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 -