Evaluating the functional form of perceptual learning with trial-by-trial analysis

Abstract

The functional form of the learning curve is one of the fundamental characterizations of perceptual learning. Dosher & Lu (2007) compared a number of functional forms for their ability to fit learning curves (reduced contrast thresholds in an orientation identification task) estimated from blocks of trials with a staircase procedure, and concluded that a single exponential function fit best. Recently, we showed that learning curves estimated from blocks of trials in staircase procedures are imprecise and may be biased, especially in fast learning situations (Zhang et al, 2019). A more detailed evaluation of the functional form of perceptual learning with trial-by-trial data is necessary. In this study, we developed a generative model in which the threshold in each trial is determined by the learning curve generated with a candidate functional form, the probability of a correct response reflects the trial-specific psychometric function, with the predicted response drawn from the Bernoulli distribution. The quality of fit was computed as the sum of the loglikelihood across the entire learning curve. Five candidate models (exponential, power, Apex, summed exponentials, cascade exponentials) were fit to the published experimental data from three perceptual learning tasks: contrast detection (n=41; Zhang et al., 2018), Vernier offset discrimination (n=16; Zhang et al., 2018), and orientation identification (n=78; Liu, Dosher and Lu, 2010, 2012), using the Bayesian information criterion (BIC, Schwarz, 1978) for model selection. We found the preferences in pairwise comparisons are: (1) exponential 65.93%, power 5.19%, no preference 28.89%; (2) exponential 96.30%, Apex 3.70%; (3) exponential 98.52%, summed exponentials 1.48%; and (4) exponential 83.70%, cascade exponentials 14.81%, no preference 1.48%. In most cases, a single exponential function provided the best account of the learning curve in perceptual learning, implying a constant relative rate of learning.