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Analysis of RT distributions with R Emil Ratko-Dehnert WS 2010/ 2011 Session 10 –

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Last time... RT distributions in the field – Convolution – Ex-Gaussian, Ex-Wald, Gamma, Weibull – Comparing functional fits Bootstrapping Creating functions in R 2

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Ex-Gauss distribution Convolution of an exponential and a gaussian distribution Correspondance to mental processes (exp -> decision, gauss -> residual perceptual/ response processes) Very good fits to RT data 3 Ex-Gauss

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Ex-Wald distribution Is the convolution of a Wald and an exponential distribution Represents decision and response components as a diffusion process (Schwarz, 2001) Neurally plausible (single cell recordings) + parameters can be interpreted psychologically Very good fits to RT data and highly successful in modelling RTs in various cognitive fields 4 Ex-Wald

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Diffusion Process 5 A B z drift rate ~N(ν,η) time Evidence Boundary separation Mean drift ν Respond „A“ Respond „B“ Information space Ex-Wald

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Gamma distribution Series of exponential distributions α = average scale of processes, β = reflects approximate number of processes Above average fits Suitable for three-stage exponential models 6 Gamma

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Weibull Distribution Like a series of races the weibull distribution renders an asymptotic description of their minima γ should lie between 1 (exp.) and 3.6 (gauss.) Decent functional fits, appropriate for processes which can be modelled as races 7 Weibull

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Palmer et al. (2009) Compared functional fits for three different search tasks (feature, conjunction, spatial config.) H 0 : fit to normal distribution All proposed distributions could reject H 0, but not equally well 1: Ex-Gauss, 2: Ex-Wald, 3: Gamma, 4:Weibull 8

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FUNCTIONAL FORMS OF RANDOM VARIABLES 9 II

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So far... We looked at densities and (cumulative) distribution functions for analysis of RTs As all densities for RTs are unimodal and rightskewed they can be inappropriate for analysis Similarly all CDF are sigmoidal, so they might not be adequate to compare 10

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RTs in ms 11

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Survivor function The survivor function F(t) is the probability that the lifetime of an object is at least t In oder words: the probability that failure occurs after t 12

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Hazard function The hazard function h(t) gives the likelihood that an event will occur in the next small interval dt in time, given that it has not occured before that point in time Thus, it is the conditional probability: 14

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Connection to survivor function When F(t) is differentiable, the hazard function can be expressed as a function of the survivor function F(t): 15

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Ex: h(t) of Ex-Gauss and Wald 16

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Cumulative Hazard Function Accumulted hazard over time Is an alternative (but equivalent) represen- tation of the hazard h(t) cf. Density Distribution 18

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Literature Ashby, Tein & Balakrishnan, 1993 Bloxom, 1984 Colonius, 1988 Luce, 1986 Maddox, Ashby & Gottlob, 1998 Thomas, 1971 Burbeck & Luce,

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ESTIMATION THEORY 21 III

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Next steps Theoretical analysis of distributions and their discrimination is important but in research practice another aspect also paramount „good“ estimation of densities, distribution and hazard functions are the first step to analyse RT data 22

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AND NOW TO 23

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