Annotated Mplus Output Censored Regression

This page shows an example of censored regression with footnotes explaining the output. First an example is shown using Stata, and then an example is shown using Mplus, to help you relate the output you are likely to be familiar with (Stata) to output that may be new to you (Mplus). We suggest that you view this page using two web browsers so you can show the page side by side showing the Stata output in one browser and the corresponding Mplus output in the other browser.

This example is drawn from the Mplus User's Guide (example 3.2) and we suggest that you see the Mplus User's Guide for more details about this example. We thank the kind people at Muthén & Muthén for permission to use examples from their manual.

Example Using Stata

Here is a probit regression example using Stata with two continuous predictors x1 and x2 used to predict a binary outcome variable, u1.

infile u1 x1 x3 using http://www.ats.ucla.edu/stat/mplus/output/ex3.2.dat, clear
summarize u1

Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
u1 |      1000    .9240341    1.113079          0A   6.579389
tobit u1 x1 x3, ll(0)

Tobit regression                                  Number of obs   =       1000
LR chi2(2)      =     697.44
Prob > chi2     =     0.0000
Log likelihood = -1142.8851                       Pseudo R2       =     0.2338

------------------------------------------------------------------------------
u1 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 |   1.074801D   .0419657    25.61   0.000     .9924498    1.157152
x3 |   .4947541D   .0378985    13.05   0.000     .4203842     .569124
_cons |   .5154865E   .0405066    12.73   0.000     .4359986    .5949743
-------------+----------------------------------------------------------------
/sigma |   1.071333F   .0316242                      1.009276    1.133391
------------------------------------------------------------------------------
Obs. summary:        376  left-censored observations at u1<=0
624     uncensored observations
0 right-censored observations
estat ic

------------------------------------------------------------------------------
Model |    Obs    ll(null)   ll(model)     df          AIC         BIC
-------------+----------------------------------------------------------------
. |   1000   -1491.605   -1142.885B      4      2293.77C    2313.401C
------------------------------------------------------------------------------


The output is labeled with superscripts to help you relate the later Mplus output to this Stata output. To summarize the output, both predictors in this model, x1 and x2, are significantly related to the outcome variable, u1.

Mplus Example

Here is the same example illustrated in Mplus based on the ex3.2.dat data file. Note that by using estimator=wls; (weighted least squares) the results are shown in a probit metric. Had we specified something like estimator=ml; (maximum likelihood) then the results would be shown in a logit scale.

TITLE:
this is an example of a censored
regression for a censored dependent
variable with two covariates
DATA:
FILE IS ex3.2.dat;
VARIABLE:
NAMES ARE y1 x1 x3;
CENSORED ARE y1 (b);
ANALYSIS:
ESTIMATOR = MLR;
MODEL:
y1 ON x1 x3;
SUMMARY OF ANALYSIS

<some output omitted to save space>
Number of observations                                        1000

<some output omitted to save space>
SUMMARY OF CENSORED LIMITS
Y1                 0.000A

THE MODEL ESTIMATION TERMINATED NORMALLY

TESTS OF MODEL FIT

Loglikelihood

H0 Value                       -1142.885B

Information Criteria

Number of Free Parameters              4
Akaike (AIC)                    2293.770C
Bayesian (BIC)                  2313.401C
(n* = (n + 2) / 24)

MODEL RESULTS
Estimates     S.E.  Est./S.E.
Y1         ON
X1                 1.075D    0.043     25.101
X3                 0.495D    0.037     13.344

Intercepts
Y1                 0.515E    0.040     12.810

Residual Variances
Y1                 1.148F    0.067     17.235

1. This indicates that the variable y1 is censored at 0. This is derived from the data, where Mplus notes that the lowest value of y1 is 0 (it seeks the lowest value because the input specification indicated the censoring was from below). Note how this corresponds to the results of the Stata summarize command that found the minimum value of y1 to be 0.
2. This is the log likelihood of the model. Note how this corresponds to the ll(model) from the Stata estat ic command.
3. These are the AIC and BIC fit indices, and correspond to the values shown from the estat ic command from Stata.
4. These are the regression coefficients showing the relationship between x1 x2 and y1. Such coefficients are interpreted in the same way as an OLS regression coefficient. The difference is that these coefficients attempt to estimate how estimate how strong the coefficient would have been had the censoring not taken place. Note the correspondence between these coefficients and those from Stata.
5. This is the intercept, the predicted value when all predictors are held constant at 0. Note the correspondence to the value shown in the Stata output.
6. This is the residual variance in y1 after accounting for the predictors, and would be analogous to the MSE from an OLS regression. In the Stata output this is reported as /sigma and is reported as a standard deviation (as opposed to a variance). Squaring the value from Stata yields 1.071333^2 = 1.1477544, corresponding to the result from Mplus.

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