The class of binary multinomial processing tree (BMPT) models is characterized by binary links at non terminal nodes, each associated with a parameter. The parameters are functionally independent and each is free to vary in the open unit interval. Previous work has shown that this class is statistically closed under some types of parametric constraints. By statistically closed is meant that when a certain parametric constraint is imposed, the constrained model, while not a BMPT, is nevertheless statistically equivalent to a model which is a BMPT. The closure theorems studied involve both dimension reducing constraints (Hu & Batchelder, 1994, Psychometrika) and order constraints (Knapp & Batchelder, 2004, J. Math. Psych.). These results allow certain statistical hypotheses to be handled within a general MPT inference scheme based on the EM-algorithm. This paper generalizes BMPT models to allow non terminal nodes to have multiple links. Multi-link MPT models cover a number of applications in cognitive modeling, e.g. source monitoring, and they are typical of tree models in statistical genetics, e.g. the ABO blood group model. For multi-link MPT models, the parameters are functionally independent pdfs with spaces corresponding to simplexes of various dimensionalities. The paper provides statistical closure theorems for dimension reducing constraints as well as order constraints both within and between parameter vectors. The results presented in this paper constitute a theoretical foundation for hypothesis tests for multi-link MPT models. |