Multinomial processing trees with response times: Changing speed and accuracy by selectively influencing a vertex

In many experiments a person performs a task, such as identifying a letter, and an experimental factor, such as brightness, is manipulated. Empirically, changing the level of a factor often produces a relation, stochastic dominance, on the response time cumulative distribution functions. Specifically, for levels 1 and 2 of the factor, let H1(t) and H2(t) be the cumulative distribution functions of the correct response time. Then one often finds that for all times t, H1(t) > H2(t). We consider a Multinomial Processing Tree in which arcs have a probability of being selected and require time to be selected. It is natural to consider the effect of a factor on products of probability and time. At levels 1 and 2 of the factor, let π(1) and π(2) be the probability of a correct response. The factor produces weighted stochastic dominance if π(1)H1(t) > π(2)H2(t) for all times t. An experimental factor selectively influences a vertex in a Multinomial Processing Tree if changing the level of the factor changes parameters at a single vertex, leaving all else invariant. We consider conditions under which a factor selectively influencing a vertex in a Multinomial Processing Tree produces weighted stochastic dominance. Our assumptions allow parameters in a Multinomial Processing Tree to vary from trial to trial, and to be correlated through dependence on a common random variable. Further, the same Multinomial Processing Tree need not be used on every trial, there may be a mixture of Multinomial Processing Trees. We demonstrate results of theorems with a simulation.