Mediation: from 3 variables to 6 and 27 models

We all know the 3 variable model (X->M->Y & X->Y) with M being an intermediate variable (hence both cause and effect) is 'the simplest' possible 'statistical model', besides the bare-bone 2 variable one (X->Y).
Many have delved into its intricacies however, which are many in fact, dealing with specification alternatives. For instance, as Felix Thoemmes e.g. showed, there are 6 alternative statistically indistinguishable models that link all 3 variables (these are then all 'saturated models': no test of fit to data in SEM can be done).

Another take on this is the Conrady & Jouffe (2015) display of all 27 possible causal specifications one may contemplate when analyzing such 'simple' model (They use N1, N2, N3): this makes clear that the researcher/analyst 'gets married' to a specific causal structure s/he believes to operate behind the data, i.e. that generated it; In BayesiaLab this is obvious by asking the user for direct input ('drawing an arrow', much like in AMOS).

Of course, with >3 variables, the number of possibilities quickly get out of hand! Some can be ruled out by theory and prior findings, others by 'hunches', others by 'strong beliefs' (more so the case in practice, I'd say).

Conrady, S., & Jouffe, L. (2015). Bayesian Networks and BayesiaLab: A Practical Introduction for Researchers. http://www.bayesia.com/book
Thoemmes, F. (2015). Reversing Arrows in Mediation Models Does Not Distinguish Plausible Models. Basic and Applied Social Psychology, 37(4), 226-234. doi:10.1080/01973533.2015.1049351 www.tandfonline.com/doi/pdf/10.1080/01973533.2015.1049351

nested_CF-s

Here is the worked example from the article below; they showed how to 'process' such data, in the classical manner (standard adjustment), and in the causal modeling way (structured adjustment). Very simple side-by-side application.

Kaufman, J. S., & Kaufman, S. (2001). Assessment of Structured Socioeconomic Effects on Health. Epidemiology, 12(2), 157-167. doi:10.2307/3703617

Change patterns

Notice anything interesting in this?

There are some interesting cyclical patterns here... For context related to SEMNET, see below.

Changes-leading-to-changes

A quick illustration of how talking about changes in a variable X leading to changes in another variable Y can be simplified.

Changes-leading-to-changes makes language difficult, because we are used to talk about ‘imaginary changes in X leading to imaginary changes in Y’ to describe Y(X) relations from cross-sectional/timeless regressions X->Y, whereas in reality there are no actual changes to talk about in the same-time Y-on-X regression, only potential ones.

The problem comes when talking about what a model 'change-in-X->change-in-Y' tells us; here, you run into the problem of having cases that belong to 4 different quadrants of a scatter plot change-in-Y(change-in-X), see figure above: those increasing in both (Q2), those decreasing in both (Q4), and those increasing in 1: Y/X and decreasing in the other one: X/Y (Q1 / Q3, respectively).
You can still talk about all of these, with some help from peeking at where the averages for change-in-X & change-in-Y are (see the big green dot), whether they are positive or negative.

A simple way of describing these graphs where the changesX->changesY slope is >0 , is to say that if one increases in X, then we expect their Y to also increase, even though the 2 graphs show that the Y actual mean shows an average Y increase, then an average Y decrease: the only difference is that the slope was 'pushed down' by the 2 means, as the slope naturally crosses the (MeanX, MeanY) dot.

PatientDoctor

Here's an adaptation of Dave Kenny's social relations approach to patient-doctor relations, some of these may be forced... I add his original look for proper context

Kenny, D. A. (1994). Interpersonal perception: A social relations analysis: Guilford Press.

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: