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An extension of SIC predictions to the Wiener coactive model

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Abstract

The survivor interaction contrasts (SIC) is a powerful measure for distinguishing among candidate models of human information processing. One class of models to which SIC analysis can apply are the coactive, or channel summation, models of human information processing. In general, parametric forms of coactive models assume that responses are made based on the first passage time across a fixed threshold of a sum of stochastic processes. Previous work has shown that the SIC for a coactive model based on the sum of Poisson processes has a distinctive down–up–down form, with an early negative region that is smaller than the later positive region. In this note, we demonstrate that a coactive process based on the sum of two Wiener processes has the same SIC form.

Highlights

► The survivor interaction contrast distinguishes among classes of cognitive models. ► A coactive model using a sum of Poisson processes has a distinctive down–up–down SIC. ► We demonstrate that a coactive process, using two Wiener processes, has the same SIC form.

Introduction

One of the fundamental goals in modeling cognitive processing is determining how multiple sources of information are processed together. One major component of this determination is the distinction between parallel processing, in which sources are processed simultaneously, and serial processing, in which the sources are processed one at a time (e.g., Sternberg, 1966, Townsend, 1974). A special case of parallel processing is of particular interest to psychologists, coactive processing, in which information is accumulated in parallel, then pooled (Bernstein, 1970, Grice et al., 1984, Miller, 1982, Schwarz, 1989, Schwarz, 1994, Townsend and Ashby, 1983). Distinguishing among these processing types based on observable data can be difficult. However, under certain conditions, the Survivor Interaction Contrast (SIC) predicted by each processing type are distinct (Townsend & Nozawa, 1995). SIC predictions for independent serial and parallel systems are based on very general theorems; however the SIC prediction for the coactive model is specific to a Poisson accumulator model. In this paper we show that the predicted SIC for coactive models given by the sum of two Wiener processes is qualitatively the same as the predicted SIC of the Poisson coactive model. By demonstrating that the same SIC form is predicted by a coactive model based on a different stochastic process, we hope to add credence to the claim that this SIC is a signature of coactive processing in general.

Coactive processing models are generally used to describe systems in which information is gathered from multiple sources in parallel and is pooled toward a single decision. This type of model has been alternately described in the literature as ‘coactive’, (Miller, 1982, Townsend and Ashby, 1983, Townsend and Nozawa, 1995) ‘superposition’ (Schwarz, 1989, Schwarz, 1994) and ‘energy summation/integration’ (Bernstein, 1970, Nickerson, 1973). The key feature is that the summed activation level across sources is compared to a single threshold.

The coactive model is often used to model performance when there are multiple sources of information that contain redundant information about the appropriate response. Participants are normally faster and more accurate when there is redundant information than when an individual source is presented (e.g., Hershenson, 1962, Kinchla, 1974). This phenomenon is known as the redundant target effect (e.g., Miller, 1982). As it turns out, a redundant target effect is not necessarily enough to indicate coactive processing. Raab (1962) demonstrated that a redundant target effect can be produced by an independent, separate decision, parallel model due to statistical facilitation alone.1 Essentially his argument was that the probability that either of two processes is finished is higher than the probability that one specific process has finished. Nonetheless, there are methods of ruling out statistical facilitation as an explanation of the redundant target effect.

Miller (1982) developed one method of ruling out statistical facilitation as an explanation for faster response times in a redundant target design. He showed that, under certain assumptions, a parallel, separate decision model must have a smaller CDF of completion times in the redundant trials (FAB(t)) than the sum of the CDFs of completion times in single target trials (FA(t),FB(t)), FAB(t)FA(t)+FB(t). If Eq. (1) is violated for some t, then this is taken as evidence of coactive processing (see Maris & Maris, 2003, for a statistical test). One issue with this test is that it conflates the workload capacity with architecture (e.g., Townsend & Wenger, 2004). For example, if there are more resources dedicated to processing A and B when they are presented together rather than apart, then even an independent parallel model of processing could predict violations of Eq. (1). The applicability of the Miller inequality depends on the assumption of context invariance, i.e., the processing time of one source of information does not depend on the presence of another source of information.

To rule out an independent parallel model that may have more resources available when both sources are presented together, we can measure the SIC. The SIC is defined as the contrast between changes in processing speed of one source of information to changes in processing speed of the other source. Unlike the Miller inequality, this measure is based on equal processing loads for each component of the contrast. We use S(t) to denote the survivor function of a random variable and F(t) to denote the cumulative distribution function, i.e., S(t)=Pr{T>t}=1Pr{Tt}=1F(t). The different distributions associated with the processing speeds are indicated by subscripts on S(t) and F(t), so for example the survivor function of response times when the first source is processed at high speed and the second source is processed at the lower speed is denoted by SHL(t). Using this notation, the SIC is given by, SIC(t)=[SHH(t)SHL(t)][SLH(t)SLL(t)]=[FLH(t)FLL(t)][FHH(t)FHL(t)].

Data for the calculation of the SIC is often elicited within the double factorial paradigm. In this design two sources of information can independently be present or absent and, when present, can be independently presented at two levels of salience.2 This gives four different redundant target conditions, one for each of the four high–low salience combinations on the two channels, which are used to calculate the SIC from Eq. (2). By comparing the target present and absent conditions we can also test the Miller inequality and the capacity coefficient (Townsend and Nozawa, 1995, Townsend and Wenger, 2004). The capacity coefficient is another way to test for coactive processing (Townsend & Wenger, 2004) and is related to the Miller inequality (Townsend & Eidels, submitted for publication).

Independent parallel models, like independent serial models, predict specific SIC forms depending on whether processing of one or both sources must be completed before a response is made. We refer to models of processing in which processing of only one source must be completed as first-terminating, or OR systems. We refer to models of processing in which processing of both sources must be completed as exhaustive, or AND systems.

The mean interaction contrast (MIC; Eq. (3)) is also important in distinguishing among certain processing types. With M indicating the mean response time and the subscripts as defined above, the MIC is given by, MIC=[MHHMHL][MLHMLL]. The MIC was originally used as a test of serial independent processing. Sternberg (1969) showed that, assuming selective influence of the salience manipulations, this type of processing would lead to MIC=0. Schweickert and Townsend (1989) extended the use of the MIC by showing Parallel-AND models have a negative MIC. Townsend and Nozawa (1995) further extended these results to Parallel-OR processes and the Poisson coactive model which both predict MIC>0. Note that, due to the linearity of the integral, the fact that response times are always positive, and the integral of the survivor function of a positive random variable is its expected value, the integrated SIC is equal to the MIC, i.e., 0SIC(t)dt=MIC.

The SIC and MIC predictions of the four standard models are shown in Fig. 1. In the upper left panel the SIC for the Parallel-AND model is shown; it is always negative, so the MIC is also negative. The Serial-AND model prediction is shown in the upper right panel: first negative, then positive with equal positive and negative areas so the MIC is zero. On the bottom row the Parallel- and Serial-OR model predictions are shown. The Parallel-OR model is always positive with a positive MIC and the Serial-OR model SIC and MIC are zero.

In contrast, Townsend and Nozawa (1995) show that a specific case of the coactive model, based on the Poisson process (cf. Schwarz, 1989) predicts an SIC of the form depicted in Fig. 2. In this case, the SIC is negative for early times, then positive for later times, much like the SIC for the Serial-AND model. The difference between this coactive SIC and the Serial-AND SIC is in the relative positive and negative areas under the SIC: the coactive SIC has more positive than negative area whereas the Serial-AND SIC has equal positive and negative areas. Thus, the integrated SIC, i.e., the MIC, can distinguish between these two models. Note that despite the similarities in the form of the SIC, the Serial-AND model is a very different process than a coactive process.

Although these differences are qualitative, there are statistical tests available for determining if the collected data are enough to rule out any class of models. Houpt and Townsend (2010) show that a version of the Kolmogorov–Smirnov test can be used to check the positive and negative parts of an empirical SIC are significantly different from zero. This test is similar to the Maris and Maris (2003) test of the Miller inequality.

In the next section, we show that the distinguishing properties of the Poisson coactive SIC (down–up–down form and positive MIC) generalize to another class of coactive models, defined as the sum of two Wiener processes.

The Wiener process is formally defined as a stochastic process W(t) such that:

  • 1.

    W(0)=0,

  • 2.

    {W(t),t0} has stationary and independent increments,

  • 3.

    for every t>0,W(t) is normally distributed with mean 0 and variance t (Ross, 1996, pg. 357).

In contrast, a Poisson process, N(t), is defined by:
  • 1.

    N(0)=0,

  • 2.

    {N(t),t0} has stationary and independent increments,

  • 3.

    the number of events in any interval of length t is Poisson distributed with mean λt (Ross, 1996, pg. 60).

Note that the Poisson process takes on discrete values while the Wiener process is real valued. Furthermore, the Poisson process can only increase as time increases, whereas the Wiener process increases and decreases.

Section snippets

Theory

The coactive model we consider here consists of two processing streams modeled by two (possibly dependent) Wiener processes which are summed together. In general each channel can have its own drift rate (ν1;ν2) and diffusion coefficient (σ12;σ22). Furthermore, the processes can have arbitrary correlation (q12). Processing continues until the summed activation reaches a fixed, positive threshold (α). Note that this is the model of coactive processing proposed by Schwarz (1994).

Let {X1(t),t0}

Conclusions

The coactive model has been of particular interest, mainly as a model of redundant target effects. Determining whether a redundant target effect is due merely to statistical facilitation or to coactive processing of the stimulus information can be difficult. The Miller inequality, Eq. (1), is one possible method, but can fail if the assumption of context invariance fails. The SIC is a powerful measure for distinguishing among certain classes of information processing systems. Information about

Acknowledgments

This work was supported by NIH-NIMH MH057717-07 and AFOSRFA9550-07-1-0078. We would like to thank Jay Myung, Denis Cousineau and Devin Burns for their comments on the manuscript.

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