A Statistical Test for the Capacity Coefficient

Wright State University CORE Scholar Psychology Faculty Publications Psychology 7-16-2011 A Statistical Test for the Capacity Coefficient Joseph W. Houpt Wright State University - Main Campus, joseph.houpt@wright.edu James T. Townsend Follow this and additional works at: http://corescholar.libraries.wright.edu/psychology Part of the Cognition and Perception Commons, Cognitive Psychology Commons, Quantitative Psychology Commons, and the Statistical Models Commons Repository Citation Houpt, J. W., & Townsend, J. T. (2011). A Statistical Test for the Capacity Coefficient. The 44th Annual Meeting of the Society for Mathematical Psychology. http://corescholar.libraries.wright.edu/psychology/8 This Presentation is brought to you for free and open access by the Psychology at CORE Scholar. It has been accepted for inclusion in Psychology Faculty Publications by an authorized administrator of CORE Scholar. For more information, please contact corescholar@www.libraries.wright.edu.

A Statistical Test for the Capacity Coefficient Joseph W. Houpt James T. Townsend INDIANA UNIVERSITY BLOOBIIHGTON Math Psych '11 Medford, MA July 16, 2011

~ The workload capacity coefficient (C(t)) is a measure of how a persons performance changes with changes in workload. ~ Until now there have been no non-parametric statistical tests for C(t). ~ Develop a statistic test for the capacity coefficient for both OR and AND tasks. ~ Null hypothesis: C(t) = 1 ~ Adapt the Nelson-Aalen Estimator for the cumulative reverse hazard function. ~ Define unbiased and consistent estimators of OR and AND UCIP performance that are Gaussian Processes in the limit.

The Motivation ~ Can we dedicate the same amount of resources to processing each source when there are more sources?

The Motivation ~ 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 mcreases: Lim ited capacity. N=l N=2

The Motivation ~ 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 capacit y. N=l N=2

The Motivation ~ 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 mcreases: Super capacity. N=l N=2

The Measure ~ Fix a baseline for performance: Unlimited Capacity, Independent, Parallel (UCIP) model.

The Measure ~ Fix a baseline for performance: Unl imited Capacity, Independent, Parallel (UCIP) model. ~ No change due to increased workload

The Measure ~ Fix a baseline for performance: Unlimited Capacity, Independent, Parallel (UCIP) model. ~ No change due to increased workload ~ Stochastic independence

The Measure ~ Fix a baseline for performance: Unlimited Capacity, Independent, Pa rallel (UCIP) model. ~ No change due to increased workload ~ Stochastic independence

The Measure ~ Fix a baseline for performance: Unlimited Capacity, Independent, Parallel (UCIP) model. ~ No change due to increased workload ~ Stochastic independence First-Terminating (OR) Exhaustive (AND) (Town send & Nozawa, 1995) (Townsend & Wenger, 2004) SAs(t) SA(t)Ss(t) FAs(t) FA(t)Fs(t) log [SAs(t)] log [SA(t)Ss(t)] log [FAs(t)] log [FA(t)Fs(t)] HAs(t) HA(t) + Hs(t) KAs(t) KA(t) + Ks(t)

The Measure Limited Capacity Worse than predicted by a UCIP model

The Measure Limited Capacity Worse than predicted by a UCIP model - Unlimited Capacity Predicted by a UCIP model - -------

The Measure Limited Capacity Unlimited Capacity Super Capacity Worse than predicted Predicted by a UCIP Better than predicted by a UCIP model model by a UCIP model

The Nelson-Aalen Estimator Cumulative Hazard Function H(t) =lot ~(;(s) 1 ds =-log [1- F(t)] RT =the set of sample response times Y (t) = # of responses that have not occurred as of immediately before t

The Nelson-Aalen Estimator Cumulative Hazard Function ~ Unbiased: E [ H(t)- H(t)] = 0 ~ Consistent: limsize of RT--too H(t) = H(t) ~ [H(t)- H(t)J =?-Gaussian Process (e.g., Aalen, Borgan, & Gjessing, H. K., 2008)

The Nelson-Aalen Estimator Cumulative Reverse Hazard Function 00 K(t) = 1 ;~:~ ds =log [F(t)] G(t) = # of responses that have occurred up to and including t 1 k(t) = L G t t;e{rt> t} ( I)

The Nelson-Aalen Estimator Cumulative Reverse Hazard Function ~ Unbiased: E [ K(t)- K(t)J = 0 ~ Consistent: limsize ofrt---+oo K(t) = K(t) ~ [K(t)- K(t)] ::::} a Gaussian Process

UCI P Performance OR Process m 1 ~ Fucrp(t) = Il[1 ~ Fi(t)] i=l log (1 FiJCJP( t)) ~ log (D[l- F;( t)[) Hucrp(t) = L m i=l log (1 ~ Fi(t)) = L m i=l Hi(t). Hucrp(t) = L m Hi(t) Var ( Hucrp(t)) = L m L Y. 2 1 (t ) i=l i=l tjert(i)< t I J

UCI P Performance AND Process m Fucrp(t) =IT Fi(t) i=l log (FiJCJP(t)) ~log (D. F;(t)) Kucrp(t) = L m i=l log (Fi(t)) = L m i=l Ki(t). Kucrp(t) = L m m 1 Ki(t) Var ( kucrp(t)) =L L G-2(t ) i=l i=l tj ERT(i) < t I J

Test Statistic OR Task k ~ Lor(tj) _ ~ ~ Lor(tj) Zor( t) = ~ Y, ( ) ~ ~ Y: ( ) tje{rt(r)<t} r tj i=l tje{rt (i)<t} i tj Lor(tj) + L L Lor(tj) L Yr (t ) Y. (t ) 1 tje{rt(r) < t} J i=l tje{rt(i)<t} J Var [ Zor (t)] = 2 2 k

Test Statistic OR Task k ~ Lor(tj) _ ~ ~ Lor(tj) Zor( t) = ~ Y, ( ) ~ ~ Y: ( ) tje{rt(r)<t} r tj i=l tje{ RT(i) < t} i tj Lor(tj) + L L Lor(tj) L Yr (t ) Y. (t ) 1 tje{rt(r) < t} J i=l tje{rt(i)<t} J Var [Zor (t)] = 2 2 k Uor = Zor(t) rv N(O 1) Var [Zor (t)] '

Test Statistic AND Task Zand(t) Uand = Var [Zand(t)] "-' N(O, 1)

Weighting Function Lor(t) = Land(t) = Yr (t)(i:t 1 Yi( t)) ; r (t)+i:l Y;(t) if for all i, ~(t) > 0 { otherwise Gr (t)(i:t- 1 Gi(t)) ; r (t)+i:l Gi(t) if for all i, Gi(t) > 0 { otherwise

OR: Exponential Race Model Cor (t) < 1 Power as a Function of Trials and C(t) 1.0 0.8 O.S 0.4 Power 0.2 0.0

OR: Exponential Race Model Cor (t) > 1 Power as a Function of Trials and C(t) 1.0 Power 0.8 O.S 0.4 0.2 0.0

The Task pp PA AP AA

The Results OR Ca acit OR Task AND Task BJ -10.75 *** 2.88 ** RS -4.67 *** 2.93 ** JS -8.41 *** 3.16 *** MB -7.12*** 2.87 ** RM -9.69 *** 3.06 ** LB -3.02 ** 2.70 ** AND Capacity JG -5.54 *** 3.01 ** WY -6.08 *** 2.99 ** AW -5.75 *** 2.88 ** * p < 0.05 ** p < 0.01 *** p 0.001 < "

~ Adapted the Nelson-Aalen Estimator for the cumulative reverse hazard function.

~ Adapted the Nelson-Aalen Estimator for the cumulative reverse hazard function. ~ Defined unbiased and consistent estimators of 0 R and AND UCIP performance that are Gaussian Processes in the limit.

~ Adapted the Nelson-Aalen Estimator for the cumulative reverse hazard function. ~ Defined unbiased and consistent estimators of 0 R and AND UCIP performance that are Gaussian Processes in the limit. ~ Developed a statistical test for the capacity coefficient for both OR and AND tasks. ~ Null hypothesis: C(t) = 1