DECOMPOSITIONS OF RESPONSE-TIMES - AN ALMOST GENERAL-THEORY

Response time (RT) whose distribution depends on two external factors can sometimes be presented as an algebraic combination of two random v ariables, component times, each of which is selectively influenced by one of these factors. The algebraic operation connecting these compone nt times (such as addition or ''minimum of the two'') is referred to a s the decomposition rule. We consider a broad subclass of associative and commutative decomposition rules, and for any operation from this s ubclass we construct a decomposition test, a relationship between obse rvable RTs that must hold ii these RTs are decomposable by means of th is operation. The decomposition tests are constructed under The assump tion that RT components are either stochastically independent or perfe ctly positively stochastically interdependent (in which case they are increasing functions of a common random variable). The decomposition t ests generalize the summation test proposed by Ashby & Townsend (1980) and Roberts & Sternberg (1992) for additive decompositions into stoch astically independent components. Under the assumption of perfect posi tive stochastic interdependence, a successful decomposition test is no t only necessary but also sufficient for the RT decomposability by mea ns of the corresponding operation. Under the assumption of stochastic independence, it is possible that a decomposition test is successful b ut RTs cannot be decomposed by any operation. Cinder both assumptions, however, a successful decomposition test recovers the true decomposit ion rule essentially uniquely. For a given decomposition rule, the com ponent times themselves cannot be determined uniquely, and the stochas tic relationship between them generally has to be assumed rather than recovered from the decomposition tests. (C) 1995 Academic Press, Inc.