Mplus Data Analysis Examples: Tobit Models



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Mplus Data Analysis Examples
Tobit Analysis

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. Tobit models are designed to make improved estimates when there is either left- or right-censoring.

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) and can be analyzed using tobit analysis.

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 and whether the student is enrolled in a public or private school. 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.

Description of the Data

Let's pursue Example 3 from above.

We have a hypothetical data file, tobitex.dat with 200 observations. The academic aptitude variable is apt, the reading and writing test scores are read and write respectively. The variable public is a zero-one variable with the ones indicating a public school student.

Let's look at the data.

title: Mplus DAE for censored regression
data: file is "d:\tobitex.dat";
variable:
names are id apt read math public;
usevar apt read math public;
analysis:
  type = basic;
plot: type is plot1;

RESULTS FOR BASIC ANALYSIS

     SAMPLE STATISTICS

           Means
              APT           READ          MATH          PUBLIC
              ________      ________      ________      ________
      1       651.060        52.230        52.645         0.545

           Covariances
              APT           READ          MATH          PUBLIC
              ________      ________      ________      ________
 APT        10290.147
 READ         621.051       105.123
 MATH         586.408        63.615        87.768
 PUBLIC        12.997        -0.272        -0.137         0.249

           Correlations
              APT           READ          MATH          PUBLIC
              ________      ________      ________      ________
 APT            1.000
 READ           0.597         1.000
 MATH           0.617         0.662         1.000
 PUBLIC         0.257        -0.053        -0.029         1.000

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a tobit analysis, let's consider some other methods that you might use.

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. If we compare the predicted values from an OLS model with a tobit model we would find that the tobit predicted values have a range of 491.33-840.12 while the OLS predicted have a range of 497.76-825.22.
Truncated Regression - Beginners often confuse the concepts of truncation with censoring. With censored data we have all of the observations 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. It would be inappropriate to analyze the data in our example using a truncated regression model.

Tobit Analysis

NOTE: This example was done using Mplus version 4.21. The syntax may not work with earlier versions of Mplus.

We use the usevar statement to indicate that we are not using all of the variables in the data set in the current model. We have omitted the missing statement because we have no missing data in this data set. Mplus allows for both left- and right-censoring. We use the (a) option on the censored statement to indicate that we have right censoring. If we had left-censoring, we would have used the (b) option instead. The default estimation method is MLR - maximum likelihood parameter estimates with standard errors and a chi-square test statistic that are robust to non-normality and non-independence of observations when used with type = complex, according to the Mplus 4 manual. The MLR standard errors are computed using a sandwich estimator. This is what we generally call robust standard errors. To get the "regular" standard errors, we use the estimator = ml on the analysis statement.

title: Mplus DAE for censored regression
data: file is "d:\tobitex.dat";
variable:
names are id apt read math public;
usevar apt read math public;
censored are apt (a);
analysis:
estimator = ml;
model:
apt on read math public;

MODEL RESULTS
                   Estimates     S.E.  Est./S.E.

 APT        ON
    READ               3.682    0.687      5.356
    MATH               4.558    0.754      6.046
    PUBLIC            62.163   10.574      5.879

 Intercepts
    APT              188.395   32.751      5.752

 Residual Variances
    APT             5421.543  570.530      9.503

Sample Write-Up of the Analysis

Each of the predictor variables in the model, read, math and public, was statically significant. A unit change in read and math lead to a 3.68 and 4.56 increase in the predicted aptitude, respectively. Attending a public school increased the predicted aptitude by 62.16 points as compared with private school attendance.