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David H. Foster, K. Zychaluk; Model-free estimation of the psychometric function. Journal of Vision 2009;9(8):30. doi: 10.1167/9.8.30.
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© ARVO (1962-2015); The Authors (2016-present)
The psychometric function is central to the theory and practice of psychophysics. It describes the relationship between stimulus level and a subject's response, usually represented by the probability of success in a certain number of trials at that stimulus level. The psychometric function itself is, of course, not directly accessible to the experimenter and must be estimated from observations. Traditionally, this function is estimated by fitting a parametric model to the experimental data, usually the proportion of successful trials at each stimulus level. Common models include the Gaussian and Weibull cumulative distribution functions. This approach works well if the model is correct, but it can mislead if not. In practice, the correct model is rarely known. Here, a nonparametric approach based on local linear fitting is advocated. No assumption is made about the true model underlying the data except that the function is smooth. The critical role of the bandwidth is explained, and a method described for estimating its optimum value by cross-validation. A wide range of data sets were fitted by the local linear method and, for comparison, by several parametric models. The local linear method usually performed better and never worse than the parametric ones. As a matter of principle, a correct parametric model will always do better than a nonparametric model, simply because the parametric model assumes more about the data, but given an experimenter's ignorance of the correct model, the local linear method provides an impartial and consistent way of addressing this uncertainty.
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