Statistical Equivalence Theory And Its Application In Hypothesis Testing Of General Processing Tree Models

With the development of cognitive measurement technology,cognitive psychometric models have become important parts of cognitive psychology researches,which foster the development of cognitive psychometric theory and make cognitive psychometric more accurate.Multinomial processing tree(MPT)models are substantively motivated mathematical stochastic models for categorical data,typically representing responses of participants in cognitive tasks.The MPT model,which is also a cognitive psychometrics model for human potential cognitive processes,is an effective statistical model for modeling and analyzing potential cognitive processes based on the development of generated logic.At present,the MPT model has been successfully applied in cognitive psychology,cognitive neurology,and other fields.From a structural viewpoint,MPT models can be split into binary MPT(BMPT)models and multi-link MPT(MMPT)models.Many scholars have conducted extended studies on the representation form,parameter types,individual differences and other aspects of the model,which is collectively called general processing tree(GPT)models.Hypothesis testing of the GPT model is to test the significance of some differences in latent cognitive abilities and is implemented through the parametric constraints of the model.Although many scholars have studied it,there are still some problems to consider.Firstly,the parametric constraint types and re-parameterization process of the GPT model need to be systematically and deeply discussed.Although the equality and order constraints of the GPT model have been discussed in previous studies,their breadth and depth still need to be further explored,that is,the types of the parametric constraints of the GPT model are not systematically studied in breadth,and the statistical equivalence and common characteristics of the re-parameterization process in depth need to be further explored.Secondly,the statistical equivalence of hypothesis testing of the GPT model has yet to be studied,especially the MMPT model.The existing method of hypothesis testing of the MMPT model is to equivalently convert to the BMPT model,but this usually leads to the structure of the equivalent model being too complex and many model parameters losing their original meanings.Finally,the uniqueness of the string encoding in the GPT model and the string processing of hypothesis testing need to be solved.Currently,except for the simple equality and order constraints of GPT models,which can be automatically implemented by a computer,all other parameter constraints need to be manually constructed by the modeler.The main reason is the lack of an operational model encoding and decoding algorithm,so the unique problem of string encoding and decoding of GPT models also needs to be explored.Therefore,this paper focuses on the statistical equivalence theory of hypothesis testing of GPT models and their application.The main research contents are as following:First,this paper summarizes and integrates the basic concepts of GPT models,such as four elements,parametric categorical model,mathematical equivalence,statistical equivalence,split transformation,and gives a formal description,which provides a formal specification for the discussion of subsequent research in this paper.At the same time,the identifiability of the GPT model,the uniqueness of the model equation and the construction theorem of statistical equivalence are considered.Secondly,parametric constraint types of GPT models are systematically discussed,and four representative tree models are proposed based on the common features of parametric constraint re-parameterization.In this paper,parametric constraint types of GPT hypothesis testing are systematically discussed from three dimensions,namely,the parametric constraint relation,the relationship between the constraint parameter vectors and the number of constraint parameters.And the re-parameterization process of each parameter constraint type and its recursive characteristics is discussed respectively.According to these shared characteristics,four representative tree models for parametric constraints of the GPT model are proposed.The recursive nesting of the representative tree model can facilitate the equivalent transformation of parameter constraints of the GPT model,and can provide a growing toolbox for statistical analysis of the GPT model.Thirdly,the parametric constraint representation theorem and the equivalent transformation process of the statistical closure of the GPT model are reached.Depending on various types of GPT model parametric constraint and its parametric statistics equivalent transformation process,the formal definition and representation theorem of parameter constraints are given.The ideas of statistical equivalent transformation(order constraints→product constraints→constant constraints→no constraint)and the basic steps of the hypothesis testing of GPT models are concluded.Then the GPT model with parametric constraints is statistically equivalent to the GPT model without no constraints.This shows that MMPT models have statistical closure under some simple parametric constraints.Fourthly,a new encoding and decoding algorithm of string language of GPT models is proposed,and the recursive definition,encoding and decoding rules of GPT models are given,as well as the subtree discriminant theorem and node tree discriminant theorem of string language of GPT models are proof.The new algorithm not only makes the GPT model encoding and decoding strings unique but also makes the string language capture the whole family of the GPT model well.According to the recursive characteristics of re-parameterization processing of parametric constraints of the GPT model,a recursive substitution rule is deduced based on the string words of the representative tree models.The new substitution rule can realize the statistical equivalent transformation of the parameter constraints of GPT models by recursive substitution of string words.In this paper,the original algorithm of the GPT model not only expanded the GPT model string coding theory,is advantageous to the GPT model of computer programming,storage and transport.In addition,modular coding of representative tree models also provides feasibility and theoretical support for computer automatic implementation of parametric constraints of the GPT model and enriches the theory and technology of GPT model statistical analysis.Finally,the feasibility and practicability of the study are verified by the GPT model analysis of three psychological research examples,namely,,,and the cognitive measurement of four mixed operations.In the fisrst instance,through the anslysis of the source monitoting of the picture superiority effect shows equivalent processing of equality and order constraints of the GPT model within the experiment group.Example 2 is age differences in source monitoring analysis for literary texts and shows the equivalent process of hypothesis test of GPT models within the same experiment group and between experiment groups.At the same time,in Example 1 and Example 2,a novel hypothesis test is proposed on the basis of verifying the conclusions of existing studies.In Example 3,the GPT model is used to measure the cognitive ability of the four mixed operations of the primary school students,and the implementation process of string encoding in the GPT model is also demonstrated.The results of the model analysis show that the parameter constraint reparameterization process of GPT models is not only statistically equivalent,but also can obtain the estimated value and the confidence interval of the quantitative index of the order constraint.It follows that the original analysis and specific statistical measurements of the researcher can be supported and enhanced by the GPT model,or at least can be used as a supplement to traditional empirical measurements.To sum up,this paper systematically studied the statistical theory and application of GPT models,that is,we discuss the parametric constraint types of the GPT model and its parameterization process and summarize the four representative tree models of parametric constraints equivalent transformation,propose a new string encoding and decoding algorithm of GPT models,and the recursive substitution rules based on of the string word module of the representative tree model is summarized.These studies provide theoretical support for computer transmission,coding of the GPT model and automatic execution of hypothesis testing of the GPT model,expand the statistical analysis theory and statistical modeling technology of the GPT model,deepen the application and popularization of the GPT model,and enrich the model analysis theory of cognitive psychometric model.