Tree Inference: Response time in Multinomial Processing Trees, representation and uniqueness of parameters

Abstract

Multinomial Processing Trees are widely used in Psychology to model probabilities of responses in classes such as correct and incorrect. Some models include an additional measure such as response time, but predictions can become complex. Here we develop testable predictions using a method traditional for response times: manipulating experimental factors that selectively influence processes. In a Multinomial Processing Tree, each vertex represents a process, such as memory retrieval. An arc descending from a vertex represents a possible process outcome; for example, success or failure. Each arc has associated with it the probability the outcome it represents will occur. We assume that also associated with each arc is the time required for the outcome to occur. A factor that changes parameter values on arcs descending from a single vertex selectively influences that vertex. Suppose each of two factors selectively influences a different vertex in a Multinomial Processing Tree. There are only two ways the two vertices can be arranged. Either there is a path from one vertex to the other, and parameters on some arc on this path are changed by a factor, or there is not. Here we consider two Multinomial Processing Trees, important representatives of the two vertex arrangements. For each we develop testable necessary and sufficient conditions. Parameter values in the Multinomial Processing Trees may not be unique. If two sets of parameter values both lead to the same predictions, the values are related by admissible transformations, which we derive along with degrees of freedom.