Multinomial goodness-of-fit tests arise in a diversity of milieu. The long history of the problem has spawned a multitude of asymptotic tests. If the sample size relative to the number of categories is small, the accu...
详细信息
Multinomial goodness-of-fit tests arise in a diversity of milieu. The long history of the problem has spawned a multitude of asymptotic tests. If the sample size relative to the number of categories is small, the accuracy of these tests is compromised. In that case, an exact test is a prudent option. But such tests are computationally intensive and need efficient algorithms. This paper gives a conceptual overview, and empirical comparisons of two avenues, namely the network and fast Fourier transform (FFT) algorithms, for an exact goodness-of-fit test on a multinomial. We show that a recursive execution of a polynomial product forms the basis of both these approaches. Specific details to implement the network method, and techniques to enhance the efficiency of the FFT algorithm are given. Our empirical comparisons show that For exact analysis with the chi-square and likelihood ratio statistics, the network-cum-polynomial multiplication algorithm is the more efficient and accurate of the two.
An efficient algorithm for Fisher's exact test on unordered 2 x J contingency tables is proposed based on the network algorithm described by Mehta and Patel (1980, 1983). When either all or some of the column sums...
详细信息
An efficient algorithm for Fisher's exact test on unordered 2 x J contingency tables is proposed based on the network algorithm described by Mehta and Patel (1980, 1983). When either all or some of the column sums are equal, this method substantially reduces the computational effort needed to obtain the exact p-value. The principal computational efficiency is gained from the idea that the network size could be reduced by multiplying the probabilities of equivalent tables in which the first row entries are permutation of each other rather than creating them repeatedly and summing up their probabilities. The ranges of the nodes in the network and the ranges of the arcs emanating from each node to the nodes at the succeeding stages are redefined, so that the algorithm creates only the representatives of equivalent tables which are permutationally distinct without missing any possible table. An equation to calculate the numbers of equivalent tables is given. This method is also applicable to exact Pearson chi-squared and exact likelihood ratio tests. Some numerical examples are presented to compare the computational time of the improved algorithm with the original network algorithm. (C) 1997 Elsevier Science B.V.
We present a comparison of two efficient algorithms for exact analysis of an unordered 2 x K table. First, by considering conditional generating functions, we show that both the network algorithm of Mehta and Patel (J...
详细信息
We present a comparison of two efficient algorithms for exact analysis of an unordered 2 x K table. First, by considering conditional generating functions, we show that both the network algorithm of Mehta and Patel (J. Amer. Statist. Assoc. 78 (1983)) and the fast Fourier transform (FFT) algorithm of Baglivo et al. (J. Amer. Statist. Assoc. 82 (1992)) rest on the same foundation. This foundation is a recursive polynomial relation, We further show that the network algorithm is equivalent to a stage-wise implementation of this recursion while the FFT algorithm is based on performing the same recursion at complex roots of unity. Our empirical results for the Pearson X(2), likelihood ratio, and Freeman-Halton statistics show that the network algorithm, or equivalently, the recursive polynomial multiplication algorithm is superior to the FFT algorithm with respect to computing speed and accuracy.
Yao and Tritchler (1993, Biometrics 49, 233-236) proposed an exact test for conditional independence in a series of 2 x 2 tables, and presented a fast Fourier transform (FFT) algorithm to compute exact significance le...
详细信息
Yao and Tritchler (1993, Biometrics 49, 233-236) proposed an exact test for conditional independence in a series of 2 x 2 tables, and presented a fast Fourier transform (FFT) algorithm to compute exact significance levels for it. The purpose of this note is to provide a general perspective on computational methods applicable to this problem. We also assess the power and efficiency comparisons done by them.
The investigation of interaction in a series of 2 × 2 tables is warranted in a variety of research endeavors. Though many large-sample approaches for such investigations are available, the exact analysis of the p...
详细信息
We formulate the problem of exact inference for Kendall’s S and Spearman’s D algebraically, using a general recursion formula developed by Smid for the score S with ties in both rankings. Analogous recursion formula...
详细信息
The odds ratio is widely used as a measure of association in epidemiologic studies and clinical trials. We consider calculation of exact confidence limits for the common odds ratio in a series of independent 2 × ...
详细信息
The odds ratio is widely used as a measure of association in epidemiologic studies and clinical trials. We consider calculation of exact confidence limits for the common odds ratio in a series of independent 2 × 2 tables and propose three modifications of the network algorithm of Mehta, Patel and Gray: (1) formulating and dealing with the problem in algebraic instead of graph theoretic terms, (2) performing convolutions on the natural scale instead of the logarithmic scale, and (3) using the secant method instead of binary search to compute roots of polynomial equations. Enhancement of computational efficiency, exceeding an order of magnitude, afforded by these modifications is empirically demonstrated. We also compare the modified method with one based on the fast Fourier transform (FFT). Further, we show that the FFT method can also result in considerable loss of numerical accuracy. The modifications proposed in this article yield an algorithm that is not only fast and accurate but that combines conceptual simplicity with ease of implementation. Anyone with a rudimentary knowledge of computer programming can implement it and quickly compute exact confidence intervals for relatively large data sets even on microcomputers. Thus it should help make exact analysis of the common odds ratio more common.
This article proposes an efficient numerical algorithm for small-sample exact inferences in contingency tables having ordinal classifications. The inferences, which apply conditional on the observed marginal totals, a...
详细信息
This article proposes an efficient numerical algorithm for small-sample exact inferences in contingency tables having ordinal classifications. The inferences, which apply conditional on the observed marginal totals, also provide an exact analysis for the log-linear model of linear-by-linear association for cell probabilities. An exact test of independence has a one-sided P value equal to the null probability that model-based maximum likelihood estimates of odds ratios are at least as large as the observed estimates. The conditional nonnull distribution yields confidence intervals for odds ratios having a linear-by-linear structure. The computations utilize an extension of the network algorithm proposed by Mehta and Patel (1983). [ABSTRACT FROM AUTHOR]
Recently, Goldberg proposed a new approach to the maximum network flow problem. The approach yields a very simple algorithm running in O(n3)<span style="display: inline-bloc
Evolution algorithms for combinatorial optimization have been proposed in the 70's. They did not have a major influence. With the availability of parallel computers, these algorithms will become more important. In...
详细信息
Evolution algorithms for combinatorial optimization have been proposed in the 70's. They did not have a major influence. With the availability of parallel computers, these algorithms will become more important. In this paper we discuss the dynamics of three different classes of evolution algorithms: network algorithms derived from the replicator equation, Darwinian algorithms and genetic algorithms inheriting genetic information. We present a new genetic algorithm which relies on intelligent evolution of individuals. With this algorithm, we have computed the best solution of a famous travelling salesman problem. The algorithm is inherently parallel and shows a superlinear speedup in multiprocessor systems.
暂无评论