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. Townsend 1 , Haiyuan Yang 1 , Mario Fific 2 1 Indiana University Bloomington 2 Grand 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 one: 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 of 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 other 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: S ( t ) = Pr{ T > t } = 1 ! F ( t ) SIC ( t ) = [ S LL ( t ) ! S LH ( t )] ! [ S HL ( t ) ! S HH ( t )] Here, the subscripts indicate the salience of each source of information.

The Survivor Interaction Contrast � Assuming selective influence (Townsend & Nozawa, 1995; Dzhafarov, Schweickert & Sung, 2004)

The Serial Exhaustive SIC • Recall the definition of the SIC… SIC ( t ) = [ S LL ( t ) ! S LH ( t )] ! [ S HL ( t ) ! S HH ( t )] • The SIC for the Serial-AND process is given by 2 2 SIC SerAND Pr{ T X + T Y " t } = ! X , Y

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 t ! (0, " ) F XH ( t ) ! F XL ( t ) is log concave. F YH ( t ) ! F YL ( t ) The convolution of two unimodal functions is unimodal if at least one of them is logarithm concave (Ibragimov, 1956).

(Bagnoli & Bergstrom, 2005 )

What about N > 2

N Processes ¡ • Recall the definition of the 2-process SIC … 2 P ( T " t ) SIC 2 = ! X , Y • The n-process SIC is defined as… SIC n = ! X 1,... Xn n P ( T " t ) • Here is an example… SIC 3 = {[ S LLL ( t ) ! S LLH ( t )] ! [ S LHL ( t ) ! S LHH ( t )]} ! {[ S HLL ( t ) ! S HLH ( t )] ! [ S HHL ( t ) ! S HHH ( t )]}

N Processes: Parallel-OR • Theorem Parallel-OR processing implies overadditivity for all N. N=2 N=3 N=4 … ¡ ¡ n n SIC par . OR P (min( X 1,..., Xn ) > t ) = ! X 1 ... X n n n P ( " X i > t ) = ! X 1 ... X n i = 1 n # 1 = [ P ( X nL > t ) # P ( X nH > t )] $ SIC par . OR posi%ve ¡

N Processes: Parallel-AND • Theorem Parallel-AND processing implies underadditivity for even N and overadditivity for odd N. N=2 N=3 N=4 … ¡ ¡ n n SIC par . AND P (max( X 1 ,..., X n ) > t ) = ! X 1 ,,, X n n n = ! X 1 ... X n P ( " X i > t ) i = 1 n # 1 = [ P ( X nL < t ) # P ( X nH < t )] $ SIC par . AND nega%ve ¡

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. ¡ n n SIC ( t ) P ( X X  X t ) = Δ + + + ≥ ser . AND X ,... X 1 2 n 1 n ∞ n 1 [ f ( t ) f ( t )] SIC − ( t t ) dt = − − × − ∫ nH n nL n ser . AND n n − ∞

N Processes: Serial-AND • Theorem III N-Stage Serial-AND processing implies that if for every process is Normal distributed, then the SIC has F iH ( t i ) ! F iL ( t i ) N-1 zeros. N=2 N=3 N=4 … ¡ ¡

Conclusion ¡ … ¡ ¡ N=3 N=4 N=2 ¡ ¡ 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 Serial OR -­‑ 0.1 -­‑ 0.1 -­‑ 0.1 -­‑ 0.2 -­‑ 0.2 -­‑ 0.2 -­‑ 0.3 -­‑ 0.3 -­‑ 0.3 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 0.2 0.1 Serial AND 0.0 -­‑ 0.1 -­‑ 0.2 -­‑ 0.3 0 200 400 600 800 1000 1200 1400 0.2 0.2 0.2 0.1 0.1 0.1 Parallel OR 0.0 0.0 0.0 -­‑ 0.1 -­‑ 0.1 -­‑ 0.1 -­‑ 0.2 -­‑ 0.2 -­‑ 0.2 -­‑ 0.3 -­‑ 0.3 -­‑ 0.3 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 0.2 0.2 0.2 0.1 0.1 0.1 Parallel AND 0.0 0.0 0.0 -­‑ 0.1 -­‑ 0.1 -­‑ 0.1 -­‑ 0.2 -­‑ 0.2 -­‑ 0.2 -­‑ 0.3 -­‑ 0.3 -­‑ 0.3 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400

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 RT N=4 MAL MA L FAV V SA SAV V NAM M FAS S • Only “NO”/target absent responses are analyzed – processing exhaustive

Serial exhaustive N=4 1234 ¡ 4 ¡way ¡ 123x ¡ 12x4 ¡ 1x34 ¡ x234 ¡ 3 ¡way ¡ 2 ¡way ¡ 0.2 0.2 0.2 xx34 ¡ x2x4 ¡ 0.1 0.1 x23x ¡ 0.1 0.0 0.0 0.0 -­‑ 0.1 -­‑ 0.1 -­‑ 0.1 -­‑ 0.2 -­‑ 0.2 -­‑ 0.2 -­‑ 0.3 -­‑ 0.3 0 200 400 600 800 1000 1200 1400 -­‑ 0.3 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 0.2 0.2 1xx4 ¡ 1x3x ¡ 0.1 0.1 0.0 0.0 -­‑ 0.1 -­‑ 0.1 -­‑ 0.2 -­‑ 0.2 -­‑ 0.3 0 200 400 600 800 1000 1200 1400 -­‑ 0.3 0 200 400 600 800 1000 1200 1400 0.2 12xx ¡ 0.1 0.0 -­‑ 0.1 -­‑ 0.2 -­‑ 0.3 0 200 400 600 800 1000 1200 1400