpa at zero is out of the range of plausible values for gpa.

h. Std. Err. - These are the standard errors of the individual regression coefficients.  They are used in both the calculation of the t test statistic, superscript i, and the confidence interval of the regression coefficient, superscript k.

i. t -  The test statistic t is the ratio of the Coef. to the Std. Err. of the respective predictor.  The t value is used to test against a two-sided alternative hypothesis that the Coef. is not equal to zero.

j. P>|t| - This is the probability the t test statistic (or a more extreme test statistic) would be observed under the null hypothesis that a particular predictor's regression coefficient is zero, given that the rest of the predictors are in the model.  For a given alpha level, P>|t| determines whether or not the null hypothesis can be rejected. If P>|t| is less than alpha, then the null hypothesis can be rejected and the parameter estimate is considered statistically significant at that alpha level.

gpa - The t test statistic for the predictor gpa is (111.3085/15.19665) = 7.32 with an associated p-value of <0.001.  If we set our alpha level to 0.05, we would reject the null hypothesis and conclude that the regression coefficient for gpa has been found to be statistically different from zero given topnotch is in the model.

topnotch -The t test statistic for the predictor topnotch is (46.65774/15.75356) = 2.96 with an associated p-value of 0.003.  If we set our alpha level to 0.05, we would reject the null hypothesis and conclude that the regression coefficient for topnotch has been found to be statistically different from zero given gpa is in the model.

_cons - The t test statistic for the intercept, _cons, is (205.8515/51.24073) = 4.02 with an associated p-value of < 0.001.  If we set our alpha level at 0.05, we would reject the null hypothesis and conclude that _cons has been found to be statistically different from zero given gpa and topnotch are in the model and evaluated at zero.

k. [95% Conf. Interval] - This is the Confidence Interval (CI) for an individual coefficient given that the other predictors are in the model.  For a given predictor with a level of 95% confidence, we'd say that we are 95% confident that the "true" coefficient lies between the lower and upper limit of the interval.  The CI is equivalent to the t test statistic: if the CI includes zero, we'd fail to reject the null hypothesis that a particular regression coefficient is zero given the other predictors are in the model with alpha level of zero.  An advantage of a CI is that it is illustrative; it provides a range where  the "true" parameter may lie.

l. /sigma - This is the estimated standard error of the regression.  This value, 111.4882, is comparable to the root mean squared error that would be obtained in an OLS regression.

m. Obs. summary -  This indicates how many of the observations in the dataset are censored.  Here, we see that none of the records are left censored (all are greater than 200) and 25 of the records are right-censored (greater than or equal to 800).

The content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California.