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Mplus Discussion > Growth Modeling of Longitudinal Data >
 Harry Garst posted on Thursday, October 03, 2002 - 8:02 am
Dear Bengt and Linda,

I have a question about multivariate (also called cross-domain) growth curve modeling.
Do I have to specify regression paths between the intercept factor and the slope factor of the SAME repeatedly measured variable or can I specify correlated residuals? In Muthen (page 472, 1997) and Curran, Stice & Chassin (page 135, 1997) there are cross-lagged paths from the intercept factors to the other slope factors, but only correlations between the residuals of the intercept factor and the slope factor of the same variable. In many datasets, the choice between both strategies would have a huge impact on the regression effects of the cross-lagged paths. This is because the starting points of the trajectories usually have strong affects on the subsequent changes (reflected by the slope factor). Partialling out the intercept factor changes the interpretation of the slope factor: it is the linear change if all persons would have started from the same position (assuming a linear relationship between intercept and slope factors). Actually I am not worried about controlling for ‘regression to the mean effects’ because both intercept and slope factors have been estimated from ALL data points. But there are other arguments which might be more valid: there may be causal effects: e.g. depressed people may be rejected by others which will make them even more depressed, rich people may have more opportunities to make money and will be more wealthier etc. Moreover, there may be floor and ceiling effects. If you are up you can only go down and vice versa. Or there may be boundary conditions which would have a compensating influence. My question is: do we have to partial out pre-existing differences in starting points?
Thanks in advance.

Harry Garst

Muthén, B. (1997). Latent variable modeling with longitudinal and multilevel data. In A. Raftery (ed), Sociological Methodology (pp. 453-480). Boston: Blackwell Publishers.

Curran, P.J., Stice, E. & Chassin, L. (1997). The relation between adolescent and peer alcohol use: A longitudinal random coefficients model. Journal of Consulting and Clinical Psychology, 65, 130-140.
 bmuthen posted on Thursday, October 03, 2002 - 9:19 am
You can correlate the intercept and growth factors also with cross-domain modeling. You do not need to regress the slope on the intercept.
 Harry Garst posted on Friday, October 04, 2002 - 12:21 am
Yes, I know we can. My question is: what are the pros and cons of partialling out pre-existing differences in starting points (regress the slope on the intercept both referring to the same repeatedly measured variable. You did not do so in your article. However, in many datasets this choice will dramatically change the substantive interpretation of the results. Is this problem a special case of Lord's paradox?

Harry Garst
 bmuthen posted on Friday, October 04, 2002 - 9:45 am
This is a tough question and is in my view driven by the substantive context and cannot be settled statistically. If from a subject-matter point of view, theory makes it plausible to see growth as causally dependent on initial status, then one would want to regress the slope on the intercept. And, therefore covariate effects on the slope will be partial effects controlling for initial status (so may go from significant to insignificant). In addition to your examples, my favorite example is from achievement measurement, where those who start high may be in a better position to understand instruction and therefore grow faster. A covariate such as home resources may only influence the slope indirectly through improving the initial status.

In conventional growth modeling, regressing slope on intercept has not been done typically. But this may be more a function of software than sound reasoning. SEM has made the choice visible. From a multilevel angle, pages 361-364 of the new Raudenbush-Bryk edition, the slope-regressed-on-intercept topic is discussed for the single growth process situation.

This is an interesting question and I invite other Mplus Discussion readers to contribute.
 Anonymous posted on Tuesday, November 16, 2004 - 6:57 am
I would like to continue the previous discussion a little bit because I have similar kind of questions as Harry had. I have modelled the change of children´s problem behaviors using LGM. There is a floor effect in the assessment tool (= children´s self-rated problem behavior), i.e. vast majority of children do not show any (or show only few)symptoms of problem behavior. Moreover, across time there is a decreasing trend in problem behaviors (due to the social desirability, I suppose). That means that those who report problem behaviors, show decrease in their symptoms across time, in average, (and those who do not show any symptoms remain where they are, in average).

Now I am interested in to predict the changes in problem behavior. If I estimate the path from the initial level to the trend (strong negative path)and then add some covariates, there are statistically significant and very interpretable effects of covariates. However, if I just let the level and trend of problem behavior to correlate with each others (i.e. not control for the impact of initial status), no effects of covariates are evident. I first thought that this phenomena is very understandable due to the floor effect of the measurement tool and I really thought that I understand what is happening here. Also, it seemed to me very reasonable to control for the impact of the initial status due to reasons mentioned above. Now, however, I am not anymore so sure about this phenomena and how to interpret the results. Could you please help me with this - is it reasonable to control for the initial status of the construct in this kind of case? Thank you for any help.
 bmuthen posted on Tuesday, November 16, 2004 - 3:57 pm
I am positively inclined to regressing the slope also on initial status to see how much of the covariate effect influences the slope indirectly via the intercept and how much influences the slope directly. It seems reasonable in your application. But I would also add that for your data you may want to consider "2-part" growth modeling as an alternative. 2-part modeling makes a more serious attempt at modeling the zeros - i.e. the many children with no problem behavior. This gives you a growth model for having problem behavior or not and - among those who exhibit problems -a growth model for how much problem behavior there is. Covariates are allowed to have different effects for these 2 different growth models, which may be called for from a substantive point of view. Ignoring this differential covariate effect may be confounding the picture that you are seeing. The 2-part analysis described in the Olsen-Schafer article on our web site and can be carried out in Mplus Version 3 as shown in the User's Guide examples. Here, the choice of regressing on initial status or not may give a different outcome.
 Kim Henry posted on Monday, August 25, 2008 - 5:13 pm
Are any readers aware of a good citation for the legitimacy of this approach? I recently received a review of a manuscript that employed this technique. The reviewer insisted that it was “inappropriate since both factors are defined by the same set of manifest indicators; LGM approaches typically allow for covariance between the slope and intercept, but one cannot infer a predictive path between predictors due to the nature of the latent variable loadings.”
 Bengt O. Muthen posted on Tuesday, August 26, 2008 - 3:34 pm
The reference we give when teaching on this is the Bloomquist (1977) blood pressure analysis in Journal of the American Statistical Association. I am sure there are many others. When centering at the first time point, the random intercept growth factor is defined as the systematic part of the variation at that time point (it is the only factor influencing that time point) so it preceeds the change thereafter which is orchestrated by the slope growth factor. That temporal order is why I think the regression of slope onh intercept is ok. This is not in my mind invalidated by the intercept factor also influencing later time points - it is natural that a starting point influences later status - since it doesn't influence change.
 Bengt O. Muthen posted on Thursday, August 28, 2008 - 2:58 pm
Two other references, from education, are

Seltzer, M., Choi, K. & Thum, Y.M. (2003). Examining relationships between where students start and how rapidly they progress: Using new developments in growth modeling to gain insight into the distribution of achievement within schools. Educational Evaluation and Policy Analysis, 25, 263-286.

Choi, K., Seltzer, M., Herman, J. & Yamashiro, K. (2007). Children left behind in AYP and non-AYP schools: Using student progress and the distribution of student gains to validate AYP. Educational Measurement: Issues and Practice, 26, 3, 21-32.

Choi and Seltzer also have a 3-level version of this forthcoming in JEBS, where the slope of the regression of slope on intercept varies across schools.
 fritz posted on Friday, May 22, 2009 - 5:48 am
Dear all,
I've got a question regarding "regression towards the mean". I'm working on a LGCM and I'm interested in "catch-up" effects. I'm now concerned that a negative correlation between intercept and slope, which would indicate such an effect, is only a regression artifact.

On the other hand, many books introducing LGCM-techniques say that we can use this correlation without even mentioning regression towards the mean. Moreover, in his initial message (see above) Harry Garst stated that "Actually I am not worried about controlling for ‘regression to the mean effects’ because both intercept and slope factors have been estimated from ALL data points." Does this mean that regression to the mean is no problem in such models? And if so, is there a reference I can quote?

To sum my question up:
1. Do I have to be concerned about regression to the mean in LGCM?
2. If so, how can I diagnose and, maybe, adjust for this problem?

Any recommendations or good references (of course, I found a variety of general literature regarding regression to the mean but not in the context of LGCM) are appreciated.
Thanks in advance!
 Linda K. Muthen posted on Friday, May 22, 2009 - 5:35 pm
See a paper by David Rogosa which has the words 10 myths in the title. I think this discusses regression to the mean in growth models.
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