多项式加工树模型次序约束的量化分析及其应用补充材料

此补充材料是与研究文献“多项式加工树模型次序约束的量化分析及其应用”（以下简称本研究）相关的形式化定义、形式化结论和实例源代码及操作步骤。该补充材料的目的主要有两个，其一是为了本研究理论的完整性和规范性，在本补充材料中给出了相关定义和结论的形式化描述(A1-A3)；其二是为了使作者能够理解次序约束的量化过程能够重复获得本研究实例的结果，在补充材料A4中给出相关的源代码和操作过程。该补充材料主要分四部分，其分别为A1：MPT模型的形式化定义；A2：结论1 MPT模型参数向量间两参数次序约束的量化分析；A3：结论2 MPT模型参数向量内两参数次序约束的量化分析；A4：MPT模型实例分析的源代码、分析过程及原始结果。其主要内容如下：首先，在本研究MPT模型定义的文字描述的基础上，在此补充材料中给出了MPT模型的形式化定义，以便于更加规范准确地描述模型的定义（A1）。其次，针对我们归纳出的MPT模型次序约束量化分析结论，为了表明结论的准确性和严谨性，在这里分别给出了这两个结论的形式化表述和推导，即MPT模型参数向量间两参数次序约束量化分析（A2 结论1）、MPT模型参数向量内两参数次序约束量化分析（A3 结论2）。需要注意的是，虽然形式化表述更加准确、简洁，但含有较多的符号，相对有些抽象，对纯心理学背景的同仁来说可能不够直观。为此，我们在正文中以文字和实例的形式对相关结论进行了表述，而在此处将形式化表述作为正文的一个理论补充。形式化结果及其推导以在线补充材料的形式提供给大家，以便使感兴趣的读者能够更准确地把握本研究的理论结果。最后，为了让读者能够重复本研究实例的结果，我们在A4中补充了相关的模型分析软件GPT.exe、程序源代码、分析结果原始代码、计算过程和相关操作步骤概述。 感谢您的阅读，希望本补充材料能够为您理解本研究并重复其结果提供有效帮助，同时希望本研究能够起到抛砖引玉的作用。如有需要或疑问请联系我们。

This supplementary material contains formal definitions, formal conclusions, sample source code, and operational procedures related to the research paper Quantitative Analysis of Order Constraints in Polynomial Processing Tree Models and Their Applications (hereinafter referred to as this study). The two primary objectives of this supplementary material are: first, to enhance the completeness and standardization of the theory underlying this study by providing formal descriptions of relevant definitions and conclusions (A1-A3) within this document; second, to enable readers to replicate the results of the case studies in this study by understanding the quantitative process of order constraints, with relevant source code and operational procedures presented in Supplementary Material A4. This supplementary material is divided into four main sections: A1: Formal Definition of the MPT Model; A2: Quantitative Analysis of Two-Parameter Order Constraints Between MPT Model Parameter Vectors (Conclusion 1); A3: Quantitative Analysis of Two-Parameter Order Constraints Within MPT Model Parameter Vectors (Conclusion 2); A4: Source Code, Analysis Process, and Original Results for Case Analysis of the MPT Model. First, based on the textual description of the MPT model definition in this study, this supplementary material provides the formal definition of the MPT model to enable a more standardized and accurate characterization of the model (A1). Second, for the quantitative analysis conclusions of MPT model order constraints summarized by our team, formal expressions and derivations of these two conclusions are provided herein to demonstrate the accuracy and rigor of the findings, specifically the quantitative analysis of two-parameter order constraints between MPT model parameter vectors (A2, Conclusion 1) and the quantitative analysis of two-parameter order constraints within MPT model parameter vectors (A3, Conclusion 2). It should be noted that while formal expressions are more accurate and concise, they contain a large number of symbols and are relatively abstract, which may not be sufficiently intuitive for colleagues with a pure psychology background. To address this, we have presented the relevant conclusions in text and example formats within the main article, while the formal expressions are provided here as a theoretical supplement to the main text. Formal results and their derivations are made available as online supplementary materials to allow interested readers to more precisely grasp the theoretical findings of this study. Finally, to enable readers to replicate the results of the case studies in this study, we have supplemented relevant model analysis software GPT.exe, program source code, original analysis result codes, calculation procedures, and an overview of relevant operational steps in A4. Thank you for your reading. We hope that this supplementary material will provide effective assistance for you to understand this study and replicate its results, and we also hope that this study will serve as a modest opening to inspire further research. Please contact us if you have any needs or questions.