What are Multinomial Binary Tree Models?

Multinomial models are statistical models which allow for the estimation of hypothetical parameters that represent probabilies of unobservable events. Multinomial Binary Tree (MBT) models cover a broad class of multinomial models where parameters define link probabilities in a binary processing tree. Each bifurcation is associated with one parameter that determines the link probabilities. The probability of a branch is the product of the link probabilities along this branch. Each branch must be associated with a category for which the number of observations are known but there can be more than one branch associated with the same category.
 
 
 

A Simple Example

Suppose we want to examine the structure of interitem associations in serial learning. In an experiment, participants learn a list of items and are then given a cued serial recall task with one item (A) as cue and the task to provide the first (B) and second (C) successor of that item. 
Responses are grouped into four categories depending on whether item B and/or item C are reproduced correctly. Suppose we want to predict these frequencies by the interitem associations between items A, B, and C. A simple model, according to which only forward associations exist between these items, requires 3 states and therefore 3 parameters: pAB, pBC, and pAC are probabilities of associations between items A and B, B and C, and A and C, respectively. 
Depending on which combination of states a person is in, different outcomes would be expected. Consider for instance, the two sets of states which would lead to both items B and C being reproduced correctly (henceforth category Fcc). One possibility is that there is an association between items A and B on the one hand and an association between items B and C on the other. The other possibility is that there is an association between items A and B on the one hand and an association between items A and C on the other. Both sets of states would lead to correct responses for both items B and C. The probability for category Fcc is therefore pAB*pBC+pAB*(1-pBC)*pAC. In that way the entire model can be described by an equation system, and it can be illustrated by the binary tree model depicted below.

This is a very simple example of a multinomial binary tree model. The model definition file for this example and a file with hypothetical data are distributed with the AppleTree application. 

In practice MBT models tend to be more complicated, partly because they are often so called joint multinomial models. Joint MBT models are models where several processing trees exist which may use a common set of parameters. An example of a joint MBT model is provided with the ProcessDiss.eqn model definition file which is also distributed with the AppleTree application.

MBT models allow for the estimation of paramter values representing link probabilities. The parameters are estimated in way that the correspondence (as defined by a loss function) between empirical category frequencies and hypothetical category frequencies (as a function of parameter values) is maximal. For the statistical properties of MBT models, please refer to Riefer and Batchelder (1988), and Hu and Batchelder (1994). Xiangen Hu also provides a WWW site dedicated to General Processing Tree models. The algorithms used in AppleTree are described in a forthcoming article in Behavior Research Methods, Instruments, & Computers (Rothkegel, in press).

Previous | Back to Main Page | Next