ABSTRACT
As the old saying goes, “If it looks like a duck, walks like a duck, and quacks like a duck, it’s probably a duck.” The goal of this chapter is to determine whether we indeed have a dynamical data duck or just a wet and sloppy mess. More specifically, we’ll be looking at the structural equations technique for testing NDS hypotheses. Because the vast majority of new applications of nonlinear dynamics to organizational behavior presented in this book involved the use of the structural equations technique, this chapter is necessary for clarifying the statistical reasoning that was involved throughout the applications. The structural equations methodology is an extension of analytic procedures that have been applied regularly to conventional problems in the social sciences. Structural equations are used to (a) select a model that captures the conceptual
dynamics of a system, (b) assess its degree of fit to the actual data, and (c) to compare the results against two alternative hypotheses, which are that a simple linear model provides an adequate description of the process, or that the data consists of only noise. By “noise”wemean that neither the hypothesized dynamical equation nor the linear model describes the data. One might ask here: “There are so many possible nonlinear functions, how do
we know which ones to test?” Fortunately, we have hierarchical sets of models to
work with, and a theorem about singularity. Each model in the hierarchy subsumes properties of the simpler models. Each progressively complex model adds a new feature. There are two such hierarchies to consider: (a) the catastrophes for discontinuous change, and (b) the exponential series for other dynamical processes. The models for catastrophes are the results of Thom’s (1975) classification
theorem, which is that all discontinuous changes of events can be described by one of ten elementary models. The models are hierarchical and involve one or two dependent measures (order parameters), and up to five control parameters. Beyond that point, we exceed the bounds of singularity, where there is only one response surface (or model) that will describe the configuration of events. In the event that we have something other than a discontinuous change process,
we would employ the exponential series of models (derived in Guastello, 1995a, Chapter 3). The simplest in that series is the Lyapunov function, which is an indicator of entropy and a test for chaos. The second in the series is the MayOster bifurcation model, which is relative of the logistic map. Beyond those two models we have models with multiple order parameters or transfer functions or lag terms. Arguably, the catastrophe models are dynamically more complex than those of
the exponential series, but the computation is probably more familiar. The logic of hypothesis testing was also developed with catastrophe models and dates back to the early 1980s (Guastello, 1982a, 1982b). The hypothesis testing sequence procedure will be elaborated one step at a time as follows: (a) type of data and amounts that are required; (b) probability functions, location, and scale; (c) structure of behavioral measurements; (d) the catastrophe models, which can be tested through power polynomial regression; (e) the exponential series ofmodels, which are tested through nonlinear regression; (f) catastrophes with static PDFs, and hypotheses about the control parameters; and (g) systems with more than one order parameter, phase space, and embedding.
