Exact Tests for Interaction in Several 2 × 2 Tables

Abstract

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 problem has been formulated for the probability statistic only. We present several alternative statistics applicable in this context. We also give an efficient polynomial multiplication algorithm to compute exact distributions and tail areas for the family of stratum-additive statistics. Besides the probability statistic, these include the score, likelihood ratio, and other statistics. In addition to comparing, in empirical terms, the diverse computational strategies for exact interaction analysis, we also explore the theoretical linkages between them. Data from published papers are used for illustration.     

Fast and Exact Simulation of Large Gaussian Lattice Systems in ℝ2: Exploring the Limits

The circulant embedding technique allows for the fast and exact simulation of stationary and intrinsically stationary Gaussian random fields. The method uses periodic embeddings and relies on the fast Fourier transform. However, exact simulations require that the periodic embedding is nonnegative definite, which is frequently not the case for two-dimensional simulations. This work considers a suggestion by Michael Stein, who proposed nonnegative definite periodic embeddings based on suitably modified, compactly supported covariance functions. Theoretical support is given to this proposal, and software for its implementation is provided. The method yields exact simulations of planar Gaussian lattice systems with 10⁶ and more lattice points for wide classes of processes, including those with powered exponential, Matérn, and Cauchy covariances.     

Exact Confidence Limits for the Parameter of an Exponential Distribution in the Accelerated Life Tests under Type-ⅠCensoring

Life data frequently arise in many reliability studies,such as accelerated life tests studies.This paper considers the part of life data where failure and censoring observations may exist.To develop statistical methods and theory for the analysis of these data,a new approach was proposed to obtain the exact lower and upper confidence limits for the mean life of the exponential distribution with Type-I censoring data.It is assumed that the acceleration factor is a random variable,and that the distribution of the acceleration factor is known from some empirical information or the meta analysis.A method for constructing the lower and upper confidence limits for the parameter based on an ordering relation among the sample space was proposed.Simulation studies and analyses of two examples suggest that the proposed method performed well.     

Exact Inference for Kendall's S and Spearman's ρ with Extension to Fisher's Exact Test in r × c Contingency Tables

Abstract

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 formulas are shown to hold for the score D as well as for a log transform, F, of the score used in Fisher's exact test of independence in contingency tables. A new implementation of Mehta and Patel's network algorithm is then applied to obtain exact significance levels of either S or D for observations from both continuous and discrete distributions. A simple extension is made to obtain Fisher's exact test in r x c contingency tables. Observed CPU times for contingency table problems four to six of Mehta and Patel and problems four and five of Clarkson, Fan, and Joe are roughly 2/3 of those obtained using Clarkson's et al. implementation of the network algorithm. It is shown that a hierarchy, with F > S > D, holds regarding the rate of aggregation. An algorithm for rapid lexicographic enumeration of entries in a frequency table is also given.     

Computation of the commutator of 2 × 2 matrices via five multiplications

Algorithms for computing the commutator AB ? BA of 2 × 2 matrices A and B are proposed that involve five multiplications.     

Fast Computation of ζ(3) and of Some Special Integrals Using the Ramanujan Formula and Polylogarithms

The application of the Fast E-function Evaluation (FEE) method to fast calculation of the value of (3) and of some special integrals on the basis of the Ramanujan formula and its generalization is considered.This revised version was published online in October 2005 with corrections to the Cover Date.     

A note on the ranks of 2 × 2 × 2 and 2 × 2 × 2 × 2 tensors

Using rank-1 reduction formula and the vector space spanned by the real rank-1 matrices, we present a different way to show that the maximum possible rank of the 2?×?2?×?2 tensors over the real field is 3. Following, we obtain that the maximum rank of the 2?×?2?×?2?×?2 tensors over the real field is less than or equal to 5 and propose another way to show that the maximum rank of the 2?×?2?×?2?×?2 tensors over the complex field is 4, except one special case.     

Computation of the solutions of the Fokker–Planck equation for one and two DOF systems

Uncertainty in structures may come from unknowns in the modelisation and in the properties of the materials, from variability with time, external noise, etc. This leads to uncertainty in the dynamic response. Moreover, the consequences are issues in safety, reliability, efficiency, etc. of the structure. So an issue is the gain of information on the response of the system taking into account the uncertainties [Mace BR, Worden K, Manson G. Uncertainty in structural dynamics. J Sound Vib 2005;288(3):423–9].If the forcing or the uncertainty can be modelled through a white noise, the Fokker–Planck (or Kolmogorov forward) equation exists. It is a partial differential linear equation with unknown p(X, t), where p(X, t) is the probability density function of the state X at time t.In this article, we solve this equation using the finite differences method, for one and two DOF systems. The numerical solutions obtained are proved to be nearly correct.     

Defining Relations for the Algebra of Invariants of 2×2 Matrices

We obtain defining relations of the algebra of invariants of the classical subgroups of GL ₂(C) acting by simultaneous conjugation on m-tuples of 2×2 complex matrices. The sets of defining relations look uniformly for all m2 and are derived by translation of classical results on invariant theory of orthogonal groups in the language of 2×2 matrix invariants, combined with arguments of representation theory of the general linear group GL _m (C) and ideas coming from the theory of algebras with polynomial identities.     

On the Best Polynomial Approximations of 2π-Periodic Functions and Exact Values of n-Widths of Functional Classes in the Space L 2

To solve extremal problems of approximation theory in the space L ₂, we use -moduli introduced by Ivanov. We determine the exact values of constants in Jackson-type inequalities and the exact values of n-widths of functional classes determined by these moduli.     

Exact Jackson–Stechkin-type inequalities for 2π-periodic functions in L 2 and widths of some classes of functions

We consider the problem of finding exact inequalities for the best approximations of periodic differentiable functions by trigonometric polynomials and the m-order moduli of continuity in the space L ₂ and present their applications. For some classes of functions defined by the indicated moduli of continuity, we calculate the exact values of n-widths in the space L ₂.     

Expression for the Group Inverse of the 2 × 2 Blo ck Matrix

In this paper, we present a method how to get the expression for the group inverse of 2 × 2 block matrix and get the explicit expressions of the block matrix ?A C ? BD under some conditions.     

Bounds for the Extreme Eigenvalues of Block 2 × 2 Hermitian Matrices

Let an n × n Hermitian matrix A be presented in block 2 × 2 form as , where A₁₂ ≠ 0, and assume that the diagonal blocks A₁₁ and A₂₂ are positive definite. Under these assumptions, it is proved that the extreme eigenvalues of A satisfy the bounds

On stability of closedness and self-adjointness for 2 × 2 operator matrices

Mathematical Notes - Consider an operator which is defined in Banach or Hilbert space X = X 1 × X 2 by the matrix $L = left( {begin{array}{*{20}{c}}A&;amp;B C&;amp;D end{array}}...     

Choosing the equilibrium behavior in an infinitely repeated 2 × 2 game for conservative and innovative gamers

We examine an infinitely repeated 2 × 2 game in which each gamer in every next round chooses his own pure strategy based on the results of a random experiment; the random experiment is generated by gamer’s own arbitrary mixed strategy, which with high probability (but generally different from 1) gives the gamer a choice of strategy appropriate to his type of behavior: conservative or innovative.