Signed difference analysis: Testing for structure under monotonicity

Signed difference analysis (SDA), introduced by Dunn and James (2003), is used to derive testable consequences from a psychological model in which each dependent variable is presumed to be a monotonically increasing function of a linear or nonlinear combination of latent variables. SDA is based on geometric properties of the combination of latent variables that are preserved under arbitrary monotonic transformation and requires estimation neither of these variables nor of the monotonic functions. The aim of the present paper is to connect SDA to the mathematical theory of oriented matroids. This serves to situate SDA within an existing formalism, to clarify its conceptual foundation, and to solve outstanding conjectures. We describe the theory of oriented matroids as it applies to SDA and derive tests for both linear and nonlinear models. In addition, we show that state-trace analysis is a special case of SDA which we extend to models such as additive conjoint measurement where each dependent variable is the same unspecified monotonic function of a linear combination of latent variables. Lastly, we show how measurement error can be accommodated based on the model-fitting approach developed by Kalish et al. (2016).