Changes and their causes

Here are two specifications of change scores, as latent variables however, that can be used in subsequent analyses of changes. Note that the causal assumptions behind them differs slightly, and I prefer the causal logic of the 1st: the final values depend on initial values and on some mechanism of change, i.e. the causal mechanism 'precedes' the final observed values, not the other way around.

 

One can defend the view that 'changes are a result of both initial and final values', but with less grounding I'd say. This is supposed to help with the posting SEMNET: (does it?)

This 2-wave simplified example of course would not be enough to model a pattern like the one below, which, however, LCS can fit pretty well with attention to nuance and detail... J.J. McArdle's work has lots of dynamic applications like this. Some issues are: what window of time to model, and how to 'initialize' the process.

mixtures

Here are some examples taken from Jeff Harring's 3 day (!!!) workshop "INTRODUCTION TO FINITE MIXTURE MODELS; he pointed me to the 1st mixture modeling idea, published in 1894 (!):

Pearson, K. (1894). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 185, 71-110.

* Here are also other ways of gauging if one has more than 1 population in their data: 

or by looking at the random slopes and intercepts: 

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.