Performance metrics for an application-driven selection and optimization of psychophysical sampling procedures

Arbitrarily Accurate Analytical Approximations for the Error Function

A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be improved by utilizing the approximation erf(x)≈1 for |x|>xo and with xo optimally chosen. Two generalizations are possible; the first is based on demarcating the integration interval into m equally spaced subintervals. The second, is based on utilizing a larger fixed subinterval, with a known integral, and a smaller subinterval whose integral is to be approximated. Both generalizations lead to significantly improved accuracy. Furthermore, the initial approximations, and those arising from the first generalization, can be utilized as inputs to a custom dynamic system to establish approximations with better convergence properties. Indicative results include those of a fourth-order approximation, based on four subintervals, which leads to a relative error bound of 1.43 × 10−7 over the interval [0, ∞]. The corresponding sixteenth-order approximation achieves a relative error bound of 2.01 × 10−19. Various approximations that achieve the set relative error bounds of 10−4, 10−6, 10−10, and 10−16, over [0, ∞], are specified. Applications include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for exp(−x2) that have significantly higher convergence properties than a Taylor series approximation. Fourth, the definition of a complementary demarcation function eC(x) that satisfies the constraint eC2(x)+erf2(x)=1. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to an error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal that is modeled by the error function.

Enhancing simulations with intra-subject variability for improved psychophysical assessments

Psychometric properties of perceptual assessments, like reliability, depend on stochastic properties of psychophysical sampling procedures resulting in method variability, as well as inter- and intra-subject variability. Method variability is commonly minimized by optimizing sampling procedures through computer simulations. Inter-subject variability is inherent to the population of interest and cannot be influenced. Intra-subject variability introduced by confounds (e.g., inattention or lack of motivation) cannot be simply quantified from experimental data, as these data also include method variability. Therefore, this aspect is generally neglected when developing assessments. Yet, comparing method variability and intra-subject variability could give insights on whether effort should be invested in optimizing the sampling procedure, or in addressing potential confounds instead. We propose a new approach to estimate intra-subject variability of psychometric functions by combining computer simulations and behavioral data, and to account for it when simulating experiments. The approach was illustrated in a real-world scenario of proprioceptive difference threshold assessments. The behavioral study revealed a test-retest reliability of r = 0.212. Computer simulations without considering intra-subject variability predicted a reliability of r = 0.768, whereas the new approach including an intra-subject variability model lead to a realistic estimate of reliability (r = 0.207). Such a model also allows computing the theoretically maximally attainable reliability (r = 0.552) assuming an ideal sampling procedure. Comparing the reliability estimates when exclusively accounting for method variability versus intra-subject variability reveals that intra-subject variability should be reduced by addressing confounds and that only optimizing the sampling procedure may be insufficient to achieve a high reliability. This new approach allows computing the intra-subject variability with only two measurements per subject, and predicting the reliability for a larger number of subjects and retests based on simulations, without requiring additional experiments. Such a tool of predictive value is especially valuable for target populations where time is scarce, e.g., for assessments in clinical settings.