Tree inference: Uniqueness of multinomial processing trees representing response time when two factors selectively influence processes

Multinomial Processing Trees are widely used to model response probability and sometimes to model response time and other measures. Information about the structure of a Multinomial Processing Tree can be discovered by manipulating experimental factors that selectively influence its vertices. A factor selectively influences a vertex if changing the level of the factor changes values of parameters associated with arcs descending from that vertex, leaving all else invariant. If two factors, each with a finite number of levels, selectively influence two different vertices in a Multinomial Processing Tree, an infinite number of Multinomial Processing Trees can be constructed that account for the response probability and response time. Under relatively unrestrictive assumptions, we show that all are equivalent, for these two factors, to one of two relatively simple Multinomial Processing Trees. We also show that if response probabilities and response times are produced by averaging over a mixture of different trees, the mixture is equivalent to one of the two relatively simple trees.