Spatial interaction between two or more classes or species has important implications in various fields and causes multivariate patterns such as segregation or association. Segregation occurs when members of a class or species are more likely to be found near members of the same class or conspecifics; while association occurs when members of a class or species are more likely to be found near members of another class or species. The null patterns considered are random labeling (RL) and complete spatial randomness (CSR) of points from two or more classes, which is called \emph{CSR independence}, henceforth. The clustering tests based on nearest neighbor contingency tables (NNCTs) that are in use in literature are two-sided tests. In this article, we consider the directional (i.e., one-sided) versions of the cell-specific NNCT-tests and introduce new directional NNCT-tests for the two-class case. We analyze the distributional properties; compare the empirical significant levels and empirical power estimates of the tests using extensive Monte Carlo simulations. We demonstrate that the new directional tests have comparable performance with the currently available NNCT-tests in terms of empirical size and power. We use four example data sets for illustrative purposes and provide guidelines for using these NNCT-tests.
E-variables are tools for retaining type-I error guarantee with optional stopping. We extend E-variables for sequential two-sample tests to general null hypotheses and anytime-valid confidence sequences. We provide implementations for estimating risk difference, relative risk and odds-ratios in contingency tables.
There are two popular general approaches for the analysis and visualization of a contingency table and a compositional data set: Correspondence analysis (CA) and log ratio analysis (LRA). LRA includes two independently well developed methods: association models and compositional data analysis. The application of either CA or LRA to a contingency table or to compositional data set includes a preprocessing centering step. In CA the centering step is multiplicative, while in LRA it is log bi-additive. A preprocessed matrix is double-centered, so it is a residuel matrix; which implies that it affe…
The analysis of contingency tables is a powerful statistical tool used in experiments with categorical variables. This study improves parts of the theory underlying the use of contingency tables. Specifically, the linkage disequilibrium parameter as a measure of two-way interactions applied to three-way tables makes it possible to quantify Simpsons paradox by a simple formula. With tests on three-way interactions, there is only one that determines whether the partial interactions of all variables agree or whether there is at least one variable whose partial interactions disagree. To date, ther…
Exact tests greatly improve the analysis of contingency tables when marginals are low. For instance, researchers often use Fisher's exact test, which is conditional, or Barnard's test, which is unconditional but needs to deal with a nuisance parameter. Here, we describe the m-test, an exact, unconditional test for the study of d x m binomial contingency tables. When comparing binomial trials, the m-test is related to Barnard's exact test. However, the nuisance parameter is integrated over all its possible values, instead of maximized or otherwise estimated. According to Monte Carlo simulations…
We obtain sharp asymptotic estimates on the number of $n \times n$ contingency tables with two linear margins $Cn$ and $BCn$. The results imply a second order phase transition on the number of such contingency tables, with a critical value at \ts $B_{c}:=1 + \sqrt{1+1/C}$. As a consequence, for \ts $B>B_{c}$, we prove that the classical \emph{independence heuristic} leads to a large undercounting.
Contingency tables are useful objects in statistics for representing 2-way data. With fixed row and column sums, and a total of $n$ entries, contingency tables correspond to parabolic double cosets of $S_n$. The uniform distribution on $S_n$ induces the Fisher-Yates distribution, classical for its use in the chi-squared test for independence. A Markov chain on $S_n$ can then induce a random process on the space of contingency tables through the double cosets correspondence. The random transpositions Markov chain on $S_n$ induces a natural `swap' Markov chain on the space of contingency tables;…
It is an important inferential problem to test no association between two binary variables based on data. Tests based on the sample odds ratio are commonly used. We bring in a competing test based on the Pearson correlation coefficient. In particular, the Odds ratio does not extend to higher order contingency tables, whereas Pearson correlation does. It is important to understand how Pearson correlation stacks against the odds ratio in 2x2 tables. Another measure of association is the canonical correlation. In this paper, we examine how competitive Pearson correlation is vis-à-vis odds ratio i…
We assessed several agreement coefficients applied in 2x2 contingency tables, which are commonly applied in research due to dicotomization by the conditions of the subjects (e.g., male or female) or by conveniency of the classification (e.g., traditional thresholds leading to separations in healthy or diseased, exposed or non-exposed, etc.). More extreme table configurations (e.g., high agreement between raters) are also usual, but some of the coefficients have problems with imbalanced tables. Here, we not only studied some especific estimators, but also developed a general method to the study…
We consider the asymptotic distribution of a cell in a 2 x ... x 2 contingency table as the fixed marginal totals tend to infinity. The asymptotic order of the cell variance is derived and a useful diagnostic is given for determining whether the cell has a Poisson limit or a Gaussian limit. There are three forms of Poisson convergence. The exact form is shown to be determined by the growth rates of the two smallest marginal totals. The results are generalized to contingency tables with arbitrary sizes and are further complemented with concrete examples.
We consider the lattice, $\mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(\cdot)$, on $\mathcal{L}$. We derive from the supermodularity of $n(\cdot)$ some generalized Fréchet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $n(\cdot)$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also appl…
