Independence In Two-way Contingency Tables Research Articles

A Class of Association Measures for Categorical Variables Based on Weighted Minkowski Distance

Measuring and testing association between categorical variables is one of the long-standing problems in multivariate statistics. In this paper, I define a broad class of association measures for categorical variables based on weighted Minkowski distance. The proposed framework subsumes some important measures including Cramér’s V, distance covariance, total variation distance and a slightly modified mean variance index. In addition, I establish the strong consistency of the defined measures for testing independence in two-way contingency tables, and derive the scaled forms of unweighted measures.

The LMS for testing independence in two-way contingency tables

Summary In the statistical literature there are proposed many test measures to determine the independence of two qualitative variables in contingency tables, in particular in two-way contingency tables larger than 2×2. For statistical analysis, three of the so-called “chi-squared tests”—the T 3 test, BP test and |χ| test—were selected. These tests were compared with a logarithmic minimum test, which is the author’s proposal. Critical values for the tests were determined with the Monte Carlo method. To compare the tests, an appropriate measure of untruthfulness of H 0 was used and the power of the tests was calculated.

New Equivalence Tests for Approximate Independence in Contingency Tables

We introduce new equivalence tests for approximate independence in two-way contingency tables. The critical values are calculated asymptotically. The finite sample performance of the tests is improved by means of the bootstrap. An estimator of boundary points is developed to make the bootstrap based tests statistically efficient and computationally feasible. We compare the performance of the proposed tests for different table sizes by simulation. Then we apply the tests to real data sets.

A new approach to distribution free tests in contingency tables

We suggest an extremely wide class of asymptotically distribution free goodness of fit tests for testing independence in two-way contingency tables, or equivalently, independence of two discrete random variables. The nature of these tests is that the test statistics can be viewed as definite functions of the transformation of $$\widehat{T}_n = (\widehat{T}_{ij})=\Big (\frac{\nu _{ij}- n\hat{a}_i\hat{b}_j}{\sqrt{n\hat{a}_i\hat{b}_j}}\Big )$$ where $$\nu _{ij}$$ are frequencies and $$\hat{a}_i, \hat{b}_j$$ are estimated marginal distributions. Our method is also applicable for testing independence of two discrete random vectors. We make some comparisons on statistical powers of the new tests with the conventional chi-square test and suggest some cases in which this class is significantly more powerful.

A Bootstrap Approach for Testing Marginal Independence Between Two Categorical Variables When Subjects Have Repeated Responses

To assess independence in two-way contingency tables, the Pearson chi-square test or Fisher’s exact test are typically used. These tests assume that each subject contributes at most one count to only one table cell (e.g., sex versus blood type). In other situations, each subject may have more than one count contributing to the table and these counts may occur in different cells of the table. One may wish to test independence, adjusting for the within-subject correlation. We provide a simple nonparametric bootstrap approach and assess its performance through simulation studies. The method is illustrated on subjects with multiple mental health presentations to Emergency Departments.

Nonparametric bootstrap tests of conditional independence in two-way contingency tables

When analyzing a two-way contingency table, a preliminary question is often whether the categorical variables under study, say R and S, are independent or not. Suppose now that for each individual in the table, a continuous variable X is also known. It is then worth analyzing the table conditionally on X. Contrasting these “local” results to the global unconditional case allows one to go beyond the initial analysis and provide a better understanding of the underlying phenomenon. Recently, Geenens and Simar (2010) [11] have proposed two nonparametric procedures for testing whether R and S are conditionally independent given X, free of any constraining linearity assumptions. However, based on an average of kernel-based estimators, the asymptotic criterion they suggested shows an inflated Type I error (false positive) for small to moderate sample sizes. In this paper, we address this problem by proposing consistent bootstrap versions of the Geenens–Simar test procedures when testing for local independence. A comprehensive simulation study indeed shows the superiority of the bootstrap rejection criterion as compared to the asymptotic criterion in terms of Type I error. It also highlights the advantage of the flexibility guaranteed by the nonparametric Geenens–Simar tests when compared with parametric competitors, e.g. logistic models. The approach is finally illustrated with a real-data example.

A Test of Independence in Two-Way Contingency Tables Based on Maximal Correlation

Maximal correlation has several desirable properties as a measure of dependence, including the fact that it vanishes if and only if the variables are independent. Except for a few special cases, it is hard to evaluate maximal correlation explicitly. We focus on two-dimensional contingency tables and discuss a procedure for estimating maximal correlation, which we use for constructing a test of independence. We compare the maximal correlation test with other tests of independence by Monte Carlo simulations. When the underlying continuous variables are dependent but uncorrelated, we point out some cases for which the new test is more powerful.

A Flexible Method for Testing Independence in Two-Way Contingency Tables

A flexible approach for testing association in two-way contingency tables is presented. It is simple, does not assume a specific form for the association and is applicable to tables with nominal-by-nominal, nominal-by-ordinal, and ordinal-by-ordinal classifications.

Nonparametric tests for conditional independence in two-way contingency tables

Testing for the independence between two categorical variables R and S forming a contingency table is a well-known problem: the classical chi-square and likelihood ratio tests are used. Suppose now that for each individual a set of p characteristics is also observed. Those explanatory variables, likely to be associated with R and S , can play a major role in their possible association, and it can therefore be interesting to test the independence between R and S conditionally on them. In this paper, we propose two nonparametric tests which generalise the chi-square and the likelihood ratio ideas to this case. The procedure is based on a kernel estimator of the conditional probabilities. The asymptotic law of the proposed test statistics under the conditional independence hypothesis is derived; the finite sample behaviour of the procedure is analysed through some Monte Carlo experiments and the approach is illustrated with a real data example.

Assessing Robustness of Intrinsic Tests of Independence in Two-Way Contingency Tables

For testing nested hypotheses from a Bayesian standpoint, a desirable condition is that the prior for the alternative model concentrates mass around the smaller, or null, model. For testing independence in contingency tables, the intrinsic priors satisfy this requirement. Furthermore, the degree of concentration of the priors is controlled by a discrete parameter, t, the training sample size, which plays an important role in the resulting answer. In this article we report on the robustness of the tests of independence for small or moderate sample sizes in contingency tables with respect to intrinsic priors with different degrees of concentration around the null. We compare these tests to frequentist tests and other robust Bayes tests. For large sample sizes, robustness is achieved because the intrinsic Bayesian tests are consistent. Examples using real and simulated data are given. Supplemental materials (technical details and data sets) are available online.

A class of statistical models to weaken independence in two-way contingency tables

In this paper we study a new class of statistical models for contingency tables. We define this class of models through a subset of the binomial equations of the classical independence model. We prove that they are log-linear and we use some notions from Algebraic Statistics to compute their sufficient statistic and their parametric representation. Moreover, we show how to compute maximum likelihood estimates and to perform exact inference through the Diaconis-Sturmfels algorithm. Examples show that these models can be useful in a wide range of applications.

Tests for Independence in Two-Way Contingency Tables with Small Samples

When testing the null hypothesis of independence in a two-way contingency table, the likelihood ratio test statistic is approximately distributed as Chi-squared d for large sample sizes (N) but may not be for small samples. This paper presents expressions which match the mean of the statistic to Chi-squared d as far as N −1 and N −2 , derives a method of estimating the expressions from observed data and evaluates them using Monte Carlo simulations. It is concluded that using appropriate dividing factors, rejection rates after matching are more accurate than for either the unadjusted likelihood ratio statistic or the Pearson approximation which is the main alternative statistic. Minimum cell frequencies necessary for high test accuracy are smaller than those commonly given in textbooks.

UMP unbiased tests for multiparameter testing problems with restricted alternatives

Applications of the standard theory of UMP unbiased tests depends on conditions which in general are difficult to verify. In the present paper, however, we suggest more simple rules for applying this theory for regular exponential families of distributions. This approach leads to UMP unbiased tests for various multiparameter testing problems with restricted alternatives, and is shown to give justification for conditional tests for testing symmetry, diagonals-parameter symmetry and independence in two-way contingency tables. The derived tests are shown to possess attractive small sample properties.

The effect of marginal disuniformity on the χ 2 approximation to the distribution of Pearson's X 2 in sparse contingency tables

Let n k be a sequence of contingency tables with an increasing number of cells q k , n k be the sample size and π k be the vector of cell probabilities generating n k . Theoretical papers, like Morris ( Annals of Statistics 3, 1975) and Haberman ( Annals of Statistics 5, 1977), suggest that the distribution of test-statistics for testing parametric models for π k depends on n k , on some function of the ratio n k / q k and also on the ‘disuniformity’ of the expected frequencies in the table. In the context of testing independence in two-way contingency tables, this paper investigates the χ 2 approximation to the conditional distribution of X 2 given the margins. Attention is focussed on the effect of disuniformity on this approximation, as n k and q k increase, for different levels of n k / q k . This investigation is first carried out by analytically studying the ratio VR = var( X 2)/var(χ 2) as a function of q k , of n k / q k and of a pair of disuniformity measures for the marginal totals that enter directly the formula for the exact variance of X 2. Moreover, the χ 2 approximation is studied by means of an extensive numerical work, involving several simulations for various combinations of q k / q k and of the pair of disuniformity measures mentioned above. The main outcome is that marginal disuniformity greatly affects the χ 2 approximation, mainly by affecting the X 2 variance. Hence, the variance ratio VR may be used to assess the appropriateness of the traditional χ 2 test; this criterion is many circumstances less severe than the traditional ones.

Correlation and Association Models for Studying Measurements on Ordinal Relations

This paper describes and illustrates correlation models (correspondence analysis and canonical correlation analysis) and association models for studying the order and spacing of categories of ordinal relational variables. Both correlation models and association models study departures from independence in two-way contingency tables. One result of fitting these models is the possibility of assignment of scores to the categories of the row andlor the column variables to reflect the relative spacing oJ these categories. If the model fitting is done using statistical procedures, then restricted versions of these models allow one to test hypotheses about the spacing, linearity, or equality of the categories. Correlation and association models are especially useful for studying discrete ordinal variables, which arise quite frequently in the social and behavioral sciences. We illustrate correlation and association models using two empirical examples in which respondents used ordered

AX one degree of freedom nominal association model for testing independence in two-way contingency tables

In testing the independence of the row and column variables in a two-way contingency table, the standard Pearson and likelihood ratio chi-square statistics often have low power, especially as the dimensions of the table increase. In this paper, one degree of freedom tests of independence based on likelihood ratio and score statistics from an association model for tables with nominal row and column classifications are described. The score statistic is especially easy to use, since it can be expressed in closed form, is simple to compute, and has size and power properties which are only slightly inferior to those of the more complicated likelihood ratio statistic.

Tests for Independence in Two-Way Contingency Tables Based on Canonical Analysis

ABSTRACT: The canonical structure of a two‐way contingency table is emphasized to derive union‐intersection (Ul‐) tests for independence between two categorical variables. In this context, special attention has been paid to the so‐called step‐down procedure (which is a variant of the UI‐procedure Some properties of this procedure and computational algorithms are considered and a sequential way for identifying the adequate model is presented.

A structure theorem on bivariate positive quadrant dependent distributions and tests for independence in two-way contingency tables

In this paper, the set of all bivariate positive quadrant dependent distributions with fixed marginals is shown to be compact and convex. Extreme points of this convex set are enumerated in some specific examples. Applications are given in testing the hypothesis of independence against strict positive quadrant dependence in the context of ordinal contingency tables. The performance of two tests, one of which is based on eigenvalues of a random matrix, is compared. Various procedures based upon certain functions of the eigenvalues of a random matrix are also proposed for testing for independence in a two-way contingency table when the marginals are random.

Latent conditional independence in two‐way contingency tables: A diagnostic approach

British Journal of Mathematical and Statistical PsychologyVolume 36, Issue 1 p. 114-122 Latent conditional independence in two-way contingency tables: A diagnostic approach Zvi Gilula, Corresponding Author Zvi GilulaDepartment of Statistics, Hebrew University of Jerusalem, Jerusalem 91905, Israel.Search for more papers by this author Zvi Gilula, Corresponding Author Zvi GilulaDepartment of Statistics, Hebrew University of Jerusalem, Jerusalem 91905, Israel.Search for more papers by this author First published: May 1983 https://doi.org/10.1111/j.2044-8317.1983.tb00767.xCitations: 10 This paper discusses a diagnostic approach to determine whether an observed statistical dependence between two categorical variables can be accounted for by some latent dichotomous variables in the sense that the two observed variables are conditionally independent given such latent variables. This approach is applied to two-way contingency tables. As a part of this diagnostic approach a BAN estimation procedure for a rank 2 probability matrix is discussed. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume36, Issue1May 1983Pages 114-122 RelatedInformation

Most Possible Estimates and Maximum Likelihood Estimates

Testing for independence in two-way contingency tables [ILLEGIBLE][ILLEGIBLE] using a chi-square test that measures the goodness of fit between the observed data and the expected values. This article examines the expected values normally employed in such analyses. It first notes that the usually derived expected values are based on assumptions about the distributional form of an underlying probability density function that may not always be well-founded. Then the article presents an alternative method of estimating expected values, the most possible estimate approach, that does not rely on a priori assumptions about the distributional form. Finally, by establishing that the most possible estimates approximate maximum likelihood estimates, the robustness of the standard maximum likelihood estimator is demonstrated and the researcher is thereby assured of safety in using it without attending to distributional assumptions. The analysis is applied to 3 × 3 tables in this article.