Unification of mediation and moderation -Tyler VanderWeele

Tyler VanderWeele has published a paper in Epidemiology (posted before that at Harvard University Biostatistics Working Paper Series), that clarifies (to me to some extent) the different labels used for causal indirect effects in circulation today (notice the Baron-Kenny B.-K. effect in there!!!):

1. VanderWeele, T. J. (2014). A Unification of Mediation and Interaction: A 4-Way Decomposition. Epidemiology, 25(5), 749-761. doi: 10.1097/EDE.0000000000000121

2. VanderWeele, T. J. (2013). A unification of mediation and interaction. Harvard University Biostatistics Working Paper Series, from http://biostats.bepress.com/harvardbiostat/paper164

 
He uses some labels for some of these effects that are clarified (partly) by Bengt and Tihomir in:

1.Muthen, Advances in Latent Variable Modeling using Mplus, Storrs, Connecticut, United States, May 19, 2014, MMM 2014 slide 25 URL  

2.Muthén, B., & Asparouhov, T. (2014). Causal Effects in Mediation Modeling: An Introduction With Applications to Latent Variables. Structural Equation Modeling: A Multidisciplinary Journal, 1-12. doi: 10.1080/10705511.2014.935843

Simple comparative effectiveness

This post shows how useful AMOS can be to test MANY structural hypotheses at once, focused on whether an outcome after the intervention is better in one vs. another group, controlling for baseline differences (a Comparative Effectiveness line of questioning).
The simple idea is that you can conclude that the 2 post-intervention means are different by comparing a 2-group (intervention vs comparison) model where the 2 post-intervention means are equal to the similar model in which the 2 means (technically the intercepts however) are different: if the 'equal intercepts' model fits worse, the means cannot be deemed equal; quite basic.

Now, there are about 15 'baseline' models one can start with to test this 'equal intercepts' second model, and hence 15 such potential conclusions; we should however use only 1 to conclude whether there is/not a difference: this is the test of differences (labeled T) conducted using the best fitting baseline model, but also considering the power of that model to really detect this difference. The problem is: a WELL fitting model may be under-powered to detect a specific effect, which happened here too.
Take a look, try it for yourself, with Excel only and the [previously] free AMOS v.5 software; I can email you these files (ask at comanus@netscape.net ), but also posted here: http://bit.ly/2eGSxlq
AMOS does MODEL COMPARISONS" for ALL possible nested models, assuming one correct and testing the other ones nested in it... Pretty cool!!!

The little paper telling this story is available at:
http://digitalcommons.wayne.edu/jmasm/vol13/iss1/6/
Coman, E. N., Iordache, E., Dierker, L., Fifield, J., Schensul, J. J., Suggs, S., & Barbour, R. (2014). Statistical power of alternative structural models for comparative effectiveness research: advantages of modeling unreliability. Journal of Modern Applied Statistical Methods, 13(1).

Method of path coefficients, 1921

Here are the original 16 models published by Sewall Wright in 1921 in 'Journal of agricultural research'. Enjoy!

PS: He wasn't a 'statistician'... his title read: "Senior Animal Husbandman in Animal Genetics, Bureau of Animal Industry, United States Department of Agriculture" !

Wright, S. (1921). Correlation and causation. Part I Method of path coefficients. Journal of agricultural research, 20(7), 557-585. [CITED BY  2119, GOOGLE.SCOLAR; Aug.2018: 4,415 times!].

&

& ["Let O represent residual factors." in Wright, S. (1934). The Method of Path Coefficients, The Annals of Mathematical Statistics, 5(3), 161-215.]

&

Late plug for Duncan's 1975 book

I had many questions about the mechanics of structural modeling, particularly because as a physicist I am always trying to figure out 1st what's 'given' and what do we need to obtain/get/calculate/estimate.
I read recently in Les Hayduk's 1996 book (p. 15-16, e.g.) that Duncan has proposed a very simple and elegant procedure for obtaining model predicted variances and covariances from the structural model parameters, which is probably even easier to follow/apply than Wright's tracing rules (see a great modernized tutorial on Phillip Wood's web pages: standardized and unstandardized parts).
That simple rule allows you for instance to also "estimate the beta coefficient" in a simple regression by hand, like this:
y = b*x + e , then multiply by x: x*y = x*b*x + x*e, take expectations (a simple operation in fact, Duncan explains it, roughly speaking just sum up across the entire sample and divide by sample size):
E(x,y) = b*E(x,x) + E(x,e), which, if we assume that the covariance between predictor and error is zero (common), takes us to the "quick estimation" of b: b = Cov(x,y) /Var(x), a well known 'formula'.

Anyways, Duncan has shown that for a rather simple (& 'saturated') model like the one above, by doing this multiplication+taking expectations repeatedly, one can obtain/estimate the structural coefficients from variances & covariances (model predicted ones; but one can also go the other way around). This figure above shows however in this simple model what-depends-on-what; neat isn't it?
References:
Duncan, O. D. (1975). Introduction to structural equation models. New. York: Academic Press.
Hayduk, L. A. (1996). LISREL issues, debates, and strategies: Johns Hopkins University Press.

Stacked structural models

Les Hayduk (1996, Ch. 5, 155-189) presented these special kinds of multi-group Structural Equation models, the 'stacked SEMs' with different numbers of indicators, and of course I did not understand them well... Nothing works better than trying it out for yourself (and better so in AMOS first).

So: When you don't have a variable in ONE of the groups which appears in another, you can still use the same larger multi-group model (with the 'limping' variable in!).
[see my 1st image posted here: http://evaluatehelp.blogspot.com/2013/08/placeholders.html ]

How? Les Hayduk proposes a way, simply put filling in ZEROS for the covariances of the missing ('phantom') variable with all other variables, and 1's for their variances, like so:

AMOS can run models off such Excel data (you can just save these 3 covariance matrices as different worksheets in the same file, as I did; thanks, Les for including in your book the Lisrel syntax, it helped!). NOTE: AnxFam and and Work were NOT present in the 2nd and 3rd groups, but the 3-group model includes them too.
The results I got seem to replicate Les' results, after some AMOS arm-twisting.

The morale? What appears as complicated merits a 1st reading (Hayduk's 2nd book...), there is always stuff you can grab and run with it (even if not so far at first).

PS: as suggested on Semnet one should include a covariance between the errors of the looped variables, which tests whether relevant outside influences on these 2 variables have been mitted, which in our case seems not to be the case, all 3 covariances e.g. here were NonSignificant (NS) (adding such a covariance between L2 and L3 makes the model non-identified, why? well that's why the chapter merits reading...).

Hayduk, L. A. (1996). LISREL issues, debates, and strategies: Johns Hopkins University Press.

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.