The influence of power law distributions on long-range trial dependency of response times

Abstract

Empirical response time distributions from simple cognitive tasks are typically unimodal and positively skewed. In contrast, variance based scaling analyses, which have been used to study long-range dependency via the Hurst exponent, H>0.5, assume Gaussian response time distributions. This article presents a general method which can identify long-range trial dependency for response time series with power law distributions. The method fits an α-stable distribution to the response time series which satisfies a general version of the central limit theorem and consequently, an α-stable extension (Hq=0>1/α) of long-range dependency. The method was used to reanalyze 96 response time series from three existing data sets which included simple reaction time, word naming, choice decision, and interval estimation tasks. The results showed that all response time distributions were appropriately modelled by an α-stable distribution. Furthermore, the response time series from the simple response and word naming tasks were not long-range dependent when the α-stable definition Hq=0>1/α was used in place of the Gaussian response time distribution definition Hq=2>0.5. The deviation between the two definitions of long-range dependency was shown to be caused by divergence of the variance for response time distributions with power-law decaying tails. The study concludes that the new α-stable definition, Hq=0>1/α, of the long-range trial dependency should be used in the research of response time series instead of the Gaussian definition, Hq=2>0.5.