How Do Information Processing Systems Deal with Conflicting Information? Differential Predictions for Serial, Parallel, and Coactive Models

Abstract

In this paper, we analyze how different information-processing architectures deal with conflicting information. A robust finding in psychological research is that response times are slower when processing conflicting sources of information (e.g., naming the color of the word RED when printed in green in the well-known Stroop task) than when processing congruent sources of information (e.g., naming the color of the word GREEN when printed in green). We suggest that the effect of conflicting information depends on the processing architectures and derive a new measure of information processing called the conflict contrast function, which is indicative of how different architectures perform with conflicts at different levels of salience. By varying the salience of the conflicting information source, we show that serial, parallel, and coactive information processing architectures predict qualitatively distinct conflict contrast functions. We provide new analyses of three previously collected data sets: a detection task with Stroop color-word stimuli and two categorization experiments. Our novel measure provides convergent evidence about the underlying processing architecture in the categorization tasks and surprising results in the Stroop detection task.

Appendices

Appendix 1 Derivation of Conflict Contrast Function

Starting from the minimum time relationship implied by the parallel processing model:

$$ {S}_{AB}(t)={S}_{AY}(t)\times {S}_{XB}(t) $$

where S(t) is the survivor function for stimulus conditions AB (congruent target) or AY or XB (incongruent target), one can take the negative logarithm to convert these functions to integrated hazard functions (see, e.g., Luce 1986; Townsend and Nozawa 1995). Rearranging the function then gives for an unlimited capacity, independent, parallel self-terminating model:

$$ 1=\frac{-\log \left[{S}_{AY}(t)\times {S}_{XB}(t)\right]}{-\log \left[{S}_{AB}(t)\right]} $$

In the present case, we term this function the inverse OR capacity plus distractors function (ICPD) and note that the relationship between the derived minimum time and the observed redundant target time is exactly the opposite of what is expected under Townsend and Nozawa’s (1995) capacity function. That is, if the derived minimum time is slower than the observed redundant target time, then the ratio on the right will be less than 1, but if the derived minimum time is faster than the observed redundant target time, then the ratio on the right will be greater than 1.

The key diagnostic contrast occurs when the distractor is high salience and when the distractor is low salience.

$$ {\displaystyle \begin{array}{l}{ICPD}^{diff}(t)={ICPD}_H(t)-{ICPD}_L(t)\\ {}\kern2.999999em =\frac{-\log \left[{S}_{AY_H}(t)\times {S}_{X_HB}(t)\right]}{-\log \left[{S}_{AB}(t)\right]}-\frac{-\log \left[{S}_{AY_L}(t)\times {S}_{X_LB}(t)\right]}{-\log \left[{S}_{AB}(t)\right]}\end{array}} $$

If we allow the negative sign to cancel out and multiply through by log[S_AB(t)], then this leaves us with the difference between the log product of the survivor functions for the high and low salience single targets plus conflict sources.

$$ {\displaystyle \begin{array}{l}\log \left[{S}_{AB}(t)\right]\times {ICPD}_{diff}=\log \left[{S}_{AY_H}(t)\times {S}_{X_HB}(t)\right]-\log \left[{S}_{AY_L}(t)\times {S}_{X_LB}(t)\right]\\ {}\kern5.879996em =\log \left({S}_{AY_H}(t)\right)+\log \left({S}_{X_HB}(t)\right)-\log \left({S}_{AY_L}(t)\right)-\log \left({S}_{X_LB}(t)\right)\\ {}\kern5.879996em =\left(\log \left({S}_{AY_H}(t)\right)-\log \left({S}_{AY_L}(t)\right)\right)+\left(\log \left({S}_{X_HB}(t)\right)-\log \left({S}_{X_LB}(t)\right)\right)\end{array}} $$

Hence, the diagnostic predictions are given by the sum of the difference between the high and low salience distractors on each dimension.

We term this function the conflict contrast function:

$$ CCF(t)=\left(\log \left({S}_{AY_H}(t)\right)-\log \left({S}_{AY_L}(t)\right)\right)+\left(\log \left({S}_{X_HB}(t)\right)-\log \left({S}_{X_LB}(t)\right)\right) $$

The properties of this function provide qualitative distinctions between serial, parallel, and coactive processing models if the following assumptions hold:

• 1. The processing rate of stimulus dimension A and B do not vary as a function of the other dimensional value. This assumption is known as context invariance (Colonius 1990; Townsend and Eidels 2011).

• 2. The RT of the high salience conflict dimension is faster than the low salience conflict dimension for all t. This assumption is known as stochastic dominance (Schweickert et al. 2009; Townsend and Nozawa 1995).

Appendix 2 Intuitive Predictions for Each of the Processing Models

Parallel, Independent, Self-Terminating Model

If processing is parallel self-terminating and each channel (e.g., 1 or 2) is processed independently, then the RT is determined by the minimum channel processing time. That is,

$$ {F}_{12}^{parallel}(t)=1-\left(\left[1-{F}_1(t)\right]\times \left[1-{F}_2(t)\right]\right), $$

(4)

which gives the cumulative distribution function for the minimum time distribution (or alternatively, in terms of the survivor functions, ${S}_{12}^{parallel}(t)={S}_1(t)\times {S}_2(t)$). Note that for stimuli containing conflicting information, processing the conflict sources X and Y does not allow one to make a correct response; only the target sources A and B allow one to correctly respond (e.g., in the categorization task shown in Fig. 2). In other words, the independent, parallel self-terminating model is unperturbed by the presence of conflicting information. Hence, the above equation when applied to stimulus AY_L (or AY_H) reduces to

$$ {\displaystyle \begin{array}{l}\kern0.72em {F}_{AY_L}^{parallel}(t)=1-\left(\left[1-{F}_A(t)\right]\times \left[1-{F}_{YL}(t)\right]\right)\\ {}\kern3.359999em =1-\left[1-{F}_A(t)\right]\\ {}1-{F}_{AY_L}^{parallel}(t)=1-{F}_A(t)\\ {}\kern0.84em {S}_{AY_L}^{parallel}(t)={S}_A(t)\end{array}} $$

(5)

In words, this means that the survivor function for a given dimension A, S_A(t), is the same irrespective of the value or presence of the distractor in the other dimension. The same relationship holds for stimuli X_LB and X_HB. Hence, the discriminability of the conflicting source does not matter, and the CCF function equals 0:

$$ {\displaystyle \begin{array}{l}{CCF}^{parallel}(t)=\left(\log \left({S}_{AY_H}(t)\right)-\log \left({S}_{AY_L}(t)\right)\right)+\left(\log \left({S}_{X_HB}(t)\right)-\log \left({S}_{X_LB}(t)\right)\right)\\ {}\kern3.479999em =\left(\log \left({S}_A(t)\right)-\log \left({S}_A(t)\right)\right)+\left(\log \left({S}_B(t)\right)-\log \left({S}_B(t)\right)\right)\\ {}=0\end{array}} $$

Serial, Self-Terminating Model

The RT probability density function (pdf) of a serial, self-terminating model for the incongruent target stimuli are

$$ {f}_{AY}^{serial}(t)=p\left[{f}_A(t)\right]+\left(1-p\right)\left[{f}_Y(t)\ast {f}_A(t)\right] $$

(6)

and

$$ {f}_{XB}^{serial}(t)=p\left[{f}_X(t)\ast {f}_B(t)\right]+\left(1-p\right)\left[{f}_B(t)\right], $$

(7)

where f_A(t) and f_B(t) are the pdfs associated with processing correct sources A and B, and f_X(t) and f_Y(t) are the pdfs associated with processing the conflicting sources X and Y. AY and XB are the stimuli comprising one correct source and one conflicting source. Under a serial, self-terminating model, the pdf for these stimuli is a mixture of trials on which one dimension is processed first (with probability p) and other trials in which the other dimension is processed first (with probability 1 − p). On some of these trials, the first processed dimension will provide evidence for the OR set response (i.e., when A or B is processed first) allowing the decision to terminate. On other trials, the conflict information (i.e., X or Y) will be processed first requiring the remaining dimension to be processed before the decision can be terminated accurately. For instance, if one processes the environmental properties of whales first, then the decision will not be able to terminate accurately until the second dimension, biological properties, is processed.

For a serial self-terminating model, the discriminability of the conflicting information matters. The high discriminability conflict dimension should not slow down the stimulus as much as the low discriminability conflict dimension. To explain, in the conflict conditions (i.e., the incongruent stimuli), if the high discriminability distractor is faster than the low discriminability distractor (i.e., ${S}_{Y_H}<{S}_{Y_L}$) and if the processing rate of A (or B) does not depend on the other dimension, then ${S}_{AY_H}(t)={S}_{AY_L}(t)$ if p = 1 and ${S}_{AY_H}(t)<{S}_{AY_L}(t)$ if p < 1 because $1-\int \left[{f}_{Y_H}(t)\ast {f}_A(t)\right] dt<1-\int \left[{f}_{Y_L}(t)\ast {f}_A(t)\right] dt$, and analogously for stimuli X_HB and X_LB. Hence, since ${S}_{AY_H}(t)\le {S}_{AY_L}(t)$ and ${S}_{X_HB}(t)\le {S}_{X_LB}(t)$, then the inequalities also hold for the log of the survivor function and CCF^serial(t < 0).

For the latter inequality to hold, we require that the assumptions stated above hold for all t. Namely, we require that ${S}_{Y_H}(t)<{S}_{Y_L}(t)$ and ${S}_{X_H}(t)<{S}_{X_L}(t)$ indicating an effective manipulation of conflict discriminability (i.e., the high discriminability conflict presented alone provides stronger evidence for AND category than the low discriminability conflict presented alone). Note that we only require the ordering of survivor functions for the conflicting information (i.e., the high conflict on its own should be faster than the low conflict on its own; this may generally not be true if the high or low conflict is paired with the target information source, A or B). This assumption is typically termed stochastic dominance and is a crucial underlying assumption of techniques based on the manipulation of salience to uncover information processing architecture (Schweickert et al. 2009; Townsend and Nozawa 1995). The second assumption is that the processing rate of the target information (i.e., the A component of AY_H and AY_L and the B component of X_HB and X_LB) does not vary as a function of the salience of the conflicting information. This assumption is termed context invariance (Colonius 1990; Townsend and Eidels 2011) and plays an important role in techniques aimed at uncovering the capacity of information processing. Here, we combine both of these assumptions.

Coactive Processing Model

The intuition for the prediction of a coactive model when dealing with conflicting information is that the rate of processing will be slowed down more by high discriminability target than by a low discriminability target; consequently, ${S}_{AY_H}(t)>{S}_{AY_L}(t)$ and ${S}_{X_HB}(t)>{S}_{X_LB}(t)$ and CCF > 0. To explain, consider the effect of pooling together two conflicting information sources. The final rate of processing will depend on the relative strengths of each of these sources of information. The higher the discriminability of the conflicting source, the slower the rate of accumulation of evidence for the correct OR category response. Different versions of this pooling process have been proposed. Townsend and Nozawa (1995) applied their proofs to a version of a counter model which pooled counts from different sources into a single decision process. Houpt and Townsend (2011) proved that the same diagnostic measures held for a coactive model based on the Weiner diffusion model (e.g., Ratcliff 1978). Likewise, Fifić et al. (2010; see also Ashby 2000) proposed a process model in which the area under a bivariate normal distribution in the OR category region provided evidence for a sequential sampling model which determined the RT associated with each stimulus. (This is contrasted with serial and parallel models in which independent sequential sampling models are driven by the marginal normal distributions along each dimension). The high discriminability incongruent stimuli were predicted to be slower for the coactive model because more of the bivariate normal distribution overlapped with the AND category and not the OR category compared to the low discriminability incongruent stimuli. The CCF function shows that this relationship will hold for any coactive model so long as the assumptions of stochastic dominance and context invariance are met.

Exhaustive Processing Models

In a parallel exhaustive model, all sources must be processed to completion regardless of whether the target source has finished processing. The cumulative density functions of the incongruent targets are F_AY(t) = F_A(t) × F_Y(t) and F_XB(t) = F_X(t) × F_B(t).

In a serial exhaustive model, like a parallel exhaustive model, all sources must be processed regardless of whether the target source is processed first or not. The RT density functions for the incongruent targets are f_AY(t) = f_A(t) ∗ f_Y(t) and f_XB(t) = f_X(t) ∗ f_B(t). f_AB(t) = f_A(t) ∗ f_B(t), where the * symbol indicates the convolution integral of the two target densities.

For both exhaustive models, because of the stochastic dominance relationship between the high and low salience conflicting sources, the low discriminability conflict source will slow down the incongruent targets more than the high discriminability conflict source. Hence, in both cases, ${S}_{AY_H}(t)<{S}_{AY_L}(t)$ and ${S}_{X_HB}(t)<{S}_{X_LB}(t)$ and CCF < 0.

Appendix 3 Tutorial on Using the Conflict Contrast Function

A number of good tutorials have been recently published on the application and use of Systems Factorial Technology (SFT). We direct the reader to chapters by Algom et al. (2015), Altieri et al. (2017), and Harding et al. (2016) along with the comprehensive volume on SFT (Little et al. 2017). Along with the original theorems (Townend and Nozawa 1995; Townsend and Wenger 2004), these papers cover the underlying theory along with updated developments in SFT. However, they do not necessarily address hands-on use of the analyses. Houpt et al. (2014) have released an [R] package (sft) and comprehensive tutorial on using SFT. The sft library includes commands for computing the SIC (sic) and the CCF (conflict.contrast), along with associated statistical tests for these measures and others (see Houpt and Townsend 2010, 2012; Houpt and Little 2016). In this appendix, we provide pseudocode to illustrate the processing pipeline using preprocessed data to plot the SIC and CCF functions to create the figures for application 2. Our GitHub page contains the full data and analysis code (in MATLAB) for all of the analyses in this paper: https://github.com/knowlabUnimelb/CONFLICT_FUNCTION.

In the following Fig. 9, we analyze a datafile that has been preprocessed to remove outlying RTs and error RTs. We assume a data file with two columns: one indexing the relevant condition (i.e., HH, HL, LH, LL, AY_H, AY_L, X_HB, X_LB, AB; see Fig. 5 for reference) and another with the response time (RT). We demonstrate the calculation of the empirical survivor function using a histogram method, although other estimation methods are possible (e.g., ecdf, kernel density estimation, censored survivor functions). Figure 10 shows a schematic of how the item conditions from the design correspond to the CCF analysis.

Fig. 9

Psuedocode implementation of the SIC and CCF

Fig. 10

Illustration of how different item conditions contribute to the computation of the CCF for the high target salience level (A_H and B_H). An analogous assignment of item conditions to the equation is used for the low target salience level (A_L and B_L)