An Extension of Systems Factorial Technology (SFT) to Arbitrary - - PowerPoint PPT Presentation

An Extension of Systems Factorial Technology (SFT) to Arbitrary Numbers of Processes

James T. Townsend1, Haiyuan Yang1, Mario Fific2

1Indiana University Bloomington 2Grand Valley State University

Theory Driven Methodology

• SFT is a framework for addressing the general

question: how do different sources of information combine in mental processing?

– Are both sources used concurrently, or do we use one at a time? – How many sources are enough to respond? – Does knowledge of one source affect how we process another? – Can we dedicate the same amount of resources to processing each source when there are more sources?

Architecture

• Are both sources used concurrently, or do we

use one at a time?

– Using sources concurrently: Parallel information processing.

Architecture

• Are both sources used concurrently, or do we

use one at a time?

– Using sources one at a time: Serial information processing.

Architecture

• Are both sources used concurrently, or do we

use one at a time?

– Pooled information for a single detector: Coactive processing.

Stopping Rule

• How many sources are enough to respond?

– All of them: Exhaustive processing (AND)

Stopping Rule

• How many sources are enough to respond?

– Any of them: First-terminating processing (OR) – When there are more than two sources and not all sources are be required, but possibly more than

• ne: Self-terminating.

Stochastic Dependence

• Does knowledge of one source affect how we

process another?

– Stochastic independence of the decision times.

Stochastic Dependence

• Does knowledge of one source affect how we

process another?

– Stochastic dependence of the decision times.

Workload Capacity

• Can we dedicate the same amount of resources to

processing each source when there are more sources?

– Fewer resources available for each process as the number of sources increases: Limited capacity.

Workload Capacity

• Can we dedicate the same amount of resources to

processing each source when there are more sources?

– Unchanged amount of resources available for each process as the number of sources increases: Unlimited capacity.

Workload Capacity

• Can we dedicate the same amount of resources to

processing each source when there are more sources?

– More resources available for each process as the number of sources increases: Super capacity.

Conceptual Motivation

• What we need:
• What happens when the sources are given in isolation/

together? Presence/ Absence manipulation.

• When there are multiple sources, what is the overall

effect of speeding up and slowing down the processing

• f some subset of those sources?

Salience manipulation.

Double Factorial Paradigm (DFP)

Two manipulations:

• Presence/ Absence
• Salience

Selective Influence

• The salience manipulation selectively influences a

particular process if affects the target process and no

• ther process of interest.
• Selective influence is necessary for the measures based on

the DFP to be informative.

• Not directly testable without strong assumptions.
• One approach we often use is to test the orderings of the

responses time distributions, which is implied by selective

• influence. ¡

The Survivor Interaction Contrast

• The SIC is a measure of interaction between salience

manipulations. – Instead of just using the mean time, we use the survivor function: Here, the subscripts indicate the salience of each source of information.

S(t) = Pr{T > t} =1! F(t)

SIC(t) =[SLL(t)! SLH (t)]![SHL(t)! SHH (t)]

The Survivor Interaction Contrast

Assuming selective influence

(Townsend & Nozawa, 1995; Dzhafarov, Schweickert & Sung, 2004)

The Serial Exhaustive SIC

• Recall the definition of the SIC…
• The SIC for the Serial-AND process is given

by

SIC(t) =[SLL(t)! SLH (t)]![SHL(t)! SHH (t)]

SICSerAND

2

= !X,Y

2

Pr{TX +T

Y " t}

2-stage Serial Exhaustive SIC

• Properties (Townsend & Nozawa, 1995)

– The integral over t of the SIC is 0. – The SIC is negative for small t.

2-stage Serial Exhaustive SIC

• Theorem

The SIC is 0 only once for if or is log concave.

The convolution of two unimodal functions is unimodal if at least one of them is logarithm concave (Ibragimov, 1956).

t ! (0,")

F

YH (t)! F YL(t)

FXH (t)! FXL(t)

(Bagnoli & Bergstrom, 2005 )

What about N > 2

N Processes ¡

• Recall the definition of the 2-process SIC …
• The n-process SIC is defined as…
• Here is an example…

SIC2 = !X,Y

2 P(T " t)

SICn = !X1,...Xn

n

P(T " t)

SIC3 = {[SLLL(t)! SLLH (t)]![SLHL(t)! SLHH (t)]} !{[SHLL(t)! SHLH (t)]![SHHL(t)! SHHH (t)]}

N Processes: Parallel-OR

• Theorem

Parallel-OR processing implies overadditivity for all N.

SICpar.OR

n

= !X1...Xn

n

P(min(X1,..., Xn) > t) = !X1...Xn

n

P("

i=1 n

Xi > t) =[P(XnL > t)# P(XnH > t)]$ SICpar.OR

n#1

posi%ve ¡

N=2 N=3 N=4

… ¡ ¡

N Processes: Parallel-AND

• Theorem

Parallel-AND processing implies underadditivity for even N and overadditivity for odd N.

SICpar.AND

n

= !X1,,,Xn

n

P(max(X1,..., Xn) > t) = !X1...Xn

n

P("

i=1 n

Xi > t) =[P(XnL < t)# P(XnH < t)]$ SICpar.AND

n#1

nega%ve ¡

N=2 N=3 N=4

… ¡ ¡

N Processes: Serial-OR

• Theorem

Serial-OR processing implies SIC is 0 for all t.

N=2 N=3 N=4

… ¡ ¡

N Processes: Serial-AND ¡

• Theorem I

The integral over t of the SIC is 0 for all N.

• Theorem II

For even N, the SIC is negative for small t; For odd N, the SIC is positive for small t.

¡

∫

∞ ∞ − −

− × − − = ≥ + + + Δ =

n n n AND ser n nL n nH n n X X n AND ser

dt t t SIC t f t f t X X X P t SIC

n

) ( )] ( ) ( [ ) ( ) (

1 . 2 1 ,... .

1



N Processes: Serial-AND

• Theorem III

N-Stage Serial-AND processing implies that if for every process is Normal distributed, then the SIC has N-1 zeros.

F

iH (ti)! F iL(ti)

N=2 N=3 N=4

… ¡ ¡

Conclusion ¡

¡ ¡

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Serial OR Serial AND Parallel AND Parallel OR N=2 N=3 N=4

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Empirical study

Memory search N=4

Experiment

• Short-term memory search task (N=4)

– Stimuli: Serbian linguistic features. – Two levels of item-target dissimilarity: High dissimilarity was designated as high salience condition and low dissimilarity was designated as low salience condition. – The factors of interests were position of item in the set (1,2,3,4) x phonemic similarity (low, high)

A short-term memory search task

• Only “NO”/target absent responses are analyzed –

processing exhaustive

RT

FAV V SA SAV V NAM M FAS S MA MAL L

N=4

Serial exhaustive N=4

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4 ¡way ¡ 3 ¡way ¡ 2 ¡way ¡

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12xx ¡

Parallel exhaustive N=4

1234 ¡ 123x ¡ 12x4 ¡ 1x34 ¡ x234 ¡ 1x3x ¡ x23x ¡ 1xx4 ¡ x2x4 ¡ xx34 ¡

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12xx ¡

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Subject 2, N=4

1234 ¡ 123x ¡ 12x4 ¡ 1x34 ¡ x234 ¡ 1x3x ¡ x23x ¡ 1xx4 ¡ x2x4 ¡ xx34 ¡

4 ¡way ¡ 3 ¡way ¡ 2 ¡way ¡

12xx ¡

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Thank you! ¡

Project ¡funded ¡by ¡AFOSR ¡FA9550-­‑07-­‑1-­‑0078 ¡