PsiMLE: A maximum-likelihood estimation approach to estimating psychophysical scaling and variability more reliably, efficiently, and flexibly

Abstract

A simple and popular psychophysical model-usually described as overlapping Gaussian tuning curves arranged along an ordered internal scale-is capable of accurately describing both human and nonhuman behavioral performance and neural coding in magnitude estimation, production, and reproduction tasks for most psychological dimensions (e.g., time, space, number, or brightness). This model traditionally includes two parameters that determine how a physical stimulus is transformed into a psychological magnitude: (1) an exponent that describes the compression or expansion of the physical signal into the relevant psychological scale (β), and (2) an estimate of the amount of inherent variability (often called internal noise) in the Gaussian activations along the psychological scale (σ). To date, linear slopes on log-log plots have traditionally been used to estimate β, and a completely separate method of averaging coefficients of variance has been used to estimate σ. We provide a respectful, yet critical, review of these traditional methods, and offer a tutorial on a maximum-likelihood estimation (MLE) and a Bayesian estimation method for estimating both β and σ [PsiMLE(β,σ)], coupled with free software that researchers can use to implement it without a background in MLE or Bayesian statistics (R-PsiMLE). We demonstrate the validity, reliability, efficiency, and flexibility of this method through a series of simulations and behavioral experiments, and find the new method to be superior to the traditional methods in all respects.

Children's intuitive sense of number develops independently of their perception of area, density, length, and time

“…In order to account for the heteroscedasticity inherent in the w and age data (i.e. accuracy and w values are more variable in younger children), the normally distributed regression error was allowed to scale with 1/Age (see also Odic, Im, Eisinger, Ly, & Halberda, in press). The best‐fitting parameters are shown in Figures and and in Table .…”

The integration between nonsymbolic and symbolic numbers: Evidence from an EEG study

“…The reference dot quantity either contained 25, 30, or 39 dots, which correspondingly induced overestimation, linear-like estimation, and underestimation in participants' performance . Third, there are individual differences in the underestimation bias in adults Odic et al, 2015) and young children who have acquired symbolic number knowledge (Libertus, Odic, Feigenson, & Halberda, 2016). Altogether, these behavioral findings suggest that people are able to map between the ANS and SNS, but that this mapping is not precise and is subject to a systematic underestimation bias.…”

“…More critically, previous research found that people tend to underestimate large quantities in nonsymbolic number estimation tasks (Crollen, Castronovo, & Seron, 2011;Krueger, 1982;Odic, Im, Eisinger, Ly, & Halberda, 2015). For example, when presented with 60 dots, people more commonly estimate fewer than 60 dots in contrast to estimating more than 60 dots.…”

“…If participants were paying attention to the task and were responding reasonably, their estimate would increase as the dot quantities increase. Moreover, as the dot quantities increase, the variability of participants' estimates should also increase (Odic et al, 2015).…”

“…As the nonsymbolic number estimation task included both number and letter trials, we first compared participants' estimates on these trials types. Each participant's estimates from the number and letter trials in the behavioral nonsymbolic number estimation task were fitted using separate power functions in PsiMLE 1.0 (Odic et al, 2015). PsiMLE is an R-based package that uses a maximumlikelihood estimation approach to optimize the parameters of TA B L E 1 Arabic numerals, letters, and dot quantities used in each match and mismatch condition in the behavioral nonsymbolic estimation task and the symbolic integration EEG task Right panel: The first column represents the Arabic numeral as well as the dot quantities used to create the match trials.…”