  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.
