Bayesian adaptive estimation of threshold versus contrast external noise functions: the quick TvC method

External noise paradigms, measuring contrast threshold as a function of external noise contrast (the "TvC" function), provide a valuable tool for studying perceptual mechanisms. However, measuring TvC functions at the multiple performance criteria needed to constrain observer models has previously involved demanding data collection (often>2000 trials). To ease this task, we developed a novel Bayesian adaptive procedure, the "quick TvC" or "qTvC" method, to rapidly estimate multiple TvC functions, by adapting a strategy originally developed to estimate psychometric threshold and slope [Cobo-Lewis, A. B. (1996). An adaptive method for estimating multiple parameters of a psychometric function. Journal of Mathematical Psychology, 40, 353-354; Kontsevich, L. L., and Tyler, C. W. (1999). Bayesian adaptive estimation of psychometric slope and threshold. Vision Research, 39(16), 2729-2737]. Exploiting the regularities observed in empirical TvC functions, the qTvC method estimates three parameters: the optimal threshold c(0), the critical noise level N(c), and the common slope, eta, of log-parallel psychometric functions across external noise conditions. Before each trial, the qTvC uses a one-step-ahead search to select the stimulus (jointly defined by signal and noise contrast) that minimizes the expected entropy of the three-dimensional posterior probability distribution, p(N(c),c(0),eta). The method's accuracy and precision, for estimating TvC functions at three performance criteria (65%, 79%, and 92% correct), were evaluated using Monte-Carlo simulations and a psychophysical task. Simulations showed that less than 300 trials were needed to estimate TvC functions at three widely separated criteria with good accuracy (bias<5%) and precision (mean root mean square error <1.5 dB). Using an orientation identification task, we found excellent agreement (weighted r(2)>.95) between TvC estimates obtained with the qTvC and the method of constant stimuli, although the qTvC used only 12% of the data collection (240 vs 1920 trials). The qTvC may hold considerable practical value for applying the external noise method to study mechanisms of observer state changes and special populations. We suggest that the same adaptive strategy can be applied to directly estimate other classical functions, such as the contrast sensitivity function, elliptical equi-discrimination contours, and sensory memory decay functions.