TY - JOUR

T1 - Signed difference analysis

T2 - Testing for structure under monotonicity

AU - Dunn, John C.

AU - Anderson, Laura

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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).

AB - 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).

KW - Additive conjoint measurement

KW - Oriented matroids

KW - Sign vectors

KW - Signal detection theory

KW - Signed difference analysis

KW - State-trace analysis

UR - http://www.scopus.com/inward/record.url?scp=85050891393&partnerID=8YFLogxK

U2 - 10.1016/j.jmp.2018.07.002

DO - 10.1016/j.jmp.2018.07.002

M3 - Article

AN - SCOPUS:85050891393

VL - 85

SP - 36

EP - 54

JO - Journal of Mathematical Psychology

JF - Journal of Mathematical Psychology

SN - 0022-2496

ER -