(ALT) Models: How To? PreviousNext
Mplus Discussion > Growth Modeling of Longitudinal Data >
 Karin Landolt posted on Monday, January 15, 2007 - 9:26 am

Following the writings of Patrick J. Curran and Kenneth A. Bollen, I tried to fit an ALT (autoregressive latent trajectory) model with data on smoking and depression. There are examples for markov models and growth models in the Mplus user's guide, but I don't understand how to combine the two models using the mplus-syntax...
Here's what I tried, but the model does not converge and I suppose the cross lagged regressions are not the same as a markov model:

i1 s1 | de1@0 de2@1 de3@2 de4@3 de5@4 ;
i2 s2 | sm1@0 sm2@1 sm3@2 sm4@3 sm5@4 ;
de2 ON de1 (1);
de3 ON de2 (1);
de4 ON de3 (1);
de5 ON de4 (1);
sm2 ON sm1 (2);
sm3 ON sm2 (2);
sm4 ON sm3 (2);
sm5 ON sm4 (2);
sm2 ON de1 (3);
sm3 ON de2 (3);
sm4 ON de3 (3);
sm5 ON de4 (3);
de2 ON sm1 (4);
de3 ON sm2 (4);
de4 ON sm3 (4);
de5 ON sm4 (4);

Many thanks in advance for any advice!

 Bengt O. Muthen posted on Monday, January 15, 2007 - 10:19 am
The Markov models shown in the Mplus UG concern models with categorical latent variables and status changes across time. The Bollen-Curran book doesn't consider those models. It is very easy in Mplus to do a model such as their Figure 7.8, which I assume you are referring to. Note that in the growth part (using the | statement), their figure indicates that you should not include de1 or sm1.

If you continue to have convergence problems with these models, please send input, output, data and license number to support@statmodel.com.
 Karin Landolt posted on Tuesday, January 16, 2007 - 9:30 am
Dear Professor Muthen

Thank you for your answer. The model without the first time points in the growth part did not converge using dichotomous variables, but with other - continous - variables it did so.
Maybe there is a problem with the dichotomous variables; if I can not find out the reason for the nonconvergence, I will send data and input/output files.
 Linda K. Muthen posted on Tuesday, January 16, 2007 - 10:14 am
Please send your input, data, output, and license number to support@statmodel.com if you cannot resolve this on your own.
 Alexandre Morin posted on Wednesday, May 07, 2008 - 11:59 am
Following this discussion, is there an available example of input for the Bollen and Curran book (figures 7.8 and 7.10). From the previous example, this is what I can build for figure 7.8 (keeping the autoregressive equality constraints), is that it ?

i1 s1 | de2@1 de3@2 de4@3 de5@4 ;
i2 s2 | sm2@1 sm3@2 sm4@3 sm5@4 ;
de1 WITH sm1 i1 s1 i2 s2;
sm1 WITH i1 s1 i2 s2;
i1 WITH s1 i2 s2;
s1 WITH i2 s2;
i2 WITH s2;
de2 ON de1 (1);
de3 ON de2 (1);
de4 ON de3 (1);
de5 ON de4 (1);
sm2 ON sm1 (2);
sm3 ON sm2 (2);
sm4 ON sm3 (2);
sm5 ON sm4 (2);
sm2 ON de1;
sm3 ON de2;
sm4 ON de3;
sm5 ON de4;
de2 ON sm1;
de3 ON sm2;
de4 ON sm3;
de5 ON sm4;
 Alexandre Morin posted on Wednesday, May 07, 2008 - 12:00 pm
Or the first 2 lines can be replaced by:
i1 BY de2@1 de3@1 de4@1 de5@1 ;
s1 BY de2@1 de3@2 de4@3 de5@4 ;
i2 BY de2@1 de3@1 de4@1 de5@1;
s2 BY sm2@1 sm3@2 sm4@3 sm5@4 ;
[de2@0 de3@0 de4@0 de5@0 de2@0 de3@0 de4@0 de5@0];
[i1 s1 i2 s2];
 Bengt O. Muthen posted on Wednesday, May 07, 2008 - 5:47 pm
I would write their Figure 7.8 as

iy sy | y2@1 y3@2 y4@3;
iw sw | w2@1 w3@2 w4@3;

y2 on y1;
w2 on w1;

y3 on y2;
y4 on y3;

w3 on w2;
w4 on w3;

- although I would feel more comfortable having the time scores 0 , 1, 2 so that iy, iw are clearly interpretable.
 Alexandre Morin posted on Thursday, May 08, 2008 - 4:32 am

Thank you very much for the reply.

1) I assume from your input that iy sy iw sw y1 and w1 will all be correlated as default (as in figure 7.8) ?

2) Doesn't that leave the cross variables lagged correlations out ? (y4 ON w3 ...)?

3) Was I right about the fact that if we want to include the autoregressive equality constraints we would add (1) to y2 on y1 / y3 on y2 / y4 on y3 and (2) to
w2 on w1 / w3 on w2 / w4 on w3 ?
 Bengt O. Muthen posted on Thursday, May 08, 2008 - 9:02 am
1) Tech1 shows what's done

2) Yes, I missed those

3) Right
 Alexandre Morin posted on Thursday, May 08, 2008 - 12:08 pm
Thanks a lot Dr Muthén (and it works and Tech 1 confirms what I said).
So now if anyone else want an example of Bollen & Curran ALT figure 7.8:
iy sy | y2@1 y3@2 y4@3;
iw sw | w2@1 w3@2 w4@3;
! 0-1-2 might be used instead
y2 on y1 (1);
! the equality constraints can be taken out
y3 on y2 (1);
y4 on y3 (1);
w2 on w1 (2);
w3 on w2 (2);
w4 on w3 (2);
y2 on w1 ;
y3 on w2 ;
y4 on w3 ;
w2 on y1 ;
w3 on y2 ;
w4 on y3 ;
 Alexandre Morin posted on Thursday, May 08, 2008 - 4:01 pm
After further testing, Mplus defaults do not correlate y1 WITH w1. The other exogenous variables are correctly correlated (iy sy iw sw are intercorrelated and corrrelated with y1 and w1, but those last two are not and their means and variances not estimated).

This should thus be added to the model:
y1 WITH w1;

I just had an exchange of emails with Ken Bollen regarding this input and he confirms the fact that y1 should be correlated with w1.

Everything else is has it should be.
 Michael Spaeth posted on Friday, August 28, 2009 - 7:18 am
Following both Bollen and Curran ALT-articles (2001, 2004) one often has to treat y1 as predetermined or as exogenous variable, like it is specified in the postings above. However, Curran and Bollen treat y1 as endogenous in their bivariate ALT-model including cross-lag effects (2001) (I think, because of insignificant autoregressive coefficients in their final bivariate ALT-model). I've heard from others that one still has to treat y1 as predetermined (or impose non-linear constraints), if cross-lag effects are significant in the bivariate ALT, because of the infinite regression problem, which is still evident when cross-lagg effects are included (despite omitted autoregressive effects). So under which circumstances can I treat y1 as endogenous in ALT-models?
 Alexandre Morin posted on Friday, August 28, 2009 - 7:26 am
My opinion would be to treat it as endogenous only when you have no other choices (i.e. not enough measurment points...).

But I guess (and hope) that others may have different opinions and will share them with us.
 Michael Spaeth posted on Friday, August 28, 2009 - 8:05 am
I think so too. However, autoregressive paths in my bivariate ALT are very weak, but cross-lag paths are significant. I wonder whether this factor loading problem (mentioned by curran and bollen) concerning y1 is still evident in this case. I guess yes, because of the significant cross-lagg paths. However, I'm reluctant using the predetermined ALT-Model, because I cannot statistically compare it with my original LGM which treats y1 as endogenous.
 Linda K. Muthen posted on Friday, August 28, 2009 - 8:33 am
Perhaps you could email Bollen or Curran and ask them to comment on Mplus Discussion.
 Alexandre Morin posted on Friday, August 28, 2009 - 8:42 am
This is what I was going to recommend.

Regarding the second issue, Bollen and Curran propose a alternative LCM model that is nested within the ALT that can be used for this kind of comparison.
 Alexandre Morin posted on Wednesday, September 02, 2009 - 11:42 am
Hi again,
I wrote to Drs Bollen and Curran to ask them to look at this discussion and here is Dr Bollen answer:

"Patrick Curran and I briefly discussed the blog and your note. If we are interpreting your question correctly, the issue is when to treat the first wave of each series as endogenous and when to treat it as predetermined. If the autoregressive relations are zero for both series, then you can treat the first measure as endogenous or as predetermined and include cross-lagged effects across the series. If the autoregressive relations are nonzero, you are better off treating the first measure as predetermined."

Hope this helps
 Michael Spaeth posted on Thursday, September 03, 2009 - 2:33 am
Thank you Alexandre, this is in line with my results that implied no substantial differences in cross-lagged paths across endogenous vs. exogenous ALT-models. I guess because autoregressive effects were weak.
 Chyke Doubeni posted on Thursday, September 17, 2009 - 5:16 pm
I attempted to model a similar process - number of symptoms and number of cigarettes but the model returns a FATAL ERROR if the symptom variable is specified as a count variable (negative binomial hurdle), or I include the growth part of the model. Also the count variable cannot be correlated with the continuous variable.
The symptom variable is very skewed.

Any advice on analyzing this data - possible estimators I can use? Would it make sense to log (ln) transform the variable and model it as a continuous variable.
 Linda K. Muthen posted on Friday, September 18, 2009 - 3:25 pm
Please send your output and license number to support@statmodel.com.

I would not transform a count variable.
 chris sullivan posted on Thursday, January 05, 2012 - 3:05 pm
I am trying to estimate an ALT model as described above but get the following warning when I treat the first wave measure as predetermined (we have 4 total). It doesn't happen when it is not specified that way.

The latent variable covariance matrix (psi) is not positive definite. This could indicate a negative variance/residual variance for a latent variable, a correlation greater or equal to one between two latent variables, or a linear dependency among more than two latent variables. Check the tech4 output for more information. Problem involving variable intdel

(a) Is there anything specific that I should be looking for in Tech4? and (b) are there any possible fixes for this issue?

Thank you.
 Linda K. Muthen posted on Friday, January 06, 2012 - 10:12 am
In TECH4, you should look for negative variances and correlations greater than one.

Please send your full output and license number to support@statmodel.com.
 Michael Spaeth posted on Monday, November 26, 2012 - 1:16 pm
I really like the idea of the hybrid ALT model but I find it hard to fit this complex model (bivariate ALT in my case) to my data (including issues like predetermined x1 which sometimes causes estimation problems etc.). It is very often the case that I'm more successful when I model both "model parts" separately. For instance, I model parallel LGMs with significant intercept and slope correlations between the processes (in line with theory) followed by a separate autoregressive-cross-lagged models with significant reciprocal cross-lagged effects (also in line with theory). However, when I merge both model parts together (as it is done in ALT) things become very intransparent in my case (e.g., growth factor variances, which were significant in the bivariate LGM, become insignificant in the bivariate ALT, making growth factor correlations somewhat difficult to find and to interpret; cross-lagged paths between manifest indicators, which were significant in the bivariate autoregressive model become insignificant in the bivariate ALT...etc).

Assuming that I'm really interested in both methods of examining change (LGM and autoregressive-cross-lagged model). What would speak against the approach of modeling both models separately (and to compare the results of both methods) and what is the clear advantage of the ALT model over the "two-separate-models"- approach? What would you do?
 Bengt O. Muthen posted on Tuesday, November 27, 2012 - 9:17 am
I also find the ALT model "intransparent" as you say. For me, the problem is that you add the lagged-effect regression of yt+1 on yt on top of a growth model which also models y. It is more clear to me if instead the lagged regression was of et+1 on et where e is the residual - then the growth model influences on part of y (the part without the residual) and the lagged effect is for the other part of y. And that model is a standard one - using auto-regressive errors.

Modeling the two models separately is different to me.
 Michael Spaeth posted on Tuesday, November 27, 2012 - 9:55 am
Thank you! By the way, some authors discuss this "intransparancy" as an advantage ("cross-lagged relationships controlled for intraindividual change"). Anyway... I'm not sure how to express your suggestion regarding cross-lagged regressions between residuals in a mplus syntax. What would be the difference (in terms of mplus codes) between your proposed model and the ALT model (i.e., the syntax presented by Alexandre).
 Bengt O. Muthen posted on Tuesday, November 27, 2012 - 10:52 am
To do the auto-regression among residuals that I talk about, you make each residual into a factor, uncorrelated with everything else, and then regress these factors onto each other. Make sure to get rid of the usual residual so you don't have two.
 Michael Spaeth posted on Tuesday, December 04, 2012 - 6:48 am
Thank you, Bengt. Is the following model the correct set-up for a bivariate model that you had in mind recently (please see below)?
 Michael Spaeth posted on Tuesday, December 04, 2012 - 6:49 am
iy sy | y1@0 y2@1 y3@3 y4@5 y5@7;

iz sz | z1@0 z2@1 z3@3 z4@5 z5@7;

f1 by y1; y1@0;
f2 by y2; y2@0;
f3 by y3; y3@0;
f4 by y4; y4@0;
f5 by y5; y5@0;

f6 by z1; z1@0;
f7 by z2; z2@0;
f8 by z3; z3@0;
f9 by z4; z4@0;
f10 by z5; z5@0;

f2 on f1;
f3 on f2;
f4 on f3;
f5 on f4;
f7 on f6;
f8 on f7;
f9 on f8;
f10 on f9;

f7 on f1;
f8 on f2;
f9 on f3;
f10 on f4;
f2 on f6;
f3 on f7;
f4 on f8;
f5 on f9;

f1 with f6;
f2 with f7;
f3 with f8;
f4 with f9;
f5 with f10;

f1 with iy@0;
f1 with sy@0;
f1 with iz@0;
f1 with sz@0;
f6 with iy@0;
f6 with sy@0;
f6 with iz@0;
f6 with sz@0;
 Michael Spaeth posted on Tuesday, December 04, 2012 - 6:52 am
p.s.: I'm not sure regarding the correlations between contemporaneous residuals, beacuse you said that residual factors should be uncorrelated with everything else.
 Bengt O. Muthen posted on Tuesday, December 04, 2012 - 2:03 pm
You need to have f1-f10 uncorrelated with all 4 growth factors.

Check PSI in TECH1 to see that you don't some other unwanted correlations among factors.
 Michael Spaeth posted on Wednesday, December 05, 2012 - 1:25 am
It works, thank you!
 Kerry Lee posted on Friday, January 25, 2013 - 5:59 pm
Dear Drs Muthen,
I tried to specify an ALT model in two ways:
I S| norw2@0 norw3@1 norw4@2;
norw4 ON norw3* (1);
norw3 ON norw2* (1);
norw2 ON norw1* ;
norw1 WITH I;
norw1 WITH S;

F1 BY norw1@1;
F2 BY norw2@1;
F3 BY norw3@1;
F4 BY norw4@1;

I S| F2@1 F3@2 F4@3;


[ F2-F4@0 ];


F4 ON F3* (1);
F3 ON F2* (1);
F2 ON F1* ;

It seems that with the first method, the regression beta are placed between the residuals of the manifest variables. The second method seems closer to the model specified by Curran and Bollen (2004). I also tried the second method and replaced Ft ON Ft-1 with their indicator equivalents. The result is the same as that of method 1 (and differs from method 2). Could you please verify whether my interpretation is correct? That is, method 1 places the autoregression between the residuals?

 Linda K. Muthen posted on Monday, January 28, 2013 - 10:42 am
The following statements refer to the entire variable not the residuals:

norw4 ON norw3* (1);
norw3 ON norw2* (1);
norw2 ON norw1* ;
 Kerry Lee posted on Monday, January 28, 2013 - 7:37 pm
Daer Dr Muthen,

Thanks very much for your reply. Just trying to get a better understanding of how Mplus work. I thought by using Method 2 above, I could model regression placed on the entire variable versus on the resiuduals separately. The former by

F4 ON F3* ;
F3 ON F2* ;
F2 ON F1* ;

and the residuals by

norw4 ON norw3*;
norw3 ON norw2*;
norw2 ON norw1*

What puzzles me is that using norw(t) ON norw(t-1)* in Model 2 produces the same estimate as using Method 1 above. How should the model be specified if I want to place the regression on the residuals only?

 Linda K. Muthen posted on Tuesday, January 29, 2013 - 11:40 am
No, this is not how it works. There is a FAQ on the website that shows how to work with residuals in regressions.
 Alissa Mahler posted on Thursday, May 18, 2017 - 10:59 am

I'm trying to model an ALT model with structured residuals as detailed by Berry and Willoughby (2016). The authors provide syntax in their supplemental materials that I've been using, however I'm trying to incorporate a dichotomous variable (as a time-varying element). Part of the model requires you to constrain the observed residual variances to zero, in order to identify the structured residuals. However, I'm finding when I try to do this with my binary variable, the model will not converge.

I'm wondering if this is a syntax problem, or simply something you cannot do with binary variables?

Thank you!
 Bengt O. Muthen posted on Thursday, May 18, 2017 - 4:24 pm
I assume your binary variables are repeated outcomes. Binary variables don't have free variance/residual variance parameters so that could be the source of your problem.
 Alissa Mahler posted on Wednesday, June 07, 2017 - 1:55 pm
Hello again,

Thank you for your assistance with this model! I have another question relating to the ALT-SR model.

I'm trying to run a 2-group multiple group analysis for an ALT-SR model (which includes two continuous outcomes that are repeated over 7 waves). I'm able to estimate the latent growth curve models across the two groups without any errors, but I'm finding once I begin to add in the structured residuals the model no longer converges and the standard errors cannot be computed. Again, I'm wondering if there is something that prohibits you from conducting a multiple group analysis when running a ALT-SR model.
 Bengt O. Muthen posted on Wednesday, June 07, 2017 - 6:06 pm
This should work fine. Send output to Support along with your license number.
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