The stability of several measures of association in small contingency tables

Measures of association are often used to describe the relationship between row and column variables in two—dimensional contingency tables. It is not uncommon in biomedical research to categorize continuous variables to obtain a two—dimensional table. In these situations it is desirable that the measure of association not be too sensitive to changes in the number of categories or to the choice of cut points. To accomplish this objective we attempt to find a measure of association that closely approximates the corresponding measure of association for the underlying distribution.Measures that are close to the underlying measure for various table sizes andcutpoints are called stable measures.     

A class of statistics based on the information concept

The methodic use of Shannon's entropy as a basic concept, complementing probability, leads to a new class of statistics which provides, inter alia, a measure of mutual dissimilarity y between several frequency distributions. Application to contin-gency tables with any number of dimensions yields a dimension-less, standardised contingency coefficient which depends on the direction of inference and will combine multiplicatively with the number of observed events. This class of statistics further in-cludes a continuous modification W of the number of degrees of freedom in a table, and a measure Q of its overall information content. Numerical illustrations and comparisons with former re-sults are worked out. Direct applications include the optimal partition of a quasicontinuum into cells by maximizing Q, the ordering of unordered tables by minimising local values of y, and a tentative absolute weighting of inductive inference based on the minimal necessary shift, required by an hypothesis, between the actually observed data and a set of assumed future events.     

Three centuries of categorical data analysis: Log-linear models and maximum likelihood estimation

The common view of the history of contingency tables is that it begins in 1900 with the work of Pearson and Yule, but in fact it extends back at least into the 19th century. Moreover, it remains an active area of research today. In this paper we give an overview of this history focussing on the development of log-linear models and their estimation via the method of maximum likelihood. Roy played a crucial role in this development with two papers co-authored with his students, Mitra and Marvin Kastenbaum, at roughly the mid-point temporally in this development. Then we describe a problem that eluded Roy and his students, that of the implications of sampling zeros for the existence of maximum likelihood estimates for log-linear models. Understanding the problem of non-existence is crucial to the analysis of large sparse contingency tables. We introduce some relevant results from the application of algebraic geometry to the study of this statistical problem.     

Bayesian analysis of quasi-life tables

Quasi-life tables, in which the data arise from many concurrent, independent, discrete-time renewal processes, were defined by Baxter (1994, Biometrika 81:567–577), who outlined some methods for estimation. The processes are not observed individually; only the total numbers of renewals at each time point are observed. Crowder and Stephens (2003, Lifetime Data Anal 9:345–355) implemented a formal estimating-equation approach that invokes large-sample theory. However, these asymptotic methods fail to yield sensible estimates for smaller samples. In this paper, we implement a Bayesian analysis based on MCMC computation that works equally well for large and small sample sizes. We give three simulated examples, studying the Bayesian results, the impact of changing prior specification, and empirical properties of the Bayesian estimators of the lifetime distribution parameters. We also study the Baxter (1994, Biometrika 81:567–577) data, and uncover structure that has not been commented upon previously.     

Efficient and flexible model-based clustering of jumps in diffusion processes

Jump–diffusion processes involving diffusion processes with discontinuous movements, called jumps, are widely used to model time-series data that commonly exhibit discontinuity in their sample paths. The existing jump–diffusion models have been recently extended to multivariate time-series data. The models are, however, still limited by a single parametric jump-size distribution that is common across different subjects. Such strong parametric assumptions for the shape and structure of a jump-size distribution may be too restrictive and unrealistic for multiple subjects with different characteristics. This paper thus proposes an efficient Bayesian nonparametric method to flexibly model a jump-size distribution while borrowing information across subjects in a clustering procedure using a nested Dirichlet process. For efficient posterior computation, a partially collapsed Gibbs sampler is devised to fit the proposed model. The proposed methodology is illustrated through a simulation study and an application to daily stock price data for companies in the S&P 100 index from June 2007 to June 2017.     

Minimal Basis for a Connected Markov Chain over 3 × 3 ×K Contingency Tables with Fixed Two-Dimensional Marginals

This paper considers a connected Markov chain for sampling 3 × 3 ×K contingency tables having fixed two‐dimensional marginal totals. Such sampling arises in performing various tests of the hypothesis of no three‐factor interactions. A Markov chain algorithm is a valuable tool for evaluating P‐values, especially for sparse datasets where large‐sample theory does not work well. To construct a connected Markov chain over high‐dimensional contingency tables with fixed marginals, algebraic algorithms have been proposed. These algorithms involve computations in polynomial rings using Gröbner bases. However, algorithms based on Gröbner bases do not incorporate symmetry among variables and are very time‐consuming when the contingency tables are large. We construct a minimal basis for a connected Markov chain over 3 × 3 ×K contingency tables. The minimal basis is unique. Some numerical examples illustrate the practicality of our algorithms.     

Optimal correction for continuity in the chi-squared test in 2×2 tables

The most common asymptotic procedure for analyzing a 2 × 2 table (under the conditioning principle) is the ‰ chi-squared test with correction for continuity (c.f.c). According to the way this is applied, up to the present four methods have been obtained: one for one-tailed tests (Yates') and three for two-tailed tests (those of Mantel, Conover and Haber). In this paper two further methods are defined (one for each case), the 6 resulting methods are grouped in families, their individual behaviour studied and the optimal is selected. The conclusions are established on the assumption that the method studied is applied indiscriminately (without being subjected to validity conditions), and taking a basis of 400,000 tables (with the values of sample size n between 20 and 300 and exact P-values between 1% and 10%) and a criterion of evaluation based on the percentage of times in which the approximate P-value differs from the exact (Fisher's exact test) by an excessive amount. The optimal c.f.c. depends on n, on E (the minimum quantity expected) and on the error α to be used, but the rule of selection is not complicated and the new methods proposed are frequently selected. In the paper we also study what occurs when E ≥ 5, as well as whether the chi-squared by factor (n-1).     

The minimum discrimination information approach in analyzing categorical data

A brief review of the minimum discrimination information (MDI) approach in analyzing categorical data is presented in a question -answer format, An example is given to bring out situations in which the MDI approach is more useful. No new results are proved.