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 Anonymous posted on Wednesday, August 17, 2005 - 10:53 am
In a couple of discussion threads I've noticed questions about regression with censored, nonnormal dependent variables...

I was wondering what the underlying assumptions are for CR in MPlus? I saw it said in one thread that there were some shortcomings in the Tobit estimator and it is not used in the current version of MPlus.

We have a censored, nonnormal dependent variable which is an offending diversity index based on Agresti and Agresti (1978) and are reluctant to transform it to use some type of ordered or binary regression for fear of losing a lot of the value of this measure.

Is there an approach in MPlus that would help in ensuring that our estimates are not biased? Is there a particular estimator (s) that we could use with the MPlus censored regression?
 bmuthen posted on Wednesday, August 17, 2005 - 10:59 am
Censored-normal regression can be done by ML or WLS(MV) in Mplus. This is the standard Tobit model assuming an underlying normal (uncensored) dependent variable that is censored from below or above at a known point. Earlier versions of Mplus didn't have the censored option. There is also censored-inflated modeling available in line with zero-inflated Poisson regression. And this can be incorporated in a larger model.
 Anonymous posted on Friday, August 19, 2005 - 4:51 am
Thank you Dr. Muthen.

Just to clarify--there aren't any estimators associated with the MPlus censored reg models that are appropriate for nonnormal data?

Are there any other options available in MPlus for dealing with that type of limited DV?
 Linda K. Muthen posted on Friday, August 19, 2005 - 8:23 am
Censored variables are non-normal and this is taken into account in the estimation of the model. Robust estimators are available with censored outcomes. If you look up ESTIMATOR in the index of the Mplus User's Guide, it will take you to a table where analysis type and scale of the dependent variables are crossed. The entries in the table are the estimators available.

Two-part modeling may be another approach for censored data. See what the following paper says about that:

Olsen, M. K, & Schafer, J., L. (2001). A two-part random effects model for semicontinuous longitudinal data. Journal of the American Statistical Association, 96, 730-745.
 Michael posted on Wednesday, October 12, 2005 - 10:40 am
I'm considering using my Mplus software to do SEM with censored variables, using ordinary TOBIT analysis.

Can you recommend a user-friendly paper on what it is that TOBIT does, hopefully written for a nonstatistician?

I've seen folks like Stoolmiller use it, but still don't have a great idea about what it actually does.

 bmuthen posted on Wednesday, October 12, 2005 - 11:26 am
Here are some references for Tobit regression, which is a building block for censored SEM. The Long book may be the more accessible for a nonstatistician. Tobin's article is the origin of this.

Long, S. (1996). Regression models for categorical and limited dependent variables. Thousand Oaks: Sage.

Maddala, G.S. (1983). Limited-dependent and qualitative variables in econometrics. Cambridge: Cambridge University Press.

Tobin, J (1958). Estimation of relationships for limited dependent variables. Econometrica, 26, 24-36.

I have a paper on censored factor analysis referenced on the Mplus web site but it si more technical.
 J.W. posted on Tuesday, March 18, 2008 - 11:46 am
The Tobit model assumes that the variance of the conditional distribution is uniform. Maddala (1983) comments that in the presence of heteroscedasticity, the usual Tobit estimates are inconsistent, and that there is only limited information about the direction of the bias. Maddala (1983) extends the Tobit model to cases where heteroscedasticity is present, and where one is able to specify the functional form of the variance (e.g. variance increases linearly as a function of a specific covariate). Methods for estimating the Tobit model in the presence of heteroscedasticity have been implemented in statistics/econometric software packages such as Limdep. Is it possible to run censored regression or censored-inflated regression in Mplus without bias in the presence of heteroscedasticity? Your help will be appreciated.
 Bengt O. Muthen posted on Saturday, March 22, 2008 - 12:58 pm
I am not sure, but I wonder if you can use the constraint = z option of the variable command in conjunction with Model Constraint in line with UG ex 5.23 ("QTL" example) to model the Tobit residual variance as a function of an observed z variable.
 Annibale Cois posted on Friday, November 13, 2015 - 10:16 am
Dear Mplus team

I am trying to estimate a model where some variables are right-censored.
I Know Mplus can handle censored variables, but in my case the censoring point is not constant, but varies across subjects. I have in my dataset another binary variable that indicates which values are censored and which are not.
Is there a way in Mplus to estimate such a model?
 Bengt O. Muthen posted on Friday, November 13, 2015 - 5:51 pm
There isn't an automatic way of doing it and I don't know how it would be done, but I am not sure it is impossible in Mplus; I haven't tried it.
 Annibale Cois posted on Friday, November 13, 2015 - 10:07 pm
Thanks, Bengt

I will give some thought I I can find some way around.
 John B. Nezlek posted on Saturday, September 02, 2017 - 9:31 am
Dear Mplus:
I have what I suspect is a simple, novice analyst question.
I have just run a regression analysis with a censored inflated outcome (30% ones, the rest > 1 and < 7. I followed the example 3.3 in the Guide specifying censoring from below. All worked well.

My question concerns interpreting the output. I have inserted model results below. The outcome is ENVY, the covariates are MLP and MLS. The censor limit was 1.

MLP -0.242 0.079 -3.077 0.002
MLS 0.269 0.094 2.857 0.004

MLP 1.622 0.820 1.980 0.048
MLS 18.812 9.876 1.905 0.057

ENVY#1 -136.791 70.212 -1.948 0.051
ENVY 1.448 0.563 2.572 0.010

I assume the coefficients and intercept for ENVY are equivalent to "normal" OLS regression coefficients for people above the censor limit.
I am puzzled about how to interpret the coefficients for ENVY#1, which I assume is the logistical regression distinguishing those who are not about the censor limit from those who are. The metric is clearly different from the metric for ENVY.

Any and all advice will be appreciated.
 Bengt O. Muthen posted on Saturday, September 02, 2017 - 4:39 pm
The ENVY model is a censored-normal (Tobit) model, whereas the ENVY#1 model is a logistic model for the probability of being in the class of individuals for whom only y=0 is observed.

The ENVY model gives a non-linear conditional expectation function for individuals above the censoring point, so it isn't just OLS.

All this is explained and exemplified in Chapter 7 of our book Regression And Mediation Analysis Using Mplus:
 John B. Nezlek posted on Saturday, September 02, 2017 - 9:41 pm
Thank you Bengt for the quick reply. At the risk of overstaying my welcome, I ask the following.

In terms of the ENVY model, as I understand it, for every 1 point increase in MLP, envy decreases .242, for every 1 point increase in MLS envy increases .269, and the intercept is 1.448.

For the ENVY#, I assume that for every 1 point increase in MLP, envy# increases 1.622, for every 1 point increase in MLS envy# increases 18.812, and the intercept is -136.791.

But, what exactly do the numbers for ENVY# represent? It would appear that MLS has a stronger influence on ENVY# than MLP, but if I were to write these results up, what would I say in terms of the substantive conclusions? What does it mean that the intercept for ENVY# is -136?

I have searched high and low, but I cannot find a clear answer.

Also, you mentioned the class of y=0. There are no values of 0. The censor limit was 1. Does this mean 1 or below given that I specified (bi)?

My apologies if I am being dense regarding this.
 shonnslc posted on Sunday, May 19, 2019 - 5:16 pm

When dealing with censored variables, does the conventional univariate outlier rule still apply (>3.29 standard deviation above/below the mean)? Is it appropriate to convert a censored outlier to a less extreme value before running censored normal regression? Thanks.
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