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Nondecomposable Item Response Theory models: Fundamental measurement in psychometrics 不可分解项目反应理论模型:心理测量学的基本测量方法
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102772
Vithor Rosa Franco , Jacob Arie Laros , Marie Wiberg
{"title":"Nondecomposable Item Response Theory models: Fundamental measurement in psychometrics","authors":"Vithor Rosa Franco ,&nbsp;Jacob Arie Laros ,&nbsp;Marie Wiberg","doi":"10.1016/j.jmp.2023.102772","DOIUrl":"10.1016/j.jmp.2023.102772","url":null,"abstract":"<div><p>The main aim of the current paper is to propose Item Response Theory (IRT) models derived from the nondecomposable measurement theories presented in Fishburn (1974). More specifically, we aim to: (i) present the theoretical basis of the Rasch model<span> and its relations to psychophysics’ models of utility; (ii) give a brief exposition on the measurement theories presented in Fishburn (1974, 1975), some of which do not require an additive structure; and (iii) derive IRT models from these measurement theories, as well as Bayesian implementations of these models. We also present two empirical examples to compare how well these IRT models fit to real data. In addition to deriving new IRT models, we also discuss theoretical interpretations regarding the models’ capability of generating fundamental measures of the true scores of the respondents. The manuscript ends with prospects for future studies and practical implications.</span></p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"114 ","pages":"Article 102772"},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48197512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Geometrical properties of a generalized cone and its 2D image 广义锥及其二维图像的几何性质
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102765
Tadamasa Sawada , Zygmunt Pizlo
{"title":"Geometrical properties of a generalized cone and its 2D image","authors":"Tadamasa Sawada ,&nbsp;Zygmunt Pizlo","doi":"10.1016/j.jmp.2023.102765","DOIUrl":"10.1016/j.jmp.2023.102765","url":null,"abstract":"<div><p>The generalized cone is a simple 3D shape that is produced by sweeping a planar cross-section along a curve. Many complex and articulated 3D objects can be represented by combining generalized cones. It has been shown that generalized cones play an important role in our visual system for perceiving the shapes of these objects and recognizing them. In this study, we analyzed the geometrical properties of generalized cones and their 2D images and found that there are invariant features in the images of the generalized cones under both 2D orthographic and perspective projections that facilitate the recovery of the 3D shapes of the cones from the images. We found that the 3D translational symmetry of generalized cones can be analyzed using tools designed for 3D mirror-symmetry.</p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"114 ","pages":"Article 102765"},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42893529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Adjacencies on random ordering polytopes and flow polytopes 随机排序多面体和流动多面体上的邻接关系
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102768
Jean-Paul Doignon , Kota Saito
{"title":"Adjacencies on random ordering polytopes and flow polytopes","authors":"Jean-Paul Doignon ,&nbsp;Kota Saito","doi":"10.1016/j.jmp.2023.102768","DOIUrl":"10.1016/j.jmp.2023.102768","url":null,"abstract":"<div><p><span><span>The Multiple Choice Polytope (MCP) is the prediction range of a random utility model due to Block and Marschak(1960). Fishburn(1998) offers a nice survey of the findings on random utility models at the time. A complete characterization of the MCP is a remarkable achievement of Falmagne (1978). To derive a more enlightening proof of Falmagne Theorem, Fiorini(2004) assimilates the MCP with the flow polytope of some acyclic network. However, apart from a recognition of the facets by Suck(2002), the </span>geometric structure of the MCP was apparently not much investigated. We characterize the adjacency of vertices and the adjacency of facets. Our characterization of the edges of the MCP helps understand recent findings in economics papers such as Chang, Narita and Saito(2022) and Turansick(2022). Moreover, our results on adjacencies also hold for the flow polytope of any acyclic network. In particular, they apply not only to the MCP, but also to three polytopes which Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter (2018) introduced as extended formulations of the </span>weak order polytope, interval order polytope and semiorder polytope (the prediction ranges of other models, see for instance Fishburn and Falmagne, 1989, and Marley and Regenwetter, 2017).</p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"114 ","pages":"Article 102768"},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48214488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Stochastic choice with bounded processing capacity 具有有限处理能力的随机选择
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102771
Thierry Marchant , Arunava Sen
{"title":"Stochastic choice with bounded processing capacity","authors":"Thierry Marchant ,&nbsp;Arunava Sen","doi":"10.1016/j.jmp.2023.102771","DOIUrl":"10.1016/j.jmp.2023.102771","url":null,"abstract":"<div><p>We propose and characterize a class of stochastic decision functions for a decision-maker who has a capacity for processing at most <span><math><mi>k</mi></math></span>-alternatives at a time. When faced with a menu containing more than <span><math><mi>k</mi></math></span> alternatives, she randomly chooses a sub-menu of size <span><math><mi>k</mi></math></span> with uniform probability and selects the best alternative according to a strict ordering <span><math><mo>≻</mo></math></span>. For smaller menus, she chooses the best alternative according to <span><math><mo>≻</mo></math></span>.</p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"114 ","pages":"Article 102771"},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45439330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Comparison of type I error and statistical power between state trace analysis and analysis of variance 状态跟踪分析与方差分析的I型误差和统计能力比较
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102767
Wei Liu , Yu-Xue Jia
{"title":"Comparison of type I error and statistical power between state trace analysis and analysis of variance","authors":"Wei Liu ,&nbsp;Yu-Xue Jia","doi":"10.1016/j.jmp.2023.102767","DOIUrl":"10.1016/j.jmp.2023.102767","url":null,"abstract":"<div><p>State-Trace Analysis (STA) is a methodology for investigating the number of latent variables. Recently, a quantitative STA technique based on conjoint monotonic regression and double bootstrap method (STA-CMR) has been proposed. More discussion is needed on the type I error and the statistical power of this technique, as it adopts null hypothesis significance testing (NHST) to draw statistical inference. Because the results of STA are comparable with analysis of variance (ANOVA) in a three-factor experiment with linearity assumption, it is necessary to compare STA-CMR with ANOVA accordingly. This study investigated the type I error and the statistical power of STA-CMR and ANOVA in specific linear and nonlinear models using simulated data. Results demonstrated that both techniques were effective in the linear models, where ANOVA had a greater statistical power and STA-CMR had a more rigorous control of type I error. In the nonlinear models, although STA-CMR worked just as well as in the linear models, ANOVA completely lost its effectiveness. Besides, we found that the estimated type I error rate of STA-CMR was always smaller than the preset significance level in both linear and non-linear models. We suggest that the suppressed type I error rate may be caused by the bootstrap procedure, but the exact causes need further investigation. In conclusion, despite the suppressed type I error rate, STA-CMR can be a useful tool for determining the number of latent variables, particularly in non-linear models.</p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"114 ","pages":"Article 102767"},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47806081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The best-worst-choice polytope on four alternatives 四种选择上的最佳最差选择多晶体
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102769
Jean-Paul Doignon
{"title":"The best-worst-choice polytope on four alternatives","authors":"Jean-Paul Doignon","doi":"10.1016/j.jmp.2023.102769","DOIUrl":"10.1016/j.jmp.2023.102769","url":null,"abstract":"<div><p>Several papers co-authored by A.A.J. Marley helped in popularizing the best-worst-choice paradigm due to Finn and Louviere (1992). Inspired by Block and Marschak (1960), Marley conceived a random utility model for the choice frequencies of the best and worst alternatives in any proposed set of alternatives (Marley and Louviere, 2005). He then asked for a characterization of the prediction range of the model. The range being a convex polytope, an affine description of this polytope would provide a solution to Marley problem. For four alternatives, we show that a minimal such description consists in 26 affine equalities and 144 affine inequalities. The result derives from the Gale transform of the set of polytope vertices: the transform being a family of 24 vectors in a one-dimensional vector space, it plainly reveals the affine structure of the polytope. As far as we know, Marley problem is still open when the number of alternatives exceeds 4.</p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"114 ","pages":"Article 102769"},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46556700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Length of the state trace: A method for partitioning model complexity 状态跟踪长度：一种划分模型复杂性的方法
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-04-01 DOI: 10.1016/j.jmp.2023.102755
F. Gregory Ashby
{"title":"Length of the state trace: A method for partitioning model complexity","authors":"F. Gregory Ashby","doi":"10.1016/j.jmp.2023.102755","DOIUrl":"10.1016/j.jmp.2023.102755","url":null,"abstract":"<div><p>A novel and easy-to-compute measure is proposed that compares the relative contribution of each parameter of a mathematical model to the model’s mathematical flexibility or complexity, with respect to accounting for the results of some specific experiment. When the data space is a two-dimensional plot of the type used in standard state-trace analysis, then the model complexity contributed by a single parameter equals the length of the state trace (LOST) that results when that parameter is varied and all other parameters are held constant. For the normal, equal-variance, signal-detection model, the average LOST when the response-criterion parameter <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> is varied is about four times greater than the average LOST when the sensitivity parameter <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is varied. As a result, applying the signal-detection model to random data almost always leads to the conclusion that all the points share the same value of <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> but were generated under different values of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span>. Parameters that have non-monotonic effects on performance, such as the attention-weight parameter that is used in popular exemplar and prototype models of categorization, tend to have large LOSTs, and therefore contribute to model flexibility more than parameters that have monotonic effects on performance. Comparing LOSTs for exemplar and prototype models also leads to some deep new insights into the structure of both models.</p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"113 ","pages":"Article 102755"},"PeriodicalIF":1.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43730681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stochastic additive differences 随机加性差异
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-04-01 DOI: 10.1016/j.jmp.2022.102744
Yutaka Nakamura
{"title":"Stochastic additive differences","authors":"Yutaka Nakamura","doi":"10.1016/j.jmp.2022.102744","DOIUrl":"https://doi.org/10.1016/j.jmp.2022.102744","url":null,"abstract":"<div><p><span>Properties of a binary choice probability function </span><span><math><mi>p</mi></math></span> defined on multiattributed outcomes are studied to represent <span><math><mi>p</mi></math></span><span> as a transformation of additive difference evaluations of chosen and unchosen outcomes into the unit interval. We use an algebraic assumption to obtain an additive difference representation, but allow for restricting strict increasingness of the transformation to the subset of the domain on which transformed values are strictly between 0 and 1. We also apply a topological assumption to axiomatize the cases of homogeneous product sets in the context of finite-state decision making under uncertainty.</span></p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"113 ","pages":"Article 102744"},"PeriodicalIF":1.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49868432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the correspondence between granular polytomous spaces and polytomous surmising functions 关于粒度多胞空间与多胞猜测函数的对应关系
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-04-01 DOI: 10.1016/j.jmp.2022.102743
Xun Ge
{"title":"On the correspondence between granular polytomous spaces and polytomous surmising functions","authors":"Xun Ge","doi":"10.1016/j.jmp.2022.102743","DOIUrl":"10.1016/j.jmp.2022.102743","url":null,"abstract":"<div><p>By modifying the concept of polytomous surmise functions, this paper introduces polytomous surmising functions. Then, it is shown that there is a one-to-one correspondence <em>f</em> between granular polytomous spaces and polytomous surmising functions where polytomous surmising functions cannot be replaced with polytomous surmise functions. This result gives a correction for a correspondence between granular polytomous spaces and polytomous surmise functions. As an application of the correspondence <em>f</em>, this paper demonstrates that the pair <span><math><mrow><mo>(</mo><mi>f</mi><mo>,</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></math></span><span> of mappings forms a Galois connection where all granular polytomous spaces and all polytomous surmising functions are closed elements of this Galois connection.</span></p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"113 ","pages":"Article 102743"},"PeriodicalIF":1.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46005176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Semiorders and continuous Scott–Suppes representations. Debreu’s Open Gap Lemma with a threshold 半序和连续scott - supes表示。带阈值的德布鲁开隙引理
IF 1.8 4区 心理学
Journal of Mathematical Psychology Pub Date : 2023-04-01 DOI: 10.1016/j.jmp.2023.102754
A. Estevan
{"title":"Semiorders and continuous Scott–Suppes representations. Debreu’s Open Gap Lemma with a threshold","authors":"A. Estevan","doi":"10.1016/j.jmp.2023.102754","DOIUrl":"https://doi.org/10.1016/j.jmp.2023.102754","url":null,"abstract":"<div><p>The problem of finding a utility function for a semiorder has been studied since 1956, when the notion of semiorder was introduced by Luce. But few results on continuity and no result like Debreu’s Open Gap Lemma, but for semiorders, was found. In the present paper, we characterize semiorders that accept a continuous representation (in the sense of Scott–Suppes). Two weaker theorems are also proved, which provide a programmable approach to Open Gap Lemma, yield a Debreu’s Lemma for semiorders, and enable us to remove the open-closed and closed-open gaps of a set of reals while keeping the threshold.</p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"113 ","pages":"Article 102754"},"PeriodicalIF":1.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49868433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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