Optimal unconditional critical regions for 2 2 2 multinomial trials

Analysing a 2 2 2 table is one of the most frequent problems in applied research (particularly in epidemiology). When the table arises from a 2 2 2 multinomial trial (or the case of double dichotomy), the appropriate test for independence is an unconditional one, like those of Barnard (1947), which, although they date from a long time ago, have not been developed (because of computational problems) until the last ten years. Among the different possible versions, the optimal (Martín Andrés & Tapia Garcia, 1999) is Barnard's original one, but the calculation time (even today) is excessive. This paper offers critical region tables for that version, which behave well compared to those of Shuster (1992). The tables are of particular use for researchers wishing to obtain significant results for very small sample sizes (N h 50).     

Exact Tests for 2 × 2 Contingency Tables

Fisher's exact test, difference in proportions, log odds ratio, Pearson's chi-squared, and likelihood ratio are compared as test statistics for testing independence of two dichotomous factors when the associated p values are computed by using the conditional distribution given the marginals. The statistics listed above that can be used for a one-sided alternative give identical p values. For a two-sided alternative, many of the above statistics lead to different p values. The p values are shown to differ only by which tables in the opposite tail from the observed table are considered more extreme than the observed table.     

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).     

On the equivalence of some test criteria based on BAN estimators for the multivariate exponential family

For the general multivariate exponential family of distributions it is shown that Rao's test criterion based on efficient scores is algebraically identical to the general chi-squared criterion based on maximum likelihood estimates and, similarly, that the Wald statistic is algebraically identical to the general minimum modified chi-squared statistic using linearization; these results are valid also for the multisample versions. Thus, these are extensions to the general exponential family of the findings due to Silvey (1970) and Bhapkar (1966), respectively, for the special case of the multinomial family.It is also shown that the general forms of the chi-squared and modified chi-squared criteria reduce to their respective well-known forms for the multivariate symmetric power series distribution. This finding is, thus, an extension of results noted by Ferguson (1958) and Clickner (1976) for the special case of the multinomial distribution.     

A bivariate F distribution with marginals on arbitrary numerator and denominator degrees of freedom, and related bivariate beta and t distributions

The classical bivariate F distribution arises from ratios of chi-squared random variables with common denominators. A consequent disadvantage is that its univariate F marginal distributions have one degree of freedom parameter in common. In this paper, we add a further independent chi-squared random variable to the denominator of one of the ratios and explore the extended bivariate F distribution, with marginals on arbitrary degrees of freedom, that results. Transformations linking F, beta and skew t distributions are then applied componentwise to produce bivariate beta and skew t distributions which also afford marginal (beta and skew t) distributions with arbitrary parameter values. We explore a variety of properties of these distributions and give an example of a potential application of the bivariate beta distribution in Bayesian analysis.     

Preserving confidentiality of high-dimensional tabulated data: Statistical and computational issues

Dissemination of information derived from large contingency tables formed from confidential data is a major responsibility of statistical agencies. In this paper we present solutions to several computational and algorithmic problems that arise in the dissemination of cross-tabulations (marginal sub-tables) from a single underlying table. These include data structures that exploit sparsity to support efficient computation of marginals and algorithms such as iterative proportional fitting, as well as a generalized form of the shuttle algorithm that computes sharp bounds on (small, confidentiality threatening) cells in the full table from arbitrary sets of released marginals. We give examples illustrating the techniques.     

A powerful and interpretable alternative to the Jarque–Bera test of normality based on 2nd-power skewness and kurtosis,using the Rao's score test on the APD family

We introduce the 2nd-power skewness and kurtosis, which are interesting alternatives to the classical Pearson's skewness and kurtosis, called 3rd-power skewness and 4th-power kurtosis in our terminology. We use the sample 2nd-power skewness and kurtosis to build a powerful test of normality. This test can also be derived as Rao's score test on the asymmetric power distribution, which combines the large range of exponential tail behavior provided by the exponential power distribution family with various levels of asymmetry. We find that our test statistic is asymptotically chi-squared distributed. We also propose a modified test statistic, for which we show numerically that the distribution can be approximated for finite sample sizes with very high precision by a chi-square. Similarly, we propose a directional test based on sample 2nd-power kurtosis only, for the situations where the true distribution is known to be symmetric. Our tests are very similar in spirit to the famous Jarque–Bera test, and as such are also locally optimal. They offer the same nice interpretation, with in addition the gold standard power of the regression and correlation tests. An extensive empirical power analysis is performed, which shows that our tests are among the most powerful normality tests. Our test is implemented in an R package called PoweR.     

Behaviour of skewness, kurtosis and normality tests in long memory data

We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ${\sqrt{n}}We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ?n{\sqrt{n}}-consistent nor asymptotically normal. The normalizations needed to obtain the limiting distributions depend on the long memory parameter d. A direct consequence is that if data are long memory then testing normality with the Jarque–Bera test by using the chi-squared critical values is not valid. Therefore, statistical inference based on skewness, kurtosis, and the Jarque–Bera normality test, needs a rescaling of the corresponding statistics and computing new critical values of their nonstandard limiting distributions.     

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.     

New critical region tables for Fisher's exact test

The best-known non-asymptotic method for comparing two independent proportions is Fisher's exact text. The usual critical region (CR) tables for this test contain one or more of the following defects:they distinguish between rows and columns; they distinguish between the alternatives H = p1 < p2 and H = p1 > p2; they assume that the error for the two-tailed test is twice that of the one-tailed test; they do not use the optimal version of the test; they do not give both CRs for one and two tails at the same time. All this results in the unnecessary duplication of the space required for the tables, the construction of tables of low-powered methods, or the need to manipulate two different tables (one for the one-tailed test, the other for the two-tailed test). This paper presents CR tables which have been obtained from the most powerful version of Fisher's exact test and which occupy the minimum space possible. The tables, which are valid for one- or two-tailed tests, have levels of significance of 10%, 5% and 1% and values for N (the total size of both samples) of less than or equal to 40. This article shows how to calculate the P value in a specific problem, using the tables as a means of partial checking and as a preliminary step to determining the exact P value.     

Minimum power-divergence estimator in three-way contingency tables

Cressie et al. (2000; 2003) introduced and studied a new family of statistics, based on the φ-divergence measure, for solving the problem of testing a nested sequence of loglinear models. In that family of test statistics the parameters are estimated using the minimum φ-divergence estimator which is a generalization of the maximum likelihood estimator. In this paper we study the minimum power-divergence estimator (the most important family of minimum φ-divergence estimator) for a nested sequence of loglinear models in three-way contingency tables under assumptions of multinomial sampling. A simulation study illustrates that the minimum chi-squared estimator is simultaneously the most robust and efficient estimator among the family of the minimum power-divergence estimator.     

Independence in Contingency Tables Using Simplicial Geometry

Frequently, contingency tables are generated in a multinomial sampling. Multinomial probabilities are then organized in a table assigning probabilities to each cell. A probability table can be viewed as an element in the simplex. The Aitchison geometry of the simplex identifies independent probability tables as a linear subspace. An important consequence is that, given a probability table, the nearest independent table is obtained by orthogonal projection onto the independent subspace. The nearest independent table is identified as that obtained by the product of geometric marginals, which do not coincide with the standard marginals, except in the independent case. The original probability table is decomposed into orthogonal tables, the independent and the interaction tables. The underlying model is log-linear, and a procedure to test independence of a contingency table, based on a multinomial simulation, is developed. Its performance is studied on an illustrative example.     

A pooled Bayes test of independence for sparse contingency tables from small areas

We study the association between bone mineral density (BMD) and body mass index (BMI) when contingency tables are constructed from the several U.S. counties, where BMD has three levels (normal, osteopenia and osteoporosis) and BMI has four levels (underweight, normal, overweight and obese). We use the Bayes factor (posterior odds divided by prior odds or equivalently the ratio of the marginal likelihoods) to construct the new test. Like the chi-squared test and Fisher's exact test, we have a direct Bayes test which is a standard test using data from each county. In our main contribution, for each county techniques of small area estimation are used to borrow strength across counties and a pooled test of independence of BMD and BMI is obtained using a hierarchical Bayesian model. Our pooled Bayes test is computed by performing a Monte Carlo integration using random samples rather than Gibbs samples. We have seen important differences among the pooled Bayes test, direct Bayes test and the Cressie-Read test that allows for some degree of sparseness, when the degree of evidence against independence is studied. As expected, we also found that the direct Bayes test is sensitive to the prior specifications but the pooled Bayes test is not so sensitive. Moreover, the pooled Bayes test has competitive power properties, and it is superior when the cell counts are small to moderate.     

On the effect of long-range dependence on extreme value copula estimation with fixed marginals

ABSTRACT

We establish the existence of multivariate stationary processes with arbitrary marginal copula distributions and long-range dependence. The effect of long-range dependence on extreme value copula estimation is illustrated in the case of known marginals, by deriving functional limit theorems for a standard non parametric estimator of the Pickands dependence function and related parametric projection estimators. The asymptotic properties turn out to be very different from the case of iid or short-range dependent observations. Simulated and real data examples illustrate the results.     

Independence in multi-way contingency tables: S.N. Roy's breakthroughs and later developments

In the mid-1950s S.N. Roy and his students contributed two landmark articles to the contingency table literature [Roy, S.N., Kastenbaum, M.A., 1956. On the hypothesis of no “interaction” in a multiway contingency table. Ann. Math. Statist. 27, 749–757; Roy, S.N., Mitra, S.K., 1956. An introduction to some nonparametric generalizations of analysis of variance and multivariate analysis. Biometrika 43, 361–376]. The first article generalized concepts of interaction from 2×2×2$2\times 2\times 2$ contingency tables to three-way tables of arbitrary size and to larger tables. In the second article, which is the source of our primary focus, various notions of independence were clarified for three-way contingency tables, Roy's union–intersection test was applied to construct chi-squared tests of hypotheses about the structure of such tables, and the chi-squared statistics were shown not to depend on the distinction between response and explanatory variables. This work pre-dates by many years later developments that expressed such results in the context of loglinear models. It pre-dates by a quarter century the development of graphical models. We summarize the main results in these key articles and discuss the connection between them and the later developments of loglinear modeling and of graphical modeling. We also mention ways in which these later developments have themselves been further generalized.     

A note on dependence modeling for Bernoulli variables

Understanding and modeling multivariate dependence structures depending upon the direction are challenging but an interest of theoretical and applied researchers. In this paper, we propose a characterization of tables generated by Bernoulli variables through the uniformization of the marginals and refer to them as Q-type tables. The idea is similar to the copulas. This approach helps to see the dependence structure clearly by eliminating the effect of the marginals that have nothing to do with the dependence structure. We define and study conditional and unconditional Q-type tables and provide various applications for them. The limitations of existing approaches such as Cochran-Mantel-Haenszel pooled odds ratio are discussed, and a new one that stems naturally from our approach is introduced.     

A comparison of some conditional and unconditional exact tests for 2x2 contingency tables

In 1935, R.A. Fisher published his well-known “exact” test for 2x2 contingency tables. This test is based on the conditional distribution of a cell entry when the rows and columns marginal totals are held fixed. Tocher (1950) and Lehmann (1959) showed that Fisher s test, when supplemented by randomization, is uniformly most powerful among all the unbiased tests UMPU). However, since all the practical tests for 2x2 tables are nonrandomized - and therefore biased the UMPU test is not necessarily more powerful than other tests of the same or lower size. Inthis work, the two-sided Fisher exact test and the UMPU test are compared with six nonrandomized unconditional exact tests with respect to their power. In both the two-binomial and double dichotomy models, the UMPU test is often less powerful than some of the unconditional tests of the same (or even lower) size. Thus, the assertion that the Tocher-Lehmann modification of Fisher's conditional test is the optimal test for 2x2 tables is unjustified.     

Bayesian inference for contingency tables with given marginals

Summary In this paper we introduce a class of prior distributions for contingency tables with given marginals. We are interested in the structrre of concordance/discordance of such tables. There is actually a minor limitation in that the marginals are required to assume only rational values. We do argue, though, that this is not a serious drawback for all applicatory purposes. The posterior and predictive distributions given anM-sample are computed. Examples of Bayesian estimates of some classical indices of concordance are also given. Moreover, we show how to use simulation in order to overcome some difficulties which arise in the computation of the posterior distribution.     

Measure of departure from conditional marginal homogeneity for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, Tomizawa et al. (S. Tomizawa, N. Miyamoto, and N. Ashihara, Measure of departure from marginal homogeneity for square contingency tables having ordered categories, Behaviormetrika 30 (2003), pp. 173–193.) and Tahata et al. (K. Tahata, T. Iwashita, and S. Tomizawa, Measure of departure from symmetry of cumulative marginal probabilities for square contingency tables with ordered categories, SUT J. Math., 42 (2006), pp. 7–29.) considered the measures which represent the degree of departure from the marginal homogeneity (MH) model. The present paper proposes a measure that represents the degree of departure from the conditional MH, given that an observation will fall in one of the off-diagonal cells of the table. The measure proposed is expressed by using the Cressie–Read power-divergence or the Patil–Taillie diversity index, which is applied for the conditional cumulative marginal probabilities given that an observation will fall in one of the off-diagonal cells of the table. When the MH model does not hold, the measure is useful for seeing how far the conditional cumulative marginal probabilities are from those with an MH structure and for comparing the degree of departure from MH in several tables. Examples are given.     

Improving Path Trimming in a Network Algorithm for Fisher's Exact Test in Two-way Contingency Tables

This article provides an improvement of the network algorithm for calculating the exact p value of the generalized Fisher's exact test in two-way contingency tables. We give a new exact upper bound and an approximate upper bound for the maximization problems encountered in the network algorithm. The approximate bound has some very desirable computational properties and the meaning is elucidated from a viewpoint of differential geometry. Our proposed procedure performs well regardless of the pattern of marginal totals of data.