Tree Inference: Response Time in a Binary Multinomial Processing Tree, Representation and Uniqueness of Parameters

TL;DR

This paper explores the mathematical properties of Multinomial Processing Trees (MPTs), focusing on response times, parameter representation, and conditions for model simplification, with implications for experimental design and data analysis.

Contribution

It derives necessary and sufficient conditions for response class probabilities in binary MPTs and characterizes parameter transformations, advancing understanding of model identifiability and simplification.

Findings

Conditions for probability modeling in special binary MPT cases

Parameter transformation rules for non-unique parameters

Degrees of freedom formulas for experimental designs

Abstract

A Multinomial Processing Tree (MPT) is a directed tree with a probability associated with each arc. Here we consider an additional parameter associated with each arc, a measure such as the time required to select the arc. MPTs are often used as models of tasks. Each vertex represents a process and an arc descending from a vertex represents selection of an outcome of the process. A source vertex represents processing that begins when a stimulus is presented and a terminal vertex represents making a response. Responses are partitioned into classes. An experimental factor selectively influences a vertex if changing the level of the factor changes parameter values on arcs descending from that vertex and on no others. Earlier work shows that if each of two experimental factors selectively influences a different vertex in an arbitrary MPT it is equivalent for the factors to one of two relatively simple MPTs. Which of the two applies depends on whether the two selectively influenced vertices are ordered by the factors or not. A special case, the Standard Binary Tree for Ordered Processes, arises if the vertices are so ordered and the factor selectively influencing the first vertex changes parameter values on only two arcs descending from that vertex. Here we derive necessary and sufficient conditions for the probability and measure associated with a particular response class to be accounted for by this special case. Parameter values are not unique and we give admissible transformations for transforming one set of parameter values to another. When an experiment with two factors is conducted, the number of observations and parameters to be estimated depend on the number of levels of each factor; we provide degrees of freedom.