Abstract: Multinomial processing tree (MPT) models, a family of substantive models for categorical data, are used to measure latent cognitive capacities underlying human behavior and test relevant psychological assumptions (Batchelder, 2017). MPT models assume that certain latent cognitive processes are serial in nature and represent these processes in terms of branching trees, with the parameters being the conditional link probabilities from one stage to another. Thus, MPT models can measure latent cognitive process by calculating the value of latent parameters. They verify the difference of cognitive process through hypothesis testing of potential parameters, and overcome the defect of confusing cognitive process with traditional cognitive measurement. The MPT modeling method has recently been applied in many fields such as cognitive psychology, cognitive neuroscience, game theory, and social psychology (Batchelder, 2017). From a structural viewpoint, MPT models can be divided into binary MPT (BMPT) models and multi-link MPT (MMPT) models (Batchelder, 2017). The class of BMPT models is characterized by binary links at nonterminal nodes, in which each link is associated with a parameter. However, the class of MMPT models contains at least a multi-link nonterminal node, which is associated with a multi-dimensional parameter vector. Thus, MMPT models have more complex parametric constraints than BMPT models. The hypothesis testing of the MPT model is realized through its latent parametric constraints. The main type of parametric constraints of MPT models are constant constraints, equality constraints and order constraints. The order constraints of MPT models refer to the inequality relationship between latent parameters. Knapp and Batchelder (2004) researched the reparameterization method of the order constraints of BMPT models. This paper generalizes the reparameterization method of order constraints of BMPT models to that of MMPT model and further proposes the quantitative analysis method of the MPT model with order constraints between two parameters. In order to analyze the order constraint within the framework of MPT model, the MPT model with order constraints is statistically equivalent to an MPT model without parametric constraints. Batchelder (2017) explained that both MPT models are statistical equivalent when their parameter spaces satisfy the definition of the bijection. In the paper, two different theorems of statistical equivalence of MPT models are obtained under order constraints between two parameter vectors and within a parameter vector. In order to deal with the quantitative analysis of the order constraints of the MPT model, we firstly split the constraint parameters from the parameter vectors containing the constraint parameters by implementing the split transformation. Then, by reconstructing bijective functions, the MPT model with order constraints is statistically equivalent to the resulting MPT model with no constraints. By analyzing the resulting model, we can get the estimate and confidence interval of the quantitative index of order constraints. Finally, we applied the quantitative analysis method to the published data of the source monitoring with three sources by Batchelder et al. (1994). The results show that the quantitative method of order constraints can not only test the order relationship of latent cognitive parameters of MPT models, but also obtain the quantitative index of order constraints. In a word, the analysis method in the study not only ensures that the order constraints of MPT models can be implemented within the framework of the MPT model, but also obtains the estimate and confidence interval of the quantitative index of order constraints of MPT models. Therefore, this method provides a more meaningful explanation for the potential cognitive measurement, extends the mathematical analysis of MPT model class, and expands the equivalent transformation theory of order constraint of MPT model. In addition, quantitative analysis of multi-parameter order constraints of MPT models are also a direction of our future research.

Key words: multinomial processing tree model, hypothesis testing, order constraints, quantitative analysis

摘要： 多项式加工树（MPT）模型是一种认知测量模型，能够对潜在认知过程进行测量和检验。已有研究探讨了二链MPT模型次序约束的重新参数化问题，本研究探讨了MPT模型次序约束的量化分析方法并从二链推广到多链，同时归纳出MPT模型参数向量内和参数向量间两参数次序约束量化分析的结论。数据分析结果表明该方法不仅在MPT模型框架下验证了潜在参数次序关系，而且给出了约束的量化指标，为潜在认知测量提供更有意义的解释。

关键词: 多项式加工树模型 假设检验 次序约束 量化分析