1st Order Coordination in Structural Equation Modeling
Jonathan Butner
Structural Equation Modeling is a powerful method for the examination
of Dynamical Systems Models. SEM has the ability to separate out
measurement error from structural components of the model. Thus,
through techniques like Latent Change Score (LCS) modeling it can
separate out measurement from perturbations and perturbations from the
theoretical model of interest. The greatest limitation right now
(IMHO) is that SEM is limited to linear dynamic models. Importantly linear dynamic
models are capable of depicting nonlinear patterns of change through
time (e.g. oscillations). The issue is much more subtle having to
do with multistability and moderation.
Coordination
Coordination, specifically, is how two or more variables move together
in time. It is best understood by seeing it and hearing it. So, I
have included a bunch of youtube video links - I hope they help. Commonly, coordination is about how two or more oscillatory processes move together. However, the same idea can be done without oscillations. This is the logic we follow here.
Coordination is a stable moving together pattern and thus it implies coupled relationships between variables. Uncoupled variables can appear coordinated.
However, uncoupled variables are not displaying a stable relationship
in that if you perturb one variable that added change does not
translate over to the other variables. In coordination, the
perturbations carry through to the other variables as they keep each
other in the stable coordinated relationship.
It is through this notion of stability that we are able to identify
coordination with the repetition that occurs in oscillatory effects.
Coordination is Many Things
Phase Locking (also known as mode locking) is when the changes in one variable fully carry over to the other variable.
Entrainment is when there are periods of phase-locking like behavior with periods of slippage. Many social science phenomena are believed to be entrained.
Latent Change Score Modeling
Latent Change Score Modeling (LCS; originally called latent difference
score modeling) is a technique created by Jack McArdle that utilizes a
series of dummy (sometimes called node) latent variables that exist to
enforce a specific meaning to latent constructs. Consider a very
simple regression equation:
Post = b0 +b1(Pre) + e
Where Pre and Post are the same variable at two points in time.
When b1 is equal to one, the residual captures the difference of post
minus pre (a difference score). Latent change score modeling
capitalizes on this relationship by fixing certain parameters and
laying out a specific architecture of latent constructs so that one of
the latent constructs captures that difference. Unlike making an
observed difference, through multiple measures the latent differences
can be without measurement error. For example, in this figure we
have Y at three different time points.
The deltas are essentially latent representation of differences between
two time points by capturing what remains in the unnamed latent
constructs after being predicted by the previous time point with a b
coefficient forced to be equal to one.
It is then possible to
build various dynamical systems models by treating these latent changes
as the dependent variable in the structural portions of the
model. Any error variances in the structural portion become
representations of perturbations while error variances of the epsilons
capture measurement error. Notably, the model follows certain rules in
what does and does not get variances, error variances, and
meanstructure. There is some flexibility here, but the full
extent of what can be done is for another day.
I call models drawn from this approach 1st order dynamical systems
models because we are essentially treating a form of first order
derivative as the outcome. The common oscillatory approaches
would be 2nd order dynamical systems models because they treat a form
of second order derivative (acceleration) as the outcome. A
single 2nd order dynamical system equation implies two first order
equations. So, two coupled 2nd order equations (as is commonly
done by Boker and colleagues including my own work) imply four 1st
order equations. Thus, it is possible to link results from
different models as capturing different orders of change and
understanding how these orders relate (again, for another day).
Putting Coordination and LCS Together
In Butner, Berg, Baucom, & Wiebe (In Press) we illustrate using LCS
models to capture coordination. Through a slight modification of
bivariate LCS models, we are able to estimate a series of latent
coordinations and differentiate Phase Locking from Entrainment
including ratios of phase and the positive or negative
relationship. These models do not require any particular SEM
program. However, they contain a tremendous amount of code
duplication. Essentially the code is repeated for each pair of
consecutive time points. Since our data examples are panel
studies with 6 time points, the code is repeated 6 times for each
outcome (with the exception of behavioral self control which only has
four measures). Code for all the models discussed in the paper
are below. If you send me an email from a valid email address and
your reason for wanting the data (e.g. veracity, learning, teaching), I
will send you a link to the datafile.