The LIFEREG procedure can be used to perform a Tobit analysis. The Tobit model, described by Tobin (1958), is a regression model for leftcensored data assuming a normally distributed error term. The model parameters are estimated by maximum likelihood. PROC LIFEREG provides estimates of the parameters of the distribution of the uncensored data. See Greene (1993) and Maddala (1983) for a more complete discussion of censored normal data and related distributions. This example shows how you can use PROC LIFEREG and the DATA step to compute two of the three types of predicted values discussed there.
Consider a continuous random variable Y and a constant C. If you were to sample from the distribution of Y but discard values less than (greater than) C, the distribution of the remaining observations would be truncated on the left (right). If you were to sample from the distribution of Y and report values less than (greater than) C as C, the distribution of the sample would be left (right) censored.
The probability density function of the truncated random variable is given by

where is the probability density function of Y. PROC LIFEREG cannot compute the proper likelihood function to estimate parameters or predicted values for a truncated distribution. Suppose the model being fit is specified as follows:

where is a normal error term with zero mean and standard deviation .
Define the censored random variable as






This is the Tobit model for leftcensored normal data. is sometimes called the latent variable. PROC LIFEREG estimates parameters of the distribution of by maximum likelihood.
You can use the LIFEREG procedure to compute predicted values based on the mean functions of the latent and observed variables. The mean of the latent variable is , and you can compute values of the mean for different settings of by specifying XBETA=variablename in an OUTPUT statement. Estimates of for each observation will be written to the OUT= data set. Predicted values of the observed variable can be computed based on the mean

where

and represent the normal probability density and cumulative distribution functions.
Although the distribution of in the Tobit model is often assumed normal, you can use other distributions for the Tobit model in the LIFEREG procedure by specifying a distribution with the DISTRIBUTION= option in the MODEL statement. One distribution that should be mentioned is the logistic distribution. For this distribution, the MLE has bounded influence function with respect to the response variable, but not the design variables. If you believe your data have outliers in the response direction, you might try this distribution for some robust estimation of the Tobit model.
With the logistic distribution, the predicted values of the observed variable can be computed based on the mean of ,

The following table shows a subset of the Mroz (1987) data set. In these data, Hours
is the number of hours the wife worked outside the household in a given year, Yrs_Ed
is the years of education, and Yrs_Exp
is the years of work experience. A Tobit model will be fit to the hours worked with years of education and experience as
covariates.




0 
8 
9 
0 
8 
12 
0 
9 
10 
0 
10 
15 
0 
11 
4 
0 
11 
6 
1000 
12 
1 
1960 
12 
29 
0 
13 
3 
2100 
13 
36 
3686 
14 
11 
1920 
14 
38 
0 
15 
14 
1728 
16 
3 
1568 
16 
19 
1316 
17 
7 
0 
17 
15 
If the wife was not employed (worked 0 hours), her hours worked will be left censored at zero. In order to accommodate left
censoring in PROC LIFEREG, you need two variables to indicate censoring status of observations. You can think of these variables
as lower and upper endpoints of interval censoring. If there is no censoring, set both variables to the observed value of
Hours
. To indicate left censoring, set the lower endpoint to missing and the upper endpoint to the censored value, zero in this
case.
The following statements create a SAS data set with the variables Hours
, Yrs_Ed
, and Yrs_Exp
from the preceding data. A new variable, Lower
, is created such that Lower
=. if Hours
=0 and Lower
=Hours
if Hours
>0.
data subset; input Hours Yrs_Ed Yrs_Exp @@; if Hours eq 0 then Lower=.; else Lower=Hours; datalines; 0 8 9 0 8 12 0 9 10 0 10 15 0 11 4 0 11 6 1000 12 1 1960 12 29 0 13 3 2100 13 36 3686 14 11 1920 14 38 0 15 14 1728 16 3 1568 16 19 1316 17 7 0 17 15 ;
The following statements fit a normal regression model to the leftcensored Hours
data with Yrs_Ed
and Yrs_Exp
as covariates. You need the estimated standard deviation of the normal distribution to compute the predicted values of the
censored distribution from the preceding formulas. The data set OUTEST
contains the standard deviation estimate in a variable named _SCALE_
. You also need estimates of . These are contained in the data set OUT
as the variable Xbeta
.
proc lifereg data=subset outest=OUTEST(keep=_scale_); model (lower, hours) = yrs_ed yrs_exp / d=normal; output out=OUT xbeta=Xbeta; run;
Output 51.2.1 shows the results of the model fit. These tables show parameter estimates for the uncensored, or latent variable, distribution.
Output 51.2.1: Parameter Estimates from PROC LIFEREG
Model Information  

Data Set  WORK.SUBSET 
Dependent Variable  Lower 
Dependent Variable  Hours 
Number of Observations  17 
Noncensored Values  8 
Right Censored Values  0 
Left Censored Values  9 
Interval Censored Values  0 
Number of Parameters  4 
Name of Distribution  Normal 
Log Likelihood  74.9369977 
Analysis of Maximum Likelihood Parameter Estimates  

Parameter  DF  Estimate  Standard Error  95% Confidence Limits  ChiSquare  Pr > ChiSq  
Intercept  1  5598.64  2850.248  11185.0  12.2553  3.86  0.0495 
Yrs_Ed  1  373.1477  191.8872  2.9442  749.2397  3.78  0.0518 
Yrs_Exp  1  63.3371  38.3632  11.8533  138.5276  2.73  0.0987 
Scale  1  1582.870  442.6732  914.9433  2738.397 
The following statements combine the two data sets created by PROC LIFEREG to compute predicted values for the censored distribution. The OUTEST= data set contains the estimate of the standard deviation from the uncensored distribution, and the OUT= data set contains estimates of .
data predict; drop lambda _scale_ _prob_; set out; if _n_ eq 1 then set outest; lambda = pdf('NORMAL',Xbeta/_scale_) / cdf('NORMAL',Xbeta/_scale_); Predict = cdf('NORMAL', Xbeta/_scale_) * (Xbeta + _scale_*lambda); label Xbeta='MEAN OF UNCENSORED VARIABLE' Predict = 'MEAN OF CENSORED VARIABLE'; run;
Output 51.2.2 shows the original variables, the predicted means of the uncensored distribution, and the predicted means of the censored distribution.
Output 51.2.2: Predicted Means from PROC LIFEREG
Hours  Lower  Yrs_Ed  Yrs_Exp  MEAN OF UNCENSORED VARIABLE 
MEAN OF CENSORED VARIABLE 

0  .  8  9  2043.42  73.46 
0  .  8  12  1853.41  94.23 
0  .  9  10  1606.94  128.10 
0  .  10  15  917.10  276.04 
0  .  11  4  1240.67  195.76 
0  .  11  6  1113.99  224.72 
1000  1000  12  1  1057.53  238.63 
1960  1960  12  29  715.91  1052.94 
0  .  13  3  557.71  391.42 
2100  2100  13  36  1532.42  1672.50 
3686  3686  14  11  322.14  805.58 
1920  1920  14  38  2032.24  2106.81 
0  .  15  14  885.30  1170.39 
1728  1728  16  3  561.74  951.69 
1568  1568  16  19  1575.13  1708.24 
1316  1316  17  7  1188.23  1395.61 
0  .  17  15  1694.93  1809.97 