Determining of an Estimate of the Equivalence Relation on the Basis of Pairwise Comparisons

2017 ◽  
pp. 92-101
Author(s):  
Leszek Klukowski
Keyword(s):  
Equivalence Relation ◽  
Pairwise Comparisons

Heteroscedastic Methods for Performing All Pairwise Comparisons of Regression Lines Associated With J Independent Groups

Methodology ◽  
2015 ◽  
Vol 11 (3) ◽  
pp. 110-115 ◽  
Author(s):  
Rand R. Wilcox ◽  
Jinxia Ma
Keyword(s):  
Least Squares ◽  
Pairwise Comparisons ◽  
Simple Extension ◽  
Type I ◽  
Type I Errors ◽  
Well Elderly ◽  
Using Data

Abstract. The paper compares methods that allow both within group and between group heteroscedasticity when performing all pairwise comparisons of the least squares lines associated with J independent groups. The methods are based on simple extension of results derived by Johansen (1980) and Welch (1938) in conjunction with the HC3 and HC4 estimators. The probability of one or more Type I errors is controlled using the improvement on the Bonferroni method derived by Hochberg (1988) . Results are illustrated using data from the Well Elderly 2 study, which motivated this paper.

scholarly journals Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 303
Author(s):  
Nikolai Krivulin
Keyword(s):  
Decision Problem ◽  
Optimization Problem ◽  
Pareto Frontier ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Optimal Solution ◽  
Algebraic Solution ◽  
Pairwise Comparisons ◽  
Logarithmic Scale ◽  
Idempotent Algebra

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the problem. We apply these results to the decision problem of interest and present illustrative examples.

scholarly journals The Effects of Toothbrush Wear on the Surface Roughness and Gloss of Resin Composites with Various Types of Matrices

2021 ◽  
Vol 9 (1) ◽  
pp. 8
Author(s):  
Murtadha AlAli ◽  
Nikolaos Silikas ◽  
Julian Satterthwaite
Keyword(s):  
Surface Roughness ◽  
Pairwise Comparisons ◽  
Resin Composites ◽  
Hybrid Resin ◽  
Before And After ◽  
Post Hoc ◽  
Main Effects ◽  
Toothbrush Wear

Objective: To evaluate and compare the surface roughness and gloss of a DMA-free composite and Bis-GMA-free composite with a DMA-based composite before and after toothbrushing simulation. Materials and Methods: Fifteen dimensionally standardised composite specimens of three nano-hybrid resin composites (Tetric EvoCeram, Admira Fusion, and Venus Diamond) were used. Five specimens from each composite were polished and then subjected to a toothbrushing simulator. Surface roughness (Ra) and gloss were measured before toothbrushing and after 5000, 10,000, 15,000, and 20,000 toothbrushing cycles. The data was analysed using 5 × 3 ANOVA to assess surface roughness and gloss values and pairwise comparisons in the form of Tukey post hoc tests were performed to interpret main effects. Results: For all tested materials, surface roughness increased, and gloss decreased after toothbrushing abrasion. Surface roughness (Ra) values ranged from 0.14 to 0.22 μm at baseline and increased to between 0.41 and 0.49 μm after 20,000 toothbrushing cycles. Gloss values ranged between 31.9 and 50.6 GU at baseline and between 5.1 and 19.5 GU after 20,000 toothbrushing cycles. The lowest initial Ra value was detected in Venus Diamond and the highest initial gloss value was detected in Tetric EvoCeram. Conclusions: Simulated toothbrushing abrasion led to an increase in surface roughness and a decrease in gloss for all tested materials. Venus Diamond had the smoothest surface and Tetric EvoCeram had the glossiest surface after polishing and following 20,000 cycles of toothbrushing abrasion. Admira Fusion demonstrated the roughest surface and had the lowest gloss values before and after toothbrushing abrasion.

scholarly journals A Numerical Comparison of the Sensitivity of the Geometric Mean Method, Eigenvalue Method, and Best–Worst Method

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 554
Author(s):  
Jiří Mazurek ◽  
Radomír Perzina ◽  
Jaroslav Ramík ◽  
David Bartl
Keyword(s):  
Research Method ◽  
Geometric Mean ◽  
Pairwise Comparisons ◽  
Numerical Comparison ◽  
Priority Vector ◽  
Eigenvalue Method ◽  
Fundamental Scale ◽  
Matrix Vector ◽  
Best Worst Method ◽  
Order Violation

In this paper, we compare three methods for deriving a priority vector in the theoretical framework of pairwise comparisons—the Geometric Mean Method (GMM), Eigenvalue Method (EVM) and Best–Worst Method (BWM)—with respect to two features: sensitivity and order violation. As the research method, we apply One-Factor-At-a-Time (OFAT) sensitivity analysis via Monte Carlo simulations; the number of compared objects ranges from 3 to 8, and the comparison scale coincides with Saaty’s fundamental scale from 1 to 9 with reciprocals. Our findings suggest that the BWM is, on average, significantly more sensitive statistically (and thus less robust) and more susceptible to order violation than the GMM and EVM for every examined matrix (vector) size, even after adjustment for the different numbers of pairwise comparisons required by each method. On the other hand, differences in sensitivity and order violation between the GMM and EMM were found to be mostly statistically insignificant.

Fundamental relations and identities of fuzzy hyperalgebras

2021 ◽  
pp. 1-10
Author(s):  
Narjes Firouzkouhi ◽  
Abbas Amini ◽  
Chun Cheng ◽  
Mehdi Soleymani ◽  
Bijan Davvaz
Keyword(s):  
Universal Algebra ◽  
Equivalence Relation ◽  
Polynomial Function ◽  
Equivalence Relations ◽  
Fundamental Relation ◽  
Term Function ◽  
Biological Phenomena ◽  
Definition Of

Inspired by fuzzy hyperalgebras and fuzzy polynomial function (term function), some homomorphism properties of fundamental relation on fuzzy hyperalgebras are conveyed. The obtained relations of fuzzy hyperalgebra are utilized for certain applications, i.e., biological phenomena and genetics along with some elucidatory examples presenting various aspects of fuzzy hyperalgebras. Then, by considering the definition of identities (weak and strong) as a class of fuzzy polynomial function, the smallest equivalence relation (fundamental relation) is obtained which is an important tool for fuzzy hyperalgebraic systems. Through the characterization of these equivalence relations of a fuzzy hyperalgebra, we assign the smallest equivalence relation α i 1 i 2 ∗ on a fuzzy hyperalgebra via identities where the factor hyperalgebra is a universal algebra. We extend and improve the identities on fuzzy hyperalgebras and characterize the smallest equivalence relation α J ∗ on the set of strong identities.

scholarly journals On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

scholarly journals Systematic Review and Network Meta-Analysis of Anaplastic Lymphoma Kinase (ALK) Inhibitors for Treatment-Naïve ALK-Positive Lung Cancer

Cancers ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1966
Author(s):  
Cheng-Hao Chuang ◽  
Hsiao-Ling Chen ◽  
Hsiu-Mei Chang ◽  
Yu-Chen Tsai ◽  
Kuan-Li Wu ◽  
...  
Keyword(s):  
Lung Cancer ◽  
Brain Metastasis ◽  
Anaplastic Lymphoma Kinase ◽  
Low Dose ◽  
Meta Analysis ◽  
Phase Iii ◽  
Pairwise Comparisons ◽  
Anaplastic Lymphoma ◽  
Treatment Naïve ◽  
Alk Positive

Several anaplastic lymphoma kinase inhibitors (ALKIs) have demonstrated excellent efficacy on overall survival (OS), progression-free survival (PFS), objective response rate (ORR), and also better adverse effect (AE) profiles compared to cytotoxic chemotherapy in advanced stage anaplastic lymphoma kinase (ALK) rearrangement-positive non-small cell lung cancer (NSCLC) in phase III randomized clinical trials (RCTs). We conducted this systematic review and network meta-analysis to provide a ranking of ALKIs for treatment-naïve ALK-positive patients in terms of PFS, ORR, and AEs. In addition, a sub-group analysis of treatment benefits in patients with baseline brain metastasis was also conducted. Contrast-based analysis was performed for multiple treatment comparisons with the restricted maximum likelihood approach. Treatment rank was estimated using the surface under the cumulative ranking curve (SUCRA), as well as the probability of being the best (Prbest) reference. All next-generation ALKIs were superior to crizotinib in PFS but lorlatinib and brigatinib had increased AEs. The probability of lorlatinib being ranked first among all treatment arms was highest (SUCRA = 93.3%, Prbest = 71.8%), although there were no significant differences in pairwise comparisons with high- (600 mg twice daily) and low- (300 mg twice daily) dose alectinib. In subgroup analysis of patients with baseline brain metastasis, low-dose alectinib had the best PFS (SUCRA = 87.3%, Prbest = 74.9%). Lorlatinib was associated with the best ranking for ORR (SUCRA = 90.3%, Prbest = 71.3%), although there were no significant differences in pairwise comparisons with the other ALKIs. In addition, low-dose alectinib had the best safety performance (SUCRA = 99.4%, Prbest = 97.9%). Lorlatinib and low-dose alectinib had the best PFS and ORR in the overall population and baseline brain metastasis subgroup, respectively. Low-dose alectinib had the lowest AE risk among the available ALKIs. Further head-to-head large-scale phase III RCTs are needed to verify our conclusions.

scholarly journals Multiple Granulation Rough Set Approach to Interval-Valued Intuitionistic Fuzzy Ordered Information Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 949
Author(s):  
Zhen Li ◽  
Xiaoyan Zhang
Keyword(s):  
Information Theory ◽  
Pattern Recognition ◽  
Information System ◽  
Information Systems ◽  
Rough Set ◽  
Fuzzy Set ◽  
Equivalence Relation ◽  
Target Concept ◽  
Actual Case ◽  
Interval Valued

As a further extension of the fuzzy set and the intuitive fuzzy set, the interval-valued intuitive fuzzy set (IIFS) is a more effective tool to deal with uncertain problems. However, the classical rough set is based on the equivalence relation, which do not apply to the IIFS. In this paper, we combine the IIFS with the ordered information system to obtain the interval-valued intuitive fuzzy ordered information system (IIFOIS). On this basis, three types of multiple granulation rough set models based on the dominance relation are established to effectively overcome the limitation mentioned above, which belongs to the interdisciplinary subject of information theory in mathematics and pattern recognition. First, for an IIFOIS, we put forward a multiple granulation rough set (MGRS) model from two completely symmetry positions, which are optimistic and pessimistic, respectively. Furthermore, we discuss the approximation representation and a few essential characteristics for the target concept, besides several significant rough measures about two kinds of MGRS symmetry models are discussed. Furthermore, a more general MGRS model named the generalized MGRS (GMGRS) model is proposed in an IIFOIS, and some important properties and rough measures are also investigated. Finally, the relationships and differences between the single granulation rough set and the three types of MGRS are discussed carefully by comparing the rough measures between them in an IIFOIS. In order to better utilize the theory to realistic problems, an actual case shows the methods of MGRS models in an IIFOIS is given in this paper.

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