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Link functions and random effects structures are the two important components in building flexible regression models for dependent ordinal data. The power link functions include the commonly used links as special cases but have an additional skewness parameter, making the probability response curves adaptive to the data structure. It overcomes the arbitrary symmetry assumption imposed by the commonly used logistic or probit links as well as the fixed skewness in the complementary log–log or log–log links. By employing Gaussian processes, the regression model can incorporate various dependence structures in the data, such as temporal and spatial correlations. The full Bayesian estimation of the proposed model is conveniently implemented through RStan. Extensive simulation studies are carried out for discussion in model computation, parameterization, and evaluation in terms of estimation bias and overall model performance. The proposed model is applied to the PM2.5 data in Beijing and the Berberis thunbergii abundance data in New England. The results suggest that the proposed model leads to important improvement in estimation and prediction in modeling dependent ordinal response data. [ABSTRACT FROM AUTHOR]