Cross-sectional and longitudinal: data vs. models

Gollob & Reichardt have clarified for me the issue of time (=changes) and cases (=variability, differences), and then of course McArdle has taken these explanations to another level, with analyses of changes and differences, and changes in differences, and differences in changes.
So, one can have
1. Cross-sectional Data and Cross-sectional Models
2. Longitudinal Data and Longitudinal Models 
3. Cross-sectional Data and Longitudinal Models[1: p. 82 on]

In other words, time can be built in models with cross-sectional data, see my last figure in http://evaluatehelp.blogspot.com/2013/08/placeholders.html

What I want to show here however is how time can be extracted from repeated cross-sectional data, particularly when one is interested in development and changes in the context of natural/historical development. One of the original visual introductions to this approach is in Muthen (2000) (there are certainly earlier ones...):

In the context of say students (grades 6th to 12th) surveyed every 2 years or so, one can build a dataset structured as 'repeated", i.e. with 'year' and 'cohort' fields into it, to compare changes over time as done here [4]:  http://www.getcited.org/pub/103503915 

1. Gollob, H. F., & Reichardt, C. S. (1987). Taking account of time lags in causal models. Child Development, 58(1), 80-92.
2. Gollob, H. F., & Reichardt, C. S. (1991). Interpreting and estimating indirect effects assuming time lags really matter. In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysis of change: Recent advances, unanswered questions, future directions (pp. 243-259). Washington, DC, US: American Psychological Association.
3. Muthén, B. O. (2000). Methodological issues in random coefficient growth modeling using a latent variable framework: Applications to the development of heavy drinking. In J. Rose, L. Chassin, C. Presson & J. Sherman (Eds.), Multivariate applications in substance use research (pp. 113–140). Hillsdale: Erlbaum.
4. Combining missing-by-design and mixture modeling to assess impact of a community-wide interventions. A social marketing and text messaging campaign to reduce alcohol use among high school students. E Coman, G Rots, S Suggs, S Fuxman - 2012 Modern Modeling Methods, 2012

Latent change score briefest history

Latent Change Scores, developed by Jack McArdle in 1993 [1,2] are an amazingly simple and flexible tool for analysis of changes. The original models shown in the original chapters are below:

 

and

As you can see, this didn't look as simple as the LCS tool really is, so McArdle moved to make it more appealing, as in [3] and [4]:

The LCS definition and setup can be seen in an applied manner [that's how one can define the parameters in fact] as in [5]:

This simple device is the one that allows for modeling complex hypotheses about dynamic patterns of changes, like the one in [6] below:

1. McArdle, J. J. (1991). Comments on “latent variable models for studying difference and changes” In L. Collins & J. L. Horn (Eds.), Best Methods for the Analysis of Change (pp. 164-169). Washington, D.C.: APA Press.
2. McArdle, J., J. (1991). Structural models of developmental theory in psychology. In P. V. Geert & L. Mos (Eds.), Annals of theoretical psychology (Vol. II, pp. 139-160). New York: Plenum Publishers.
3. Prindle, J. J., & McArdle, J. J. (2012). An Examination of Statistical Power in Multigroup Dynamic Structural Equation Models. Structural Equation Modeling: A Multidisciplinary Journal, 19(3), 351-371. doi: 10.1080/10705511.2012.687661
4. McArdle, J. J. (2009). Latent Variable Modeling of Differences and Changes with Longitudinal Data. Annual Review of Psychology, 60, 577-605.
5. Coman, E. N., Picho, K., McArdle, J. J., Villagra, V., Dierker, L., & Iordache, E. (2013). The paired t-test as a simple latent change score model. Frontiers in Quantitative Psychology and Measurement, 4, Article 738. doi: 10.3389/fpsyg.2013.00738 [an update 9/14/16: >4,700 reads... only 1 citation though; what would this mean...?]

6. Grimm, K. J., An, Y., McArdle, J. J., Zonderman, A. B., & Resnick, S. M. (2012). Recent Changes Leading to Subsequent Changes: Extensions of Multivariate Latent Difference Score Models. Structural Equation Modeling: A Multidisciplinary Journal, 19(2), 268-292. doi: 10.1080/10705511.2012.659627

update 7/25/18: 
Several change causal models are presented in detail in this wonderful book below, where an amazingly similar model is shown on p. 19, (Fig. 2.3 below), as an alternative of a simple autoregressive 1 (AR1) model most users can relate to (Fig. 2.2 below). The book has many other hidden gems, and deserves a close reading!

Kessler, R. C., & Greenberg, D. F. (1981). Linear panel analysis: Models of quantitative change: Elsevier.

Some known models of change in time compared visually

Time series are better known by some, but few have shown what they look like; here's the Browne  & Nesselroade's great visuals, and the LCS model at then end for comparison:

and now compare this to: