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Zhong-Lin Lu, Pan Zhang, Yukai Zhao, Ge Chen, Jia Yang, Chang-Bing Huang, Barbara Dosher; Hypothesis testing of the parameters of the learning curve based on hierarchical Bayesian. Journal of Vision 2020;20(11):1637. doi: https://doi.org/10.1167/jov.20.11.1637.
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Hypothesis testing about parameters of fitted models has been challenged by potential covariance between them. Here, we adopted the hierarchical Bayesian (HB) framework to model the parameter covariance in learning curves in perceptual learning, and test hypotheses about individual parameters for different conditions. Specifically, we analyzed data from Zhang et al (2018), who studied the effects of monetary reward on perceptual learning, comparing high, subliminal, block, low, or no reward. In that study, a two-parameter power function (initial threshold and learning rate) was fit to the learning curve of each observer. However, the covariance between parameters for initial thresholds and learning rates made it difficult to compare them across conditions. In this study, we constructed a hierarchical Bayesian model, in which five joint hyperparameter distributions of initial threshold and learning rate, one for each reward condition, were specified. The parameters of individual observers in each reward condition are drawn the corresponding hyper-distribution(s). The maximum likelihood was used to compute the distributions of the hyper parameters and parameters of each observer. We found that initial threshold and learning rate were positively correlated (ranging from 0.116 to 0.570). After controlling the covariance, ANOVA revealed that the initial thresholds and learning rates were both significantly different across the five reward conditions (all p=0). Post hoc multiple comparison found that the initial threshold was comparable among the high, no and low reward conditions, and between the no and block reward conditions. However, the learning rate was significantly different (α= 0.05) between all pairs except between the high and subliminal reward conditions. The method developed in this study can be generalized to conduct hypothesis testing about parameters of fitted models.
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