### SAS Annotated Output: Tobit Regression

This page shows an example of tobit regression analysis in SAS with footnotes explaining the output. The data in this example were gathered on undergraduates applying to graduate school and include 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 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 a censored outcome variable.

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.  We can use the dataset tobit.sas7bdat .

data tobit;
set "C:\temp\tobit";
run;

proc means data = tobit;
var gre;
run;
The MEANS Procedure

Analysis Variable : GRE
N            Mean         Std Dev         Minimum         Maximum
400     587.7000000     115.5165364     220.0000000     800.0000000

proc univariate data = tobit noprint;
histogram gre / vscale = count;
run;


To generate a tobit model in SAS, we must first add a variable to our data indicating which observations are censored.

data tobit; set tobit;
censor = (gre >= 800 or gre <= 200);
run;

proc freq data = tobit;
table censor;
run;
The FREQ Procedure

Cumulative    Cumulative
censor    Frequency     Percent     Frequency      Percent
0         375       93.75           375        93.75
1          25        6.25           400       100.00

In the output above, we see that 25 of our observations are censored while 375 are not. Next, we program our model in SAS.  This can be done with proc lifereg. We specify our model, indicating that our response is censored.

proc lifereg data = tobit;
model gre*censor(1) = gpa topnotch /d=normal;
run;
The LIFEREG Procedure

Model Information

Data Set                      WORK.TOBIT
Dependent Variable                   GRE
Censoring Variable                censor
Censoring Value(s)                     1
Number of Observations               400
Noncensored Values                   375
Right Censored Values                 25
Left Censored Values                   0
Interval Censored Values               0
Name of Distribution              Normal
Log Likelihood              -2331.431433

Number of Observations Used         400

Algorithm converged.

Type III Analysis of Effects

Wald
Effect       DF    Chi-Square    Pr > ChiSq

GPA           1       53.6486        <.0001
TOPNOTCH      1        8.7718        0.0031

Analysis of Parameter Estimates

Standard   95% Confidence     Chi-
Parameter DF Estimate    Error       Limits       Square Pr > ChiSq

Intercept  1 205.8515  51.2408 105.4213 306.2818   16.14     <.0001
GPA        1 111.3085  15.1967  81.5235 141.0934   53.65     <.0001
TOPNOTCH   1  46.6577  15.7536  15.7813  77.5342    8.77     0.0031
Scale      1 111.4882   4.1438 103.6554 119.9129

#### Tobit Regression Output

The LIFEREG Procedure

Model Information

Data Seta                      WORK.TOBIT
Dependent Variableb                   GRE
Censoring Variablec                censor
Censoring Value(s)d                     1
Number of Observationse               400
Noncensored Valuesf                   375
Right Censored Valuesg                 25
Left Censored Valuesh                   0
Interval Censored Valuesi               0
Name of Distributionj              Normal
Log Likelihoodk              -2331.431433

Number of Observations Used         400

Algorithm converged.l

Type III Analysis of Effectsm
Wald
Effect       DF    Chi-Square    Pr > ChiSq
GPA           1       53.6486        <.0001
TOPNOTCH      1        8.7718        0.0031

Analysis of Parameter Estimates

                      Standard   95% Confidence     Chi-
ParameternDFoEstimatep   Errorq      Limitsr      Squares  Pr>ChiSqt

Intercept  1 205.8515  51.2408 105.4213 306.2818   16.14     <.0001
GPA        1 111.3085  15.1967  81.5235 141.0934   53.65     <.0001
TOPNOTCH   1  46.6577  15.7536  15.7813  77.5342    8.77     0.0031
Scaleu     1 111.4882   4.1438 103.6554 119.9129

a. Dataset - This indicates the dataset used in the analysis. If a dataset is not specified in the model command, SAS uses the most recently created/modified dataset by default.

b. Dependent Variable - 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 be measured outside of this range (although the phenomenon underlying the scores, in this case aptitude, is not bounded).

c. Censoring Variable - This is the variable we defined in preparation for running the tobit model.  Values in our dataset will be considered censored based on the corresponding value of our censoring variable.

d. Censoring Value(s) - These are the values of the censoring variable that indicate a censored value in the dependent variable. In this example, the observations where censor = 1 are censored.  We indicated this in the proc lifereg command with the 1 in parenthesis after censor.

e. Number of Observations - This is the number of observations from the dataset used in the model.  If an observation is missing data in the outcome or any of the predictor variables, then it is excluded from the analysis.

f. Noncensored Values - This is the number of observations in the model that were not censored. In this example, there were 375 observations in the dataset with 200 < gre < 800.

g. Right Censored Values - This is the number of observations in the model that were right censored. In this example, there were 25 observations in the dataset with gre >= 800.

h. Left Censored Values - This is the number of observations in the model that were left censored. In this example, there were zero observations in the dataset with gre <= 200.

i. Interval Censored Values - This is the number of observations in the model that were interval censored (where the outcome variable fell in an interval that was censored). This type of censoring was not used in this model, and so there were zero observations in the dataset in this category.

j. Name of Distribution - This indicates the distribution assumed for the errors terms of the model.  In a tobit model, this distribution is normal.

k. 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.

l. Algorithm Converged. - This indicates that the SAS convergence criterion for the iterating steps used in maximizing the likelihood was met. The default criterion in SAS is the relative gradient convergence criterion, and the default precision is 10-8.

m. Type III Analysis of Effects - This is an analysis of the model variables using Type III sum of squares.  That is, the effect of a model variable does not depend on the order in which the variable is specified in the model.

n. Parameter - This lists the model parameters.  Our model includes an intercept and the specified predictor variables.

o. DF - These are the degrees of freedom associated with each of the model parameters.

p. Estimate - These are the regression coefficients. These coefficients are interpreted as you would interpret coefficients from an OLS regression: the expected GRE score changes by Estimate for each unit increase in the corresponding predictor.

Intercept- If all of the predictor variables in the model are evaluated at zero, the predicted GRE score would be Intercept = 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.

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.

q. Standard Error - These are the standard errors of the individual regression coefficients.

r. 95% Confidence Limits - 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.

s. Chi-Square - This is the Chi-Square test statistic corresponding to the hypothesis that the given parameter's estimate is equal to zero.

t. Pr>ChiSq - This is the probability the Chi-Square 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>ChiSq determines whether or not the null hypothesis can be rejected. If P>ChiSq is less than alpha, then the null hypothesis can be rejected and the parameter estimate is considered statistically significant at that alpha level.

Intercept - The Chi-Square test statistic for the intercept, Intercept is 16.14 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 the Intercept has been found to be statistically different from zero given gpa and topnotch are in the model and evaluated at zero.

gpa - The Chi-Square test statistic for the predictor gpa is 53.65 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 Chi-Square test statistic for the predictor topnotch is 8.77 with an associated p-value of 0.0031.  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.

u. Scale - 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.

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