Set Of Contingency Tables Research Articles

Can interestingness measures be usefully visualized?

Abstract The paper presents visualization techniques for interestingness measures. The process of measure visualization provides useful insights into different domain areas of the visualized measures and thus effectively assists their comprehension and selection for different knowledge discovery tasks. Assuming a common domain form of the visualized measures, a set of contingency tables, which consists of all possible tables having the same total number of observations, is constructed. These originally four-dimensional data may be effectively represented in three dimensions using a tetrahedron-based barycentric coordinate system. At the same time, an additional, scalar function of the data (referred to as the operational function, e.g., any interestingness measure) may be rendered using colour. Throughout the paper a particular group of interestingness measures, known as confirmation measures, is used to demonstrate the capabilities of the visualization techniques. They cover a wide spectrum of possibilities, ranging from the determination of specific values (extremes, zeros, etc.) of a single measure, to the localization of pre-defined regions of interest, e.g., such domain areas for which two or more measures do not differ at all or differ the most.

The effectiveness of fraud detection instruments in not-for-profit organizations

Purpose – The purpose of this study is to investigate the effectiveness of fraud detection instruments in not-for-profit (NFP) organizations. Not-for-profit organizations rely on trust and volunteer support. They are often small in size and do not have relevant expertise to prevent fraud. Such organizations are more vulnerable to fraud and, consequently, require effective fraud detection instruments. The existing literature on fraud detection is primarily descriptive and does not measure instrument performance. The authors address this research gap and provide a detailed overview of the impact of nine common fraud detection instruments. Design/methodology/approach – Data were obtained from an NFP fraud survey conducted in Australia and New Zealand. A set of contingency tables is produced to explore the relationship between the existence of a specific fraud detection instrument and actual detection of fraud. We also investigate the relationship between organization size and fraud detection strategy. Findings – The findings provide valuable insights into understanding fraud detection mechanisms. Although most fraud detection measures may not lead to more fraud detection, three highly effective instruments emerge, namely, fraud control policies, whistle-blower policies and fraud risk registers. The results also reveal that commonly used fraud detection instruments are not necessarily the most effective. This is true in a significant number of small organizations that appear to be focusing on ineffective fraud detection instruments. Practical implications – Implementation of more effective fraud detection measures will reduce the damage caused to an organization and is highly relevant for practitioners. Originality/value – The results show that differences in the effectiveness of fraud detection instruments in the NFP sector exist. This knowledge is directly applicable by related organizations to reduce fraud damage.

Minimal invariant Markov basis for sampling contingency tables with fixed marginals

In this paper we define an invariant Markov basis for a connected Markov chain over the set of contingency tables with fixed marginals and derive some characterizations of minimality of the invariant basis. We also give a necessary and sufficient condition for uniqueness of minimal invariant Markov bases. By considering the invariance, Markov bases can be presented very concisely. As an example, we present minimal invariant Markov bases for all 2 × 2 × 2 × 2 hierarchical models. The invariance here refers to permutation of indices of each axis of the contingency tables. If the categories of each axis do not have any order relations among them, it is natural to consider the action of the symmetric group on each axis of the contingency table. A general algebraic algorithm for obtaining a Markov basis was given by Diaconis and Sturmfels (The Annals of Statistics, 26, 363–397, 1998). Their algorithm is based on computing Grobner basis of a well-specified polynomial ideal. However, the reduced Grobner basis depends on the particular term order and is not symmetric. Therefore, it is of interest to consider the properties of invariant Markov basis.

Rapidly Mixing Markov Chains for Sampling Contingency Tables with a Constant Number of Rows

We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. Cryan and Dyer [J. Comput. System Sci., 67 (2003), pp. 291-310] have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which employs Markov chain methods indirectly. They leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we show that the x 2 Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris [Random Walks in Convex Sets, Ph. D. thesis, Department of Statistics, University of California, Berkeley, CA, 2000] and Morris and Sinclair [SIAM J. Comput., 34 (2004), pp. 195-226] for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste [Ann. Appl. Probab., 3 (1993), pp. 696-730] to show that the 2 x 2 chain is also rapidly mixing.

Quantum random walks on the set of contingency tables

AbstractIn this article, we define the quantum version of this classic Metropolis walks on the set of contingency tables and get several theoretical conclusions with computer simulation for two‐way contingency tables with fixed‐row sums and column sums. Also, quantum random walks for three‐way contingency tables will be defined. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Describing heterogeneous effects in stratified ordinal contingency tables, with application to multi-center clinical trials

Standard models for a set of contingency tables with ordered response categories assume a common effect within or between tables, described by a certain type of odds ratio. In practice, heterogeneity usually occurs among such odds ratios, even if its extent is minor in magnitude. This article presents models that summarize the effect while simultaneously describing the degree of heterogeneity. For cases in which the levels of the stratification factor are a sample, such as many multi-center clinical trials, we recommend the use of random effects models. These treat the true stratum-specific ordinal log odds ratios as a sample with some unknown mean and standard deviation. For the random effects distribution, we consider both normality and a nonparametric approach. In using these more realistic models permitting heterogeneity, it can be more difficult to establish significance of effects because of the extra variability inherent in the model. The primary focus is three-way contingency tables with an ordinal response and a stratification factor, but we also briefly discuss models for describing heterogeneity within contingency tables.

Sampling contingency tables

We give polynomial time algorithms for random sampling from a set of contingency tables, which is the set of m×n matrices with given row and column sums, provided the row and column sums are sufficiently large with respect to m, n. We use this to approximately count the number of such matrices. These problems are of interest in Statistics and Combinatorics. © 1997 John Wiley & Sons, Inc. Random Struct. Alg.,10, 487–506, 1997