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Mplus Discussion > Latent Variable Mixture Modeling >
 Michael Schneider posted on Saturday, July 29, 2006 - 6:53 am
I am conducting LTA analyses with three continuous variables, each measured at three measurement points.
Q1: Is there any literature about "LPTA" at all? Can I use this name in the publication of the results?
Q2: Is a multivariate normal distribution of the nine class indicators necessary/important in LPTA? If so, for what parts of the results? Is there literature about this?
Q3: Are the AIC, the BIC, and the likelihood-ratio chi-square difference tests tests valid for determining the number of classes in LPTA? Who can I cite on this in my article?
 Linda K. Muthen posted on Sunday, July 30, 2006 - 6:15 pm
Q1: I don't know of any. Perhaps a literature search would yield something.
Q2: No. The model assumes normality within classes not for the mixture of the classes. I know of no literature for this.
Q3: Yes. No citation that I know offhand.
 Scott Weaver posted on Friday, August 04, 2006 - 5:32 pm
I am trying to figure out the model that best matches and can test a set of hypotheses, and I am hoping that you or discussion board readers might have some ideas - including how to implement them in MPlus.
I have a identical sets of variables completed by both children and mothers (e.g., mother's religiosity and child's religiosity). The hypotheses involve testing whether consnance/dissonance btw these sets of variables (latent profiles) predicts child development.
Separately for children and mothers, I have conducted a LPA and found support for 3 and 4 classes, respectively. Therefore, there is a maximum of 12 possible mother-child combinations - but not all of which will probably be observed with any substantial frequency. I am looking for ideas on how to model jointly child and mother profiles such that I have a single latent class factor that can be regressed on predictors.
I can think of two approaches: (1) to specify a single latent class factor with all child and mother variables as indicators, or (2) model two '1st-order' latent class variables (one for mothers, and one for children) and also a '2nd-order' latent class variable with the two 1st-order latent class variables as indicators.
Do you have any suggestions on which approach or other approaches I should explore and, if the 'higher-order' model, guidance on the Mplus specification of the model?
Thank you!
 Bengt O. Muthen posted on Saturday, August 05, 2006 - 5:29 pm
I would probably not vote for gettting into the second-order modeling. I would choose between having 2 latent class variables, that your correlate using c1# WITH c2#, versus having a single latent class variable. The second alternative would be good if you believe that meaningful classes are define by item profiles involving both child and mother
- for example, if a child has a high profile and a mother a low profile.
 Scott Weaver posted on Sunday, August 06, 2006 - 11:22 pm
Thank you Bengt for your advice. I thought that the second-order model would work in a similar way to that of the Mover-Stayer model (8.14 in the User's guide) where the 2nd order latent class factor captures the movers and stayers (in my model, which would not have a direct path connecting 1st order mother/child latent class factors, the 2nd order latent class factor would capture the mom profile and child profile combinations.

I tried the model with 2 latent class variables, but could not get the model to terminate normally (multiple warnings - including that of an ill-conditioned fisher information matrix. I notice that results are provided for every mother by child class cell.
(1) What if not all cells have a non-zero frequency. How do I represent that in syntax.
(2) Would you recommend the Loglinear or the logistic parameterization?
(3) Since I ultimately want to see if a particular combination of mother and child profiles predicts different outcomes, how can I represent that in syntax (e.g., mother/child dyads where moms are in M-class 1 and children are in C-class 2 have higher means on outcome Y than do mother/child dyads in M-class 2 and C-class 3?
Thank you!
 Bengt O. Muthen posted on Monday, August 07, 2006 - 8:07 am
(1) zero cells are handled automatically by Mplus fixing logits. (2) For the model with 2 latent class variables, you want to correlate them by using parameterization=loglinear and say c1# with c2#. (3) The outcome means/probabilities vary across the combined classes.

I think the single latent class variable approach might be most likely to capture what you look for because it operates directly on the outcomes for mother and child and therefore looks for profile combinations across mother-child. Second best approach is the 2 latent class variables. The second-order modeling only structures the 1st-order latent class variables and therefore can't "see" the observed profile combinations. But I may be wrong.
 Bruce A. Cooper posted on Monday, September 25, 2006 - 5:48 pm
I am using Mplus 4.1 to find latent classes at three points in time, and then LTA for T1 to T2, and T2 to T3.

I have the same four symptoms (continuous vars) at each time. I have found four latent classes at time 1, and three latent classes at times 2 and 3. Unfortunately, I have a small sample (101, with some missing data at T2 & T3), so I am getting warnings and failures when I try to examine the LTA from T1 to T2, and then from T2 to T3. (I know there is no hope of combining all three times into a single model with so few cases.)

I have reduced the number of parameters to be estimated by holding the correlation matrices equal at each pair of times, and I have specified realistic starting values for the two LTA from the prior LCA analyses. I have found it necessary to specify a large number of random starts in order to get solutions for the LCA at each time, I assume because of the small sample, and also because there is at least one small latent class at each time (e.g., n=5 for the smallest class at time 1, n=16 for the smallest class at T2 and T3).

Finally, the question: I'd like to be able to set some cells to zero for the T1 to T2 LTA, likewise for the T2 to T3 LTA, but I can't figure out how to do it. I'm hoping that would help eliminate the warnings. What MODEL statement can I use to set latent class cell combination to zero?

 Bengt O. Muthen posted on Sunday, October 01, 2006 - 12:26 pm
The warnings may not be fatal, but merely refer to fixing extreme values for parameters corresponding to zero or unit probabilities. Ask for Tech1 and see if the fixed parameters are those with large values - then you are fine. Fixing transition probabilities to zero requires Model Constraint because a zero probability is obtained as a function of several logit parameters - see UG Chapter 13.
 Bruce A. Cooper posted on Thursday, October 05, 2006 - 5:27 pm
Thanks, Dr. Muthén -

The part about extreme values is true; I believe the warnings cascade from it. I have specified 3 LC at each of 2 times, based on fatigue, sleep disorder, depression, & worst pain scores.

Some LT classes do not exist or are unlikely from T1 - T2. For example, a small class at T1 reports very high levels on all symptoms. After undergoing chemotherapy, this group would not be likely to report no or very low symptoms at T2, a structural zero. Another group of patients, who all reported very high sleep disturbance and moderate fatigue at T1, but only some pain and no depression, might not be expected to report no or very low symptoms T2, a sampling zero.

I have looked at the sections you suggested in Ch.13, as well as looking over the examples in Ch.8 (esp. 8.13), but I have not been able to figure out how to specify fixed zeros for likely structural or sampling zeros, to see if the model would run without the dire warnings.

I have included a portion of the output for the cell sizes for the LC and LT patterns below, if you care to look at it.


Latent Class
1 15
2 25
3 60
1 12
2 59
3 33

11 1 0.01
12 10 0.10
13 0 0.00
21 3 0.03
22 24 0.24
23 0 0.00
31 7 0.07
32 26 0.26
33 30 0.30
 Bengt O. Muthen posted on Friday, October 06, 2006 - 7:05 pm
If the warnings stem from extreme values, then that is ignorable. These extreme values give you the zeros you observe for some latent class patterns. In other words, you don't have to bother with imposing structural zeroes.

Structural zeros can be imposed by looking at Chapter 13's loglinear parameterization. You want a zero cell and get it by fixing a "b" coefficient to a low value such as -15.
 Sara posted on Friday, October 27, 2006 - 2:19 pm
I have questions concerning the merger of Latent Profile Analysis (LPA) and Latent Transition Analysis (LTA).

I have a dataset with 6 continuous variables measured at 2 timepoints. I conducted LPA’s at both timepoints and 3 classes/subpopulations were supported at each timepoint.

Now what I want to do is estimate the probability of moving from each of the classes at time 1 to each of the classes at time 2 (c1 to c1, c1 to c2, c1 to c3, c2 to c1, and so on).

1)I thought this would be a merger of LPA and LTA. Does that sound correct? If so, can MPLUS handle this (what manual example is it)? Do you know of any published examples? If the merger of LPA and LTA is not the technique I should be looking for, what is?

2)Almost all examples of LTA I find use categorical variables. Two papers (Dolan, Jansen, van der Mass, 2004; Schittmann, Dolan, van der Mass & Neal, 2005) used continuous variables to model transition. They call this the Mixed Markov latent class (MMLC) model for continuous data. They explain that this model “can be viewed as finite mixture models, where each unique sequence of latent classes represent a distinct mixture component” (p. 462). Therefore, I wasn’t certain if this was equivalent to the combination of LPA and LTA. Is it? Also, does the MMLC model for continuous data correspond to any examples in the MPLUS manual?
 Bengt O. Muthen posted on Saturday, October 28, 2006 - 9:09 am
1) It is indeed a merger of LPA and LTA and it can be done in Mplus. Start with UGex8.13 and instead of having thresholds change over the classes you have the means of the items changing over classes. Mplus can use any type of observed item scale in combination with any other (dichotomous and continuous combined for example).

2) It sounds like MMLC is exactly what I describe in 1).
 Sara posted on Monday, October 30, 2006 - 5:57 am
Thanks for the response about LTA and LPA.
I had thought 8.13 best matched our research question.

Just to be clear, will running a model like 8.13 but with continuous variables produce (1) classes at Time1, (2) classes at Time2, and (3) classes for the various types of change (sequence of latent class membership across time which are transition probabilities)?

*Or* if I found 3 classes at each time point using LPA should I use those results to set the time specific class parameters, and just freely estimate the transition probabilities when running the LTA?

Also, the difference between 8.13 and both 8.12 and 8.14 seem slight and I feel like I may be missing something.

Is the difference between Hidden Markov Model (8.12) and LTA (8.13) simply the fact that LTA can have multiple indicators creating the classes? It seems that LTA is a type of Markov model.

Also the only difference between 8.13 and 8.14 is the addition of the "c" categorical latent variable. Does this variable use the transition probabilities to form overarching groups of "movers" and "stayers" (levels of C)?
 Bengt O. Muthen posted on Monday, October 30, 2006 - 7:03 am
Yes on your 1st question.

No on your 2nd question.

Yes, ex8.13 simply adds multiple indicators. LTA is a form of Hidden Markov - that is, Markov for latent class variables.

Yes on your last question.
 Girish Mallapragada posted on Wednesday, May 14, 2008 - 8:28 am
Dr. Muthen,

I was trying to understand some of the applications in LTA and noticed that almost always the number of latent states is equal to the number of time points in the data. Moreover, the transition probabilities are modeled as a function of intercepts and means as you state above.
My questions are:
1) Does mplus allow a model in which number of hidden states is different from number of time points?
2) the transition probabilities are a function of time-varying covariates ?
 Linda K. Muthen posted on Wednesday, May 14, 2008 - 9:15 am
Yes and yes.
 Julia Lee posted on Friday, September 30, 2011 - 8:25 am
I am analyzing my data using LTA (mover-stayer model). The data set is comprised of continuous variables. In order to make sense of the interpretation of classes, I am guessing that requesting for confidence interval is a good way to deal with the issue at the moment.
1) Is there a way I could request for confidence interval for LTA?
2) On page 647 UG Version 6, CINTERVAL is available for 3 settings. Which settings do I use if CI can be done for LTA?
3) What is the main difference between BOOTSTRAP, and BCBOOTSTRAP? Do I choose either one if I think there is nonnormality? I would appreciate some input on papers to read.

Thank you very much!
 Linda K. Muthen posted on Saturday, October 01, 2011 - 7:36 am
I don't think you need confidence intervals to interpret the classes. You can look at the item profiles for each class. If you want confidence intervals, you can use the CINTERVAL option. You can use BOOTSTRAP or BCBOOTSTRAP if you think the parameter has a non-normal distribution. I don't think you need to worry about this non-normality to interpret classes. See the Efron and MacKinoon 2004 references in the user's guide.
 Joan Reid posted on Friday, October 05, 2012 - 2:49 am
I have a question regarding the best model to use with multi-informant data. I have information from youth and parent regarding youth behavior and would like to examine whether the latent class variables resulting from the 2 sources are similar or dissimilar, and if youth are placed in similar or dissimilar classes based on informant. I have found a 4-class solution for both sets of data.

Next, I followed example 7.14 in the user's guide to correlate two latent variables (resulting in 16 possible class combinations from the latent variables with 4 classes).

Based on your responses above (to Scott Weaver on August 5/6, 2006), I think this is the best way to investigate and model the multi-informant data. Is this an appropriate approach?
 Bengt O. Muthen posted on Friday, October 05, 2012 - 8:43 am
Yes, that sounds fine.
 Karla Ausderau posted on Wednesday, October 24, 2012 - 12:47 pm
We are conducting a LPTA with four continuous variables at two time points using covariates such Proxy IQ and age in the model as well. We have decided upon a 4 profile solution.

What is the best way to examine the relationship of the covariates in the model (e.g. proxy IQ and age) to the 4 different profiles?
 Linda K. Muthen posted on Wednesday, October 24, 2012 - 1:43 pm
The multinomial logistric regression of the two categorical latent variables on the set of covariates gives the information that you want.
 Johannes Bauer posted on Friday, June 14, 2013 - 7:23 am
I am wondering if it is possible to do a LPTA with continuous latent variables as indicators. That is, a second order categorical latent variable with continuous first order factors as indicators.

If yes, should strong measurement invariance restrictions be imposed on the continuous LV measurement models? (because the categorical LV compares the latent means of subpopulations)

If no, one could estimate factor scores for the continuous LVs and use these as LPTA indicators in subsequent analyses. Should these factor scores be estimated from a model with strong measurement invariance restrictions?
 Bengt O. Muthen posted on Friday, June 14, 2013 - 11:49 am
Yes on your 1st q.

If you want to make sure that you are considering the same continuous latent variable at the different time points, the usual scalar invariance restrictions apply.

But LTA does not necessitate that the same construct is considered at different time points.
 Tait Medina posted on Monday, February 10, 2014 - 12:50 pm
A type of LTA that is discussed in the Nyland dissertation is one in which the item probabilities are constrained to be equal across time and the transition probabilities are freely estimated.

With continuous items, is a comparable model one in which the item means are constrained to be equal over time, or would the item variances need to be held invariant as well?
 Bengt O. Muthen posted on Monday, February 10, 2014 - 3:03 pm
I would say constraining the means only and not the variances since by default the latter are not varying as a function of latent class.
 Yueqi Yan posted on Thursday, July 10, 2014 - 3:43 pm
I am conducting a 3-step LPTA with 18 continuous variables at two time points. It ran well, but Mplus used the classes at Time 1 as predictors and selected the last class as reference group by default to predict how these classes change over time. My question is if there is any way that I can select the reference group by myself or I have to extract the classifications over time, choose the reference group at Time 1 by myself and then run a separate multinomial LR? Thanks!
 Bengt O. Muthen posted on Thursday, July 10, 2014 - 6:48 pm
Use SVALUES to change the starting values for the class-specific means to get the class ordering you want, using STARTS = 0.
 J.D. Haltigan posted on Tuesday, July 29, 2014 - 11:06 am

After reviewing the Kaplan supp materials regarding fixing transition probs to zero, I am still having a bit of trouble.

I am estimating pubertal initiation (so once initiated, can't revert). In the overall part of the model I have specified:

C2 on C1;
C3 on C2;
C4 on C3; !etc.

and then

c2#1 on c1#2@-15;
(i.e., can't revert to a non-puberty state from puberty; there are only two categories per class no puberty or puberty)

but I keep getting the error/message that: 'no reference to the slopes of the last class is allowed.'

Is it possible with only two categories per class to specify the non-reversion constraint?
 Linda K. Muthen posted on Tuesday, July 29, 2014 - 11:17 am
Please send the output and your license number to
 Michelle Lalonde posted on Sunday, October 18, 2015 - 6:28 pm
Hello Dr. Muthen
I am comparing a freely estimated LPTA model against models with parameter restrictions.
I tested measurement invariance by establishing equivalence restrictions relating to item-response probabilities, but am unsure how to test hypotheses regarding transitions.
I want to test whether there is change between time 1 and time 2 within my 3 profile model. I hypothesise that each profile remains stable.
I understand I must constrain the transitions I expect to occur @1, and the transitions that I expect to not occur @0. But I am not certain how to write this syntax. I had considered placing it under the model command, (ex: c2#1 ON c1#1@1), however, this poses a problem as no reference to the slope or intercept of the last class is allowed.

Could you please advise?

(syntax below)

c2#1 ON c1#1;
c2#1 ON c1#2;
c2#2 ON c1#1;
c2#2 ON c1#2;

[ope1 ton1 sod1 flo1] (1-4);
[ope1 ton1 sod1 flo1] (5-8);
[ope1 ton1 sod1 flo1] (9-12);

[ope2 ton2 sod2 flo2] (1-4);
[ope2 ton2 sod2 flo2] (5-8);
[ope2 ton2 sod2 flo2] (9-12);
 Bengt O. Muthen posted on Sunday, October 18, 2015 - 7:45 pm
Please see the LTA section of the V7Part2 short course handout from our web site at:

Mplus Version 7 workshop and Dutch Mplus Users Group, Utrecht, August 2012

Videos and handouts from 3-day Version 7 workshop.
 Stephanie Moore posted on Wednesday, April 12, 2017 - 4:12 pm
I’m running a Latent Transition Analysis with 6 continuous indicators and 4 time points. Four classes are specified at each time point. I am assuming full measurement invariance and using the 3-step method to examine covariates and other modeling extensions.

In assuming invariance, I have constrained the means to be equal which, by default, results in variances being constrained to be equal across class. However, the default in Mplus does not constrain the variances to be equal across class and across time. Therefore, the variances given in the saved start values differ across the four time points.

When specifying the LTA assuming full measurement invariance, should I include statements constraining variances to be equal across class and across time? Then, I would use the invariant means and intercepts across time when saving cprobs for the three-step method.
 Bengt O. Muthen posted on Thursday, April 13, 2017 - 4:32 pm
It is not necessary to hold variances equal across time. This is akin to not forcing residual variance invariance in factor analysis.
 davide morselli posted on Friday, August 25, 2017 - 12:31 am
Hi I am running a LPTA on two time points. As a preliminary step I ran LPA separately at T1 and T2, and then used the best fitting models to build the transition model.
Weirdly I obtain very different class sizes at T2 between the LPA and LPTA model (respectively, 201-100-8 and 158-130-27) while the the structure of the classes between T1-LPA and LPTA is fairly consistent. I was wondering why this happen and how can I fix or interpret it.

Thank you
 Bengt O. Muthen posted on Friday, August 25, 2017 - 4:35 pm
Sounds like a good position to be in. In the measurement part of your model you observe measurement invariance and in the structural part of your model you find interesting changes over time.
 Dayuma Vargas posted on Tuesday, May 08, 2018 - 3:57 pm
I have run into a similar experience as described by davide in his 25-08-2017 post while testing for invariance across time:

I ran LPAs separately at T1 and T2 (same 3 indicators measured at both times) and identified the best fitting models for each time point, which was a 4-class model at each time.

From examining the two LPAs I noticed that, although the classes look very different in terms of one of the indicators, they are similar in terms of the other two.

I decided to model the two LPAs together (C1 WITH C2) with full non-invariance as well as with partial invariance to do difference testing using loglikelihood. However, when I model the two LPAs together, the class proportions and class characteristics for T2 look very different than what I found by running the T2 LPA by itself.

(1) For interpretation of the T2 LPA, which T2 results should be used, the ones from the T2 LPA or from the joint LPAs model?

(2) Would class separation and homogeneity be re-tested for T2 using the results of the joint LPAs model given the profile changes?

(3) Webnote 15 notes that the SVALUES of the joint LPA modelling should be used to create the C1 and C2 most likely class variables for assessing the transition probabilities using the 3-step procedure. HWould not this be problematic given that the C2 most likely class variables in this case were partially based on C1?

Thank you.
 Bengt O. Muthen posted on Tuesday, May 08, 2018 - 5:39 pm
The fact that the T2 model changes a lot from running it alone vs together with T1 indicates model misfit. The misfit is likely due to correlations between the set of T1 indicators and the set of T2 indicators not being well fitted by the model. You could for instance use WITH to try to correlate indicator 1 at T1 with indicator 1 and T2, etc.
 Dayuma Vargas posted on Tuesday, May 08, 2018 - 7:13 pm
Thank you for the quick response. Quick clarification: the model misfit would be at the LPTA level, not the individual LPA level for T2, correct?

I tried introducing indicator covariances across time in the overall section of the non-invariant joint LPAs model:

y1_t1 WITH y1_t2;
y2_t1 WITH y2_t2;
y3_t1 WITH y3_t2;

When did so, class proportions for the T2 LPA still changed (compared to running that LPA by itself).

(1) Is there a way to identify potential aspects of the joint model that are causing the misfit?

(2) On comparing the LTA invariance syntax discussed in this discussion board and the syntax in webnote 15 for the joint LCAs model with invariance as part of the LTA 3-step process, I noticed a difference. In the former a C1 WITH C2 statement is included in the overall section of the model while in the latter the two latent class variables are completely independent of each other (no C1 WITH C2 in the overall section of the model or any other connection between the two LCAs, except for the model constraints). Running the joint LPAs model with and without the C1 WITH C2 statement produces some differences in class proportions and class level indicator means. Should the C1 WITH C2 statement be present in both models?

Thank you once again for your insight.
 Bengt O. Muthen posted on Thursday, May 10, 2018 - 3:02 pm
Q1: Both parts can be misfitting but yes I was referring to the LTA part.

1) Try Residual in the Output command.

2) See the article on our website:

Nylund-Gibson, K., Grimm, R., Quirk, M., & Furlong, M. (2014): A latent transition mixture model using the three-step specification. Structural Equation Modeling: A Multidisciplinary Journal, 21, 439-454.
 Daniel Lee posted on Friday, May 18, 2018 - 12:05 pm
Hi I ran a LPTA and I am not sure how to tell which profiles in the LPTA (i.e., 2 time points, 3 profiles at each time point) represent the profiles from the LPAs at each time pt. For example, if the profiles high, medium and low were identified in time points 1 and 2, I am not sure which profiles represent the low medium and high profiles in the LPTA. I would greatly appreciate your help. Thank you.
 Bengt O. Muthen posted on Friday, May 18, 2018 - 1:43 pm
You look at the estimated means in each class for each time point.
 Daniel Lee posted on Monday, May 21, 2018 - 5:10 am
Hi Dr. Muthen,
Thank you for your response. It seems like, as a precursor to conducting an LTA, measurement invariance needs to be established (i.e., by setting up equality constraints for indicator means & variances between the two LPAs...and then an LRT between models with and without the constraints).

My question is, should we assess measurement invariance with model covariates and outcomes included, or should we assess measurement invariance and then include covariates and outcomes?

Thank you, again!
 Bengt O. Muthen posted on Monday, May 21, 2018 - 5:34 pm
There are different opinions about this. See e.g. the papers

Nylund-Gibson, K. & Masyn, K. (2016). Covariates and mixture modeling: Results of a simulation study exploring the impact of misspecified effects on class enumeration. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2016.1221313

Nylund-Gibson, K., Grimm, R., Quirk, M., & Furlong, M. (2014): A latent transition mixture model using the three-step specification. Structural Equation Modeling: A Multidisciplinary Journal, 21, 439-454.
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