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: 

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