Quantile maximum likelihood estimation of response time distributions

Heathcote, Andrew; Brown, Scott; Mewhort, D. J. K. · 2002 · Psychonomic Bulletin & Review

DOI: 10.3758/bf03196299

archive: archived pipeline: cataloged verified

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Summary

This paper introduces and evaluates Quantile Maximum Likelihood (QML) estimation, a robust technique for fitting distribution functions to grouped response time (RT) data. The research addresses the limitations of existing methods for characterizing RT distribution shapes, particularly the ex-Gaussian distribution, which is widely used in cognitive psychology. While Continuous Maximum Likelihood (CML) estimation is generally efficient, it is sensitive to outliers. Conversely, moment-based estimators lack robustness and efficiency. The authors propose QML to combine the robustness of quantile-based methods with the statistical efficiency and consistency of maximum likelihood estimation. The study employs a Monte Carlo simulation to compare QML against CML. The authors generated samples from seven ex-Gaussian distributions, varying the ratio of the exponential mean to the normal standard deviation ($K = \tau/\sigma$) to represent shapes ranging from nearly normal to nearly exponential. All distributions maintained a constant mean of 1,000 and standard deviation of 100. Three sample sizes ($n = 40, 80, 160$) were tested. For each combination, 35,840 samples were generated. The authors evaluated QML using different numbers of quantiles (1, 2, 4, 8, and 16 observations per interquantile range) to assess the trade-off between robustness and information loss. Parameter estimates were obtained by maximizing the likelihood of the grouped data, using numerical integration and analytic derivatives to manage computational costs. The results demonstrate that QML is generally less biased and significantly more efficient than CML, particularly for estimating the normal mean parameter ($\mu$) in asymmetric distributions ($K = 2$ to $5$). QML estimates remained superior even when using up to 16 times fewer quantiles than data points, indicating that robustness can be achieved with minimal loss in bias or efficiency. While CML performed slightly better for nearly symmetric distributions ($K = 1/3$), QML was consistently better for the parameter ranges most representative of real choice RT data. The sampling distributions for QML estimates were largely symmetric, facilitating standard statistical analysis, whereas CML estimates for certain parameters exhibited skewness. The authors conclude that QML is the preferred method for estimating ex-Gaussian parameters, offering improved robustness against outliers without sacrificing statistical power. They provide open-source software (QMLE) to facilitate adoption, noting that while QML is computationally more intensive than CML, it remains feasible for typical empirical datasets. The paper also discusses extensions of QML to other densities and mixture models, highlighting its utility for graphical examination of fit via QQ plots and for estimating group RT distributions where individual data may be sparse.

Key finding

Quantile maximum likelihood estimation is less biased and more efficient than continuous maximum likelihood estimation for fitting ex-Gaussian distributions to response time data.

Methodology

simulation_modeling

Provenance

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Topics

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• perception reaction time