Abstract
We present a general theory of series-parallel mental architectures with selectively influenced stochastically non-independent components. A mental architecture is a hypothetical network of processes aimed at performing a task, of which we only observe the overall time it takes under variable parameters of the task. It is usually assumed that the network contains several processes selectively influenced by different experimental factors, and then the question is asked as to how these processes are arranged within the network, e.g., whether they are concurrent or sequential. One way of doing this is to consider the distribution functions for the overall processing time and compute certain linear combinations thereof (interaction contrasts). The theory of selective influences in psychology can be viewed as a special application of the interdisciplinary theory of (non)contextuality having its origins and main applications in quantum theory. In particular, lack of contextuality is equivalent to the existence of a “hidden” random entity of which all the random variables in play are functions. Consequently, for any given value of this common random entity, the processing times and their compositions (minima, maxima, or sums) become deterministic quantities. These quantities, in turn, can be treated as random variables with (shifted) Heaviside distribution functions, for which one can easily compute various linear combinations across different treatments, including interaction contrasts. This mathematical fact leads to a simple method, more general than the previously used ones, to investigate and characterize the interaction contrast for different types of series-parallel architectures.
Highlights
The notion of a network of mental processes with components selectively influenced by different experimental factors was introduced to psychology in Saul Sternberg’s (1969) influential paper
In this paper we extend this approach to other mental architectures belonging to the class of series-parallel networks, those involving other composition operations and possibly more than just two selectively influenced processes
By proving and generalizing most of the known results on the interaction contrast of distribution functions, we have demonstrated a new way of dealing with series-parallel composition (SP) mental architectures
Summary
The notion of a network of mental processes with components selectively influenced by different experimental factors was introduced to psychology in Saul Sternberg’s (1969) influential paper. It is easy to show (Dzhafarov, 2003) that the existence of a triple f , g, R for given joint distributions of (A, B) under different treatments (α, β) is equivalent to the existence of a quintuple f ′, g′, S, SA, SB , where S, SA, SB are random variables, such that f ′ (α, S, SA) = A and g′ (β, S, SB) = B. In Dzhafarov et al (2004) this representation was used to investigate different series-parallel arrangements of the hypothetical durations A and B The reason this representation has been considered convenient is that if one fixes the value S = s, f ′ (α, s, SA) = Ac and g′ (β, s, SB) = Bc are stochastically independent random variables. When we have random variables not influenced by any of these factors, we will say they selectively influenced by an empty set of factors (we could equivalently, introduce for them dummy factors, with one value each)
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