J. M. Tapia García

A consensus model for group decision making problems with linguistic interval fuzzy preference relations

Sometimes, we find decision situations in which it is difficult to express some preferences by means of concrete preference degrees. In this paper, we present a consensus model for group decision making problems in which the experts use linguistic interval fuzzy preference relations to represent their preferences. This model is based on two consensus criteria, a consensus measure and a proximity measure, and on the concept of coincidence among preferences. We compute both consensus criteria in the three representation levels of a preference relation and design an automatic feedback mechanism to guide experts in the consensus reaching process.

Comparing the asymptotic power of exact tests in 2×2 tables

Abstract A 2×2 table may arise from three types of sampling, depending on the number of previously fixed marginals, and may yield three possible, differing, probabilistic models. From the unconditional point of view each model requires a specific solution but, within each model, the calculation time increases as the test procedure chosen is more powerful and, between the models, the calculation time decreases in the number of marginals fixed. Moreover, each model yields a test which is generally more powerful than the test of any other model with a larger number of marginals fixed. The condition under which a less powerful test, of the same or a different model, can substitute a more powerful test with a loss of power lower than 2% is determined. It is concluded that the Fisher exact test can be used as an approximation to Barnards exact test for a table with 0 or 1 fixed marginals, when the sample size is ⩾100 or when the smaller sample size is ⩾80, respectively. Similarly, Barnards exact test for a table with 1 fixed marginal can be used as an approximation of the same test for a table with 0 fixed marginals, when the sample size is ⩾50.

On determining the P-value in 2 x 2 multinomial trials

Abstract One of the oldest problems in statistics is how to analyze a 2 x 2 table and it is still very much with us today. The case of zero fixed marginals, analyzed via the unconditional non-asymptotic method, is perhaps the one that has received least attention, owing to its computational complexity. This paper presents a critical review of existing publications on the topic, defines the conditions of equivalence, of symmetry, and of convexity to be satisfied by the points in the critical region (and takes advantage of the opportunity to generate, in an abbreviated manner, the sample space), defines two new criteria for ordering the elements in the sample space, analyzes the properties of symmetry of the error α (dependent on the two nuisance parameters) and their impact on the reduction of the parameter space to 1 4 (one-tailed test) or to 1 8 (two-tailed test) of its size by determining the P-value of an experimental table, shows how to minimize computations by grouping common elements, and finally, it gives various properties which allow the size of the parameter space to be reduced even more, and the number of times by which the size of the test must be determined. For purposes of comparison, it refers to the cases of one or two fixed marginals and to the assertions made by other authors.

Optimal Unconditional Asymptotic Test in 2 × 2 Multinomial Trials

Abstract The generally most powerful unconditional exact test for testing independence in a 2 × 2 multinomial trial is Barnards test, but at present it is impossible to apply in samples of even moderate size. Alternatively, it can be used as an asymptotic method. This paper evaluates 10 different approximations (as well as 48 others not included here) under the criterion that the optimal approximation is the one which produces a p-value that differs from the exact test by less than a given quantity. The best method is the one based on Pirie and Hamdans chi-squared statistics – = {|N 11 N 22 − N 12 N 21| − 0.5}2(N − 1)/{N 1• N 2• N •1 N •2} – which is valid, for ordinary significances and one(two)tailed-tests, if the minimum expected quantity E is larger than or equal to 2.5 (1.5). However, when the test is valid it may fail between 1% and 5% of times (the failures are sometimes liberal and others are conservative). If one wants to avoid failure, E will have to be rather more than 9. (Present computation capacity does not allow us to determine the exact amount.)

Analyzing Consensus Measures in Group Decision Making

Abstract In Group Decision Making (GDM) problems before to obtain a solution a high level of consensus among experts is required. Consensus measures are usually built using similarity functions measuring how close experts’ opinions or preferences are. Similarity functions are defined based on the use of a metric describing the distance between experts’ opinions or preferences. In the literature, die rent distance functions have been proposed to implement consensus measures. This paper presents analyzes the effect of the application of some die rent distance functions for measuring consensus in GDM. By using the nonparametric Wilcoxon matched-pairs signed-ranks test, it is concluded that die rent distance functions can produce significantly die rent results. Moreover, it is also shown that their application also has a significant elect on the speed of achieving consensus. Finally, these results are analyzed and used to derive decision support rules, based on a convergent criterion, that can be used to control the convergence speed of the consensus process using the compared distance functions.

Two-Tailed Unconditional Inferences on the Difference of Two Proportions in Cross-Sectional Studies

Unconditional inferences about the difference d = p 2 − p 1 between two independent proportions are usually carried out under the double binomial model. However, when the results proceed from a cross-sectional study, it is more appropriate (and exact) to carry them out under the multinomial model. These inferences have been developed in the case of tests and confidence intervals with one tail, but the case of two tails is customarily solved using the TOST (two one-sided test) method. In this article, the exact and asymptotic two-tailed inferences are developed using two real two-tailed tests. The program for carrying them out may be obtained without charge from: http://www.ugr.es/local/bioest/EQUIV_ASO.EXE.

Optimal unconditional critical regions for 2 2 2 multinomial trials

Analysing a 2 2 2 table is one of the most frequent problems in applied research (particularly in epidemiology). When the table arises from a 2 2 2 multinomial trial (or the case of double dichotomy), the appropriate test for independence is an unconditional one, like those of Barnard (1947), which, although they date from a long time ago, have not been developed (because of computational problems) until the last ten years. Among the different possible versions, the optimal (Martín Andrés & Tapia Garcia, 1999) is Barnards original one, but the calculation time (even today) is excessive. This paper offers critical region tables for that version, which behave well compared to those of Shuster (1992). The tables are of particular use for researchers wishing to obtain significant results for very small sample sizes (N h 50).