# Varieties of Selective Influence: Toward a More Complete Taxonomy and Implications for Systems Identification

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Definitions of Primary Concepts and Taxonomy

#### 2.1. Selective Influence

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

#### 2.2. Nonselective Influence

**Definition**

**5.**

**(1)**. If both direct and indirect nonselective influence are in force, the joint probability density function will be expressed in canonical form as

**(2)**. If only direct nonselective influence is afoot in process ${\mathbb{S}}_{a}$, then we have $fa({t}_{a};A,B)$ for that stage. That is, ${f}_{a}({t}_{a};A,B)$ is a function of the “other” factor B. If only direct nonselectivity in ${\mathbb{S}}_{b}$, we have

**(3)**. If only indirect nonselective influence occurs in a 1-direction serial system, it can only happen in stage ${\mathbb{S}}_{b}$ to yield ${f}_{b}({t}_{b};B|{t}_{a})$ by definition.

**(1)**. If both direct and indirect nonselective influence are in force, we have

**(2)**. If only direct nonselective influence is afoot in process ${\mathbb{S}}_{a}$, then we have ${g}_{a}({t}_{a};A,B)$ as a function of B with no dependencies on ${t}_{b}$. If only direct nonselective influence is afoot in process ${\mathbb{S}}_{b}$, then ${g}_{b}({t}_{b};B,A)$ is a function of A with no dependencies on ${t}_{a}$.

**(3)**. If only indirect nonselective influence is present in Channel ${\mathbb{S}}_{a}$, then we can write

**Definition**

**6**

**Definition**

**7.**

**(1)**${f}_{a}({t}_{a};A,B)={f}_{a}({t}_{a};A)$ and

**(2)**${f}_{b}({t}_{b};B)={\int}_{0}^{\infty}{f}_{a}({t}_{a};A,B){f}_{b}({t}_{b};B,A|{t}_{a})d{t}_{a}$.

**(1)**${g}_{a}({t}_{a};A)={\int}_{0}^{\infty}{g}_{a}({t}_{a};A,B|{t}_{b}){g}_{b}({t}_{b};B,A)d{t}_{b}$ and

**(2)**${g}_{b}({t}_{b};B)={\int}_{0}^{\infty}{g}_{b}({t}_{b};B,A|{t}_{a}){g}_{a}({t}_{a};A,B)d{t}_{a}$.

**Definition**

**8.**

**(1).**Strong marginal selectivity restricts the effect of any factor to its impact on the appropriate marginal distribution.

**(2).**Weak marginal selectivity, in our sense, demands that one or more of the experimental factors influences the joint distribution outside their impact on the pertinent marginal distribution.

## 3. Key Propositions on Satisfaction of Marginal Selectivity

**Example**

**1.**

**Proposition**

**1**

**A**). On Satisfaction of Marginal Selectivity in 1-Direction Serial Systems.

**(a)**. If direct nonselective influence of B on ${\mathbb{S}}_{a}$ is absent, then it is impossible that the marginal distribution on ${T}_{a}$ be a function of the “wrong” factor B in either the weak or strong sense. Hence, indirect nonselective influence on ${\mathbb{S}}_{a}$ is ruled out and selective influence of A on ${\mathbb{S}}_{a}$ is assured. Thus, ${f}_{a}({t}_{a};A)$ is no function of B and marginal selectivity holds for ${T}_{a}$.

**(b)**. If neither indirect nor direct nonselective influence are present in the ${\mathbb{S}}_{a}\to {\mathbb{S}}_{b}$ or the opposite direction, then strong marginal selectivity on ${T}_{b}$ is true.

**(c)**. Suppose that ${f}_{b}({t}_{b};B,A|{t}_{a})$ is a non-trivial function of A, B and ${t}_{a}$. Then, even strong marginal selectivity on ${T}_{b}$ may succeed.

**(d)**. We can construct a case where both factors cancel out and we are left with no influence, selective or nonselective at all.

**Proof.**

**(a)**: For every value of ${t}_{a}$, ${f}_{b}({t}_{b};B,A|{t}_{a})$, integration over all values of ${t}_{b}$ yields 1, leaving the marginal distribution of ${t}_{a}$ to be a function only of A. So, marginal selectivity is determined in Stage ${\mathbb{S}}_{a}$ if there is no direct nonselective influence there.

**(b)**: Obvious, because the hypothesis is tantamount to stochastic independence plus direct selective influence, that is pure selective influence is in power. This could be seen as a corollary to part (a).

**(c)**: It is helpful to picture the integral of the product ${f}_{a}({t}_{a};A){f}_{b}({t}_{b};B,A|{t}_{a})$ over all values of ${t}_{a}$ as a linear transformation $\mathbf{T}$ in continuous function space, arising from the Markov kernel ${f}_{a}({t}_{a};A)$. Then, we require that the marginal on ${T}_{b}$ be invariant over the factor A: ${f}_{b}({t}_{b};B,A)={f}_{b}({t}_{b};B)={\int}_{0}^{\infty}{f}_{b}({t}_{b};B,A|{t}_{a}){f}_{a}({t}_{a};A)d{t}_{a}$.

**(d)**: Prove with an example. Let ${f}_{a}\left({t}_{a}\right)=ABExp(-AB{t}_{a})$ and ${S}_{b}\left({t}_{b}\right|{t}_{a})=Exp(-AB{t}_{a}{t}_{b})$. Now, calculate the marginal of ${T}_{b}$:

**Proposition**

**1**

**B**). On Satisfaction of Marginal Selectivity in Parallel Exhaustive Systems.

**(a)**. If ${T}_{a}$ is afflicted by neither indirect nor direct nonselective influence, then its marginal distribution is selective. As intimated in the prologue just above, we simply impose an absence of direct and indirect nonselective influence on ${T}_{a}$. Then, integrating over ${T}_{b}$ yields a marginal on ${T}_{a}$ that is selective.

**(b)**. If neither indirect nor direct nonselective influence are present, then marginal selectivity on ${T}_{a}$ and ${T}_{b}$ is true.

**(c)**. Suppose that either ${g}_{a}({t}_{a};A,B|{t}_{b})$ is a non-trivial function of A, B and ${t}_{b}$, or ${g}_{b}({t}_{b};B,A|{t}_{a})$ is a non-trivial function of A, B and ${t}_{a}$, then even strong marginal selectivity may succeed.

**(d)**. As in Proposition 1A(d), if both channels are directly and indirectly influenced by the “other” factor, then the result could have complete freedom from either type of influence on either channel.

**Proof.**

**(b)**: Obvious. Because the hypothesis is tantamount to stochastic independence plus direct selective influence, that is, pure selective influence is in power.

**(c)**: We can employ a similar example to that used in Proposition 1. Let ${g}_{a}({t}_{a};A)=AExp(-A{t}_{a})$ and the survivor function ${\overline{G}}_{b}({t}_{b};B,A|{t}_{a})=Exp(-AB{t}_{a}{t}_{b})$. Then, the marginal survivor for ${T}_{b}$ is:

**(d)**: We perform the demonstration for ${\mathbb{S}}_{b}$. The other direction is the same. Let ${g}_{a}({t}_{a};A,B|{t}_{b})=ABExp(-AB{t}_{b}{t}_{a})$ and ${\overline{G}}_{b}({t}_{b};B,A|{t}_{a})=Exp(-AB{t}_{a}{t}_{b})$. Then, the marginal survivor for ${T}_{b}$ is:

**Definition**

**9.**

**Proposition**

**2**

**A**). On Failure of Marginal Selectivity in 1-Direction Serial Systems.

**(a)**. Direct nonselective influence only on ${\mathbb{S}}_{a}$, or only on ${\mathbb{S}}_{b}$ or both.

**(b)**. Indirect and direct nonselectivity on ${\mathbb{S}}_{b}$ with direct selective influence on ${\mathbb{S}}_{a}$.

**(c)**. It must fail when only indirect nonselectivity on ${\mathbb{S}}_{b}$ occurs. Note that we also assume ${T}_{a}$ is neither directly influenced by B or indirectly by ${T}_{b}$.

**Proof.**

**(a)**: Obvious.

**(b)**: Here, we see the intuitive statement that offset is by no means guaranteed. Again, let ${f}_{a}({t}_{a};A)=Exp(-A{t}_{a})$, but now let the conditional survivor function on ${S}_{b}$ be ${S}_{b}({t}_{b};B|{t}_{a})=Exp(-\frac{B{t}_{a}{t}_{b}^{2}}{A})$. It will be found that the marginal survivor on ${S}_{b}$ will be ${S}_{b}({t}_{b};B,A)=\frac{{A}^{2}}{B{t}_{b}^{2}+{A}^{2}}$, revealing a critical dependence in ${\mathbb{S}}_{b}$ on the wrong factor A. In fact, now, A slows down the second stage even more than in the example at the beginning of this section.

**(c)**: Due to its importance and its additional subtlety, we briefly rephrase this issue and proceed with care.

**1**represent the constant function $f\left(t\right)\equiv 1$. It is easy to see that $T\mathbf{1}=1$ is the constant function because that is just the integral over the ordinary density ${f}_{b}({t}_{b};B|{t}_{a})$ over the domain of ${t}_{b}$ from 0 to ∞.

**T**does not kill any function that is not already zero almost everywhere, then it implies that ${f}_{b}({t}_{b},B|{t}_{a})={f}_{b}({t}_{b};B)$. That is, this outcome contradicts our hypothesis that ${t}_{a}$ is effective in $fb({t}_{b};B|{t}_{a})$. Therefore, we conclude that marginal selectivity is false. □

**Proposition**

**2**

**B**). On Failure of Marginal Selectivity in Parallel Exhaustive Systems.

**(a)**. Direct nonselective influence only on ${\mathbb{S}}_{a}$, or ${\mathbb{S}}_{b}$ or both.

**(b)**. Indirect and direct nonselectivity on ${\mathbb{S}}_{b}$ with direct selective influence on ${\mathbb{S}}_{a}$; or indirect and direct nonselectivity on ${\mathbb{S}}_{a}$ with direct selective influence on ${\mathbb{S}}_{b}$

**(c)**. It must fail when only indirect nonselectivity occurs either on ${\mathbb{S}}_{a}$ or ${\mathbb{S}}_{b}$.

**Proof.**

**Proposition**

**3.**

**(a)**. In 1-direction serial systems ${\mathbb{S}}_{a}\to {\mathbb{S}}_{b}$, if indirect influence only afflicts ${\mathbb{S}}_{b}$, then the Dzhafarov definition of selective influence fails.

**(b)**In parallel systems, if indirect influence only afflicts either ${\mathbb{S}}_{a}$ or ${\mathbb{S}}_{b}$, then the Dzhafarov definition of selective influence fails.

**Proof.**

## 4. Canonically Shaped Distribution Interaction Contrasts and Marginal Selectivity

**Definition**

**10.**

**Definition**

**11.**

**(a)**. The survivor interaction contrast: Let $A,B$ be the experimental factors and write the survivor function on the underlying processing times as ${S}_{AB}\left(t\right)$. Then, the survivor interaction contrast function is defined as:

**(b)**. The distribution interaction contrast: Using a similar notation as in (A), we can exhibit the distribution interaction contrast ($DIC\left(t\right)$) as:

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**(a)**. Every parallel system with a joint distribution whose copula indicates that the experimental factors influence the distribution only through the marginals (strong marginal selectivity) predicts that the second order mixed partial derivative of the joint distribution itself will be non-negative.

**(b)**. Every parallel system enjoying strong marginal selectivity predicts that the second order mixed partial derivative of the cumulative distribution on the maximum processing time will be non-negative. Thus, this situation entails the canonical $DIC$ function.

**Proof.**

**(a)**. First, construct the copula for the joint cumulative distribution of a 2-channel parallel system: ${G}_{ab}({t}_{a},{t}_{b};A,B)$. It can be expressed as $H[{G}_{a}({t}_{a};A),{G}_{b}({t}_{b};B)]$. Then,

**(b)**. Obvious. □

**Corollary**

**1**

**Proof.**

**Corollary**

**1**

**Proof.**

**Proposition**

**6.**

**(a)**. If the experimental factors satisfy only weak marginal selectivity, then it is possible that canonical parallel predictions can fail.

**(b)**If weak marginal selectivity holds, then it may be that the canonical parallel predictions still are in force.

**Proof.**

**(a)**. Prove with a counter example. If the canonical prediction on $DIC$ holds for the parallel system when experimental factors affect the joint distribution, then for any ${H}_{ab}({t}_{a},{t}_{b};A,B)$ which is monotone in A and B (and, of course, also in ${t}_{a}$ and ${t}_{b}$), its second mixed partial derivative must be > 0.

**(b)**. Prove with an example. Let us set a parallel exhaustive system whose joint cumulative distribution follows a Gumbel’s distribution with the joint part $\alpha (A,B)=\frac{B}{A}$:

**Corollary**

**2.**

**(a)**. If the conditions in Proposition 5 are met, that is, the factors only influence the distributions through the marginal distributions, then the mean interaction contrasts are canonical: $MIC<0$.

**(b)**. If the conditions in Proposition 6 are met, that is, the factors influence the joint distribution outside of the marginal distributions, then the canonical mean interaction contrast sign may also be violated.

**Proof.**

**(a)**. First, recall that the mean processing time is just the integral from 0 to ∞ of the survivor function: $E\left(T\right)={\int}_{0}^{\infty}S\left(t\right)dt$. Moreover, as shown in Equation (2), the distributional interaction contrast is simply the negative of the survivor interaction contrast: $DIC\left(t\right)=-SIC\left(t\right)$. Then, as proved that the $DIC\left(t\right)$ is canonical and single valued under the condition of Proposition 5, so in the $SIC\left(t\right)$. Thus, the mean interaction contrast’s canonical nature ensues: $MIC<0$.

**(b)**. Prove with a counter example. Let us construct the parallel joint cumulative function following Gumbel’s distribution (see Equation (3) in the proof of Proposition 6) by having ${A}_{L}=0.005$, ${A}_{H}=0.01$, ${B}_{L}=0.1$, ${B}_{H}=0.8$ and $\theta =5$. Then, compute the MIC following Equation (1) and the result is positive (Figure 6), which contradicts the canonical prediction on $MIC$ for parallel exhaustive systems. □

**Proposition**

**7.**

**(a)**. The Dzhafarov definition of selective influence implies strong marginal selectivity.

**(b)**. Strong marginal selectivity does not imply the Dzhafarov condition.

**(c)**. The Dzhafarov condition does not imply weak marginal selectivity.

**(d)**. Weak marginal selectivity does not imply the Dzhafarov condition.

**Proof.**

**(a)**. The proof that marginal selectivity is forced by the Dzhafarov definition of selective influence ([33]) in fact entails the strong version can be seen through the fact that the latter only allows factor effects through the pairwise combining of the random variable with a factor—there is no other accompanying factorial influence.

**(b)**. Observe that lots of joint distributions in copula form may have dependencies not expressed simply via conditional independence.

**(c)**. If this was not the case, then absence of weak marginal selectivity would imply falsity of the Dzhafarov condition, but we know this is not true because strong marginal selectivity and weak marginal selectivity are incompatible.

**(d)**. If weak marginal selectivity implied the Dzhafarov condition then weak marginal selectivity would imply canonical $DIC$s, but we know this is wrong.

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Proof.**

## 5. Summary and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proposition 2A(c) in the Finite Case

**Proposition**

**A1.**

**Proof.**

## Appendix B. DICs Simulated from Two 1-Directional Serial Exhaustive Systems Using the Copula Strategy

**Figure A1.**Estimated $DIC\left(t\right)$s simulated from a 1-direction serial exhaustive system having FGM cdf with different $\alpha $ values. ${A}_{L}=3$, ${A}_{H}=12$, ${B}_{L}=1$, ${B}_{H}=4$.

## References

- Townsend, J.T.; Ashby, G.F. Stochastic Modeling of Elementary Psychological Processes; CUP Archive: Cambridge, UK, 1983. [Google Scholar]
- Sternberg, S. High-speed scanning in human memory. Science
**1966**, 153, 652–654. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Townsend, J.T. Mock parallel and serial models and experimental detection of these. In Purdue Centennial Symposium on Information Processing; Purdue University Press: West Lafayette, IN, USA, 1969; pp. 617–628. [Google Scholar]
- Townsend, J.T. A note on the identifiability of parallel and serial processes. Percept. Psychophys.
**1971**, 10, 161–163. [Google Scholar] [CrossRef] - Townsend, J.T. Some results concerning the identifiability of parallel and serial processes. Br. J. Math. Stat. Psychol.
**1972**, 25, 168–199. [Google Scholar] [CrossRef] - Treisman, A.M.; Gelade, G. A feature-integration theory of attention. Cogn. Psychol.
**1980**, 12, 97–136. [Google Scholar] [CrossRef] - Atkinson, R.; Holmgren, J.; Juola, J. Processing time as influenced by the number of elements in a visual display. Percept. Psychophys.
**1969**, 6, 321–326. [Google Scholar] [CrossRef] [Green Version] - Algom, D.; Eidels, A.; Hawkins, R.X.; Jefferson, B.; Townsend, J.T. Features of response times: Identification of cognitive mechanisms through mathematical modeling. In The Oxford Handbook of Computational and Mathematical Psychology; Oxford University Press: Oxford, UK, 2015; pp. 63–98. [Google Scholar]
- Wolfe, J.M. Visual search revived: The slopes are not that slippery: A reply to Kristjansson (2015). i-Perception
**2016**, 7, 2041669516643244. [Google Scholar] [CrossRef] [Green Version] - Little, D.R.; Eidels, A.; Houpt, J.W.; Yang, C.T. Set size slope still does not distinguish parallel from serial search. Behav. Brain Sci.
**2017**, 40, 3. [Google Scholar] [CrossRef] - Sternberg, S. Memory-scanning: Mental processes revealed by reaction-time experiments. Am. Sci.
**1969**, 57, 421–457. [Google Scholar] - Thomas, R.D. Processing time predictions of current models of perception in the classic additive factors paradigm. J. Math. Psychol.
**2006**, 50, 441–455. [Google Scholar] [CrossRef] - Dutilh, G.; Annis, J.; Brown, S.D.; Cassey, P.; Evans, N.J.; Grasman, R.P.; Hawkins, G.E.; Heathcote, A.; Holmes, W.R.; Krypotos, A.M.; et al. The quality of response time data inference: A blinded, collaborative assessment of the validity of cognitive models. Psychon. Bull. Rev.
**2019**, 26, 1051–1069. [Google Scholar] [CrossRef] - Townsend, J.T.; Nozawa, G. Spatio-temporal properties of elementary perception: An investigation of parallel, serial, and coactive theories. J. Math. Psychol.
**1995**, 39, 321–359. [Google Scholar] [CrossRef] - Sternberg, S. The discovery of processing stages: Extensions of Donders’ method. Acta Psychol.
**1969**, 30, 276–315. [Google Scholar] [CrossRef] - Townsend, J.T.; Wenger, M.J.; Houpt, J.W. Uncovering mental architecture and related mechanisms in elementary human perception, cognition and action. Stevens’ Handb. Exp. Psychol.
**2018**, 3, 429–454. [Google Scholar] - Harding, B.; Goulet, M.A.; Jolin, S.; Tremblay, C.; Villeneuve, S.P.; Durand, G. Systems factorial technology explained to humans. Tutorials Quant. Methods Psychol.
**2016**, 12, 39–56. [Google Scholar] [CrossRef] [Green Version] - Little, D.R.; Altieri, N.; Fific, M.; Yang, C.T. Systems Factorial Technology: A Theory Driven Methodology for the Identification of Perceptual and Cognitive Mechanisms; Academic Press: Cambridge, MA, USA, 2017. [Google Scholar]
- Joseph, W.H.; Daniel, R.L.; Ami, E. Developments in Systems Factorial Technology: Theory and applications. Spec. Issue J. Math. Psychol.
**2019**, 27, 11. [Google Scholar] - Dzhafarov, E.N.; Jordan, J.S.; Zhang, R.; Cervantes, V.H. Contextuality from Quantum Physics to Psychology; World Scientific: Singapore, 2015; Volume 6. [Google Scholar]
- Townsend, J.T.; Wenger, M.J. On the dynamic perceptual characteristics of gestalten: Theory-based methods. In Oxford Handbook of Perceptual Organization; Oxford University Press: Oxford, UK, 2014. [Google Scholar]
- Townsend, J.T.; Altieri, N. An accuracy–response time capacity assessment function that measures performance against standard parallel predictions. Psychol. Rev.
**2012**, 119, 500. [Google Scholar] [CrossRef] [Green Version] - Townsend, J.T. Uncovering mental processes with factorial experiments. J. Math. Psychol.
**1984**, 28, 363–400. [Google Scholar] [CrossRef] - Schweickert, R. A critical path generalization of the additive factor method: Analysis of a Stroop task. J. Math. Psychol.
**1978**, 18, 105–139. [Google Scholar] [CrossRef] [Green Version] - Schweickert, R. The bias of an estimate of coupled slack in stochastic PERT networks. J. Math. Psychol.
**1982**, 26, 1–12. [Google Scholar] [CrossRef] - Egeth, H.; Dagenbach, D. Parallel versus serial processing in visual search: Further evidence from subadditive effects of visual quality. J. Exp. Psychol. Hum. Percept. Perform.
**1991**, 17, 551. [Google Scholar] [CrossRef] - Townsend, J.T.; Ashby, G.F. Methods of modeling capacity in simple processing systems. In Cognitive Theory; Psychology Press: Hove East Sussex, UK, 1978; Volume III, pp. 200–239. [Google Scholar]
- Townsend, J.T. Truth and consequences of ordinal differences in statistical distributions: Toward a theory of hierarchical inference. Psychol. Bull.
**1990**, 108, 551. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Townsend, J.T.; Schweickert, R. Toward the trichotomy method of reaction times: Laying the foundation of stochastic mental networks. J. Math. Psychol.
**1989**, 33, 309–327. [Google Scholar] [CrossRef] - Schweickert, R.; Townsend, J.T. A trichotomy: Interactions of factors prolonging sequential and concurrent mental processes in stochastic discrete mental (PERT) networks. J. Math. Psychol.
**1989**, 33, 328–347. [Google Scholar] [CrossRef] - Townsend, J.T.; Nozawa, G. Strong evidence for parallel processing with simple dot stimuli. Bull. Psychon. Soc.
**1988**, 26. [Google Scholar] - Kujala, J.V.; Dzhafarov, E.N. Testing for selectivity in the dependence of random variables on external factors. J. Math. Psychol.
**2008**, 52, 128–144. [Google Scholar] [CrossRef] - Dzhafarov, E.N.; Kujala, J.V. The joint distribution criterion and the distance tests for selective probabilistic causality. Front. Psychol.
**2010**, 1, 151. [Google Scholar] [CrossRef] [Green Version] - Zhang, R.; Liu, Y.; Townsend, J.T. A theoretical study of process dependence for critical statistics in standard serial models and standard parallel models. J. Math. Psychol.
**2019**, 92, 102277. [Google Scholar] [CrossRef] - Houpt, J.W.; Townsend, J.T.; Jefferson, B. Stochastic foundations of elementary mental architectures. In New Handbook of Mathematical Psychology; Cambridge University Press: Cambridge, UK, 2018; Volume 2, pp. 104–128. [Google Scholar]
- Townsend, J.T.; Liu, Y.; Zhang, R. Selective influence and classificatory separability (perceptual separability) in perception and cognition: Similarities, distinctions, and synthesis. In Systems Factorial Technology; Elsevier: Amsterdam, The Netherlands, 2017; pp. 93–114. [Google Scholar]
- Townsend, J.T.; Thomas, R.D. Stochastic dependencies in parallel and serial models: Effects on systems factorial interactions. J. Math. Psychol.
**1994**, 38, 1–34. [Google Scholar] [CrossRef] - Dzhafarov, E.N. Conditionally selective dependence of random variables on external factors. J. Math. Psychol.
**1999**, 43, 123–152. [Google Scholar] [CrossRef] - Dzhafarov, E.N. Selective influence through conditional independence. Psychometrika
**2003**, 68, 7–25. [Google Scholar] [CrossRef] - Dzhafarov, E.N.; Schweickert, R. Decompositions of response times: An almost general theory. J. Math. Psychol.
**1995**, 39, 285–314. [Google Scholar] [CrossRef] - Howard, Z.L.; Garrett, P.; Little, D.R.; Townsend, J.T.; Eidels, A. A show about nothing: No-signal processes in systems factorial technology. Psychol. Rev.
**2021**, 128, 187. [Google Scholar] [CrossRef] [PubMed] - Luce, R.D.; Edwards, W. The derivation of subjective scales from just noticeable differences. Psychol. Rev.
**1958**, 65, 222. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Townsend, J.T. The mind-body equation revisited. Philos. Res. Arch.
**1975**, 4, 196–246. [Google Scholar] - Dzhafarov, E.N. Unconditionally selective dependence of random variables on external factors. J. Math. Psychol.
**2001**, 45, 421–451. [Google Scholar] [CrossRef] [Green Version] - Dzhafarov, E.N.; Schweickert, R.; Sung, K. Mental architectures with selectively influenced but stochastically interdependent components. J. Math. Psychol.
**2004**, 48, 51–64. [Google Scholar] [CrossRef] - Colonius, H.; Diederich, A. Dependency in multisensory integration: A copula-based analysis. Philos. Trans. R. Soc. A
**2019**, 377, 20180364. [Google Scholar] [CrossRef] [Green Version] - Colonius, H. An invitation to coupling and copulas: With applications to multisensory modeling. J. Math. Psychol.
**2016**, 74, 2–10. [Google Scholar] [CrossRef] - Colonius, H. Possibly dependent probability summation of reaction time. J. Math. Psychol.
**1990**, 34, 253–275. [Google Scholar] [CrossRef] [Green Version] - Townsend, J.T.; Liu, Y.; Zhang, R.; Wenger, M.J. Interactive parallel models: No Virginia, violation of miller’s race inequality does not imply coactivation and yes Virginia, context invariance is testable. Quant. Methods Psychol.
**2020**, 16, 192–212. [Google Scholar] [CrossRef] - Nelsen, R.B. An Introduction to Copulas; Springer: New York, NY, USA, 1999. [Google Scholar]
- Parzen, E. Modern Probability Theory and Its Applications; Wiley: New York, NY, USA, 1962. [Google Scholar]
- Dzhafarov, E.N.; Kujala, J.V.; Cervantes, V.H.; Zhang, R.; Jones, M. On contextuality in behavioural data. Philos. Trans. R. Soc. Math. Phys. Eng. Sci.
**2016**, 374, 20150234. [Google Scholar] [CrossRef] [PubMed] - Zhang, R.; Dzhafarov, E.N. Noncontextuality with marginal selectivity in reconstructing mental architectures. Front. Psychol.
**2015**, 6, 735. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, R.; Dzhafarov, E.N. Testing contextuality in cyclic psychophysical systems of high ranks. In International Symposium on Quantum Interaction; Springer: New York, NY, USA, 2016; pp. 151–162. [Google Scholar]
- Feller, W. An Introduction to Probability Theory and Its Applications, Volume 2; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Yang, H.; Fific, M.; Townsend, J.T. Reprint of survivor interaction contrast wiggle predictions of parallel and serial models for an arbitrary number of processes. J. Math. Psychol.
**2014**, 59, 82–94. [Google Scholar] [CrossRef] - Schweickert, R.; Fisher, D.L.; Sung, K. Discovering Cognitive Architecture by Selectively Influencing Mental Processes; World Scientific: Singapore, 2012; Volume 4. [Google Scholar]
- Kujala, J.V.; Dzhafarov, E.N. Regular Minimality and Thurstonian-type modeling. J. Math. Psychol.
**2009**, 53, 486–501. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The survivor function for ${T}_{b}$ conditioned on ${t}_{a}$ in example 1. ${S}_{b}({t}_{b}=1;B|{t}_{a})$ decreases with increases in ${t}_{a}$ and B. (

**b**). The marginal survivor function for ${T}_{b}$ in example. ${S}_{b}({t}_{b}=1;B)$ increases with A and decreases with B.

**Figure 2.**(

**a**.) The survivor function for ${T}_{b}$ in Proposition 1(c). ${S}_{b}({t}_{b}=1;B=2,A|{t}_{a})$ decreases with increase in ${t}_{a}$ and A. (

**b**.) The marginal survivor function for ${T}_{b}$ in Proposition 1(c). ${S}_{b}({t}_{b}=1;B)$ decreases with B.

**Figure 3.**$MIC$ for serial systems, parallel exhaustive systems and parallel self-terminating system.

**Figure 4.**$SIC\left(t\right)$s simulated from a serial exhaustive system and a parallel exhaustive system with the same set of parameters for the density functions of the sub-channel processing time.

**Figure 5.**The $DIC\left(t\right)$ estimated from a parallel exhaustive system that satisfies weak marginal selectivity with ${A}_{L}=3$, ${A}_{H}=12$, ${B}_{L}=1$, ${B}_{H}=4$.

**Figure 7.**$DIC\left(t\right)$ of a 1-direction serial exhaustive system that obeys the Dzhafarov’s definition.

**Figure 8.**Estimated $DIC\left(t\right)$ from the example 1-direction serial exhaustive system in Proposition 1A(c) with ${A}_{L}=3$, ${A}_{H}=8$, ${B}_{L}=4$, ${B}_{H}=6$.

**Figure 9.**Estimated $DIC\left(t\right)$s from a 1-direction serial exhaustive system having Gumbel’s bivariate exponential distribution where $q(A,B)=\frac{B}{{A}^{n}}$. ${A}_{L}=3$, ${A}_{H}=12$, ${B}_{L}=1$, ${B}_{H}=4$.

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Townsend, J.T.; Liu, Y.
Varieties of Selective Influence: Toward a More Complete Taxonomy and Implications for Systems Identification. *Mathematics* **2022**, *10*, 1059.
https://doi.org/10.3390/math10071059

**AMA Style**

Townsend JT, Liu Y.
Varieties of Selective Influence: Toward a More Complete Taxonomy and Implications for Systems Identification. *Mathematics*. 2022; 10(7):1059.
https://doi.org/10.3390/math10071059

**Chicago/Turabian Style**

Townsend, James T., and Yanjun Liu.
2022. "Varieties of Selective Influence: Toward a More Complete Taxonomy and Implications for Systems Identification" *Mathematics* 10, no. 7: 1059.
https://doi.org/10.3390/math10071059