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.     

Measure of Asymmetry for Square Contingency Tables Having Ordered Categories

For the analysis of square contingency tables with nominal categories, Tomizawa and coworkers have considered measures that represent the degree of departure from symmetry. This paper proposes a measure that represents the degree of asymmetry for square contingency tables with ordered categories (instead of those with nominal categories). The measure proposed is expressed using the Cressie–Read power-divergence or Patil–Taillie diversity index, defined for the cumulative probabilities that an observation falls in row (column) category i or below and column (row) category j (> i ) or above. The measure depends on the order of listing the categories. It should be useful for comparing the degree of asymmetry in several tables with ordered categories. The relationship between the measure and the normal distribution is shown.     

Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories

For square contingency tables that have nominal categories, Tomizawa considered two kinds of measure to represent the degree of departure from symmetry. This paper proposes a generalization of those measures. The proposed measure is expressed by using the average of the power divergence of Cressie and Read, or the average of the diversity index of Patil and Taillie. Special cases of the proposed measure include Tomizawa's measures. The proposed measure would be useful for comparing the degree of departure from symmetry in several tables.     

Power-divergence-type measure of departure from diagonals-parameter symmetry for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, Goodman considered the diagonals-parameter symmetry (DPS) model. This paper proposes a measure to represent the degree of departure from the DPS model. The proposed measure is expressed by applying Read and Cressie’s power-divergence or Patil and Taillie’s diversity index. The measure would be useful for comparing the degree of departure from the DPS model in several tables. Examples are given.     

Point-symmetry models and decomposition for collapsed square contingency tables

For square contingency tables with ordered categories, there may be some cases that one wants to analyze them by considering collapsed tables with some adjacent categories combined in the original table. This paper proposes three kinds of new models which have the structure of point-symmetry (PS), quasi point-symmetry and marginal point-symmetry for collapsed square tables. This paper also gives a decomposition of the PS model for collapsed square tables. The father's and his daughter's occupational mobility data are analyzed using new models.     

Measure of departure from marginal homogeneity using marginal odds for multi-way tables with ordered categories

For square contingency tables with ordered categories, this paper proposes a measure to represent the degree of departure from the marginal homogeneity model. It is expressed as the weighted sum of the power-divergence or Patil–Taillie diversity index, and is a function of marginal log odds ratios. The measure represents the degree of departure from the equality of the log odds that the row variable is i or below instead of i+1 or above and the log odds that the column variable is i or below instead of i+1 or above for every i. The measure is also extended to multi-way tables. Examples are given.     

Marginal homogeneity,symmetry, and independence

A contingency table of the mc form provides a convenient summary of data when c individuals in a matched set9 each belonging to a different one of c classifications, are identified as belonging to one of m categories, A study in which matched sets (c=3) of 1 case, 1 hospital control, and 1 neighborhood control are classified into one of m=4 occupational categories would be an example, Independence in the cxm tables for each of the matched sets implies symmetry in the summary mc table with consequent marginal homogeneity. Adaptation of the Mantel-Haenszel procedure for testing independence to the case of many cxm tables so as to yield a chi square with (cl)(ml) degrees of freedom (DF) provides a test of marginal homogeneity in the summary mc table. This can be viewed as a test of symmetry directed against alternatives which would make for marginal inhomogeneity and can differ     

Marginal inhomogeneity models for square contingency tables with nominal categories

For the analysis of square contingency tables with nominal categories, this paper proposes two kinds of models that indicate the structure of marginal inhomogeneity. One model states that the absolute values of log odds of the row marginal probability to the corresponding column marginal probability for each category i are constant for every i. The other model states that, on the condition that an observation falls in one of the off-diagonal cells in the square table, the absolute values of log odds of the conditional row marginal probability to the corresponding conditional column marginal probability for each category i are constant for every i. These models are used when the marginal homogeneity model does not hold, and the values of parameters in the models are useful for seeing the degree of departure from marginal homogeneity for the data on a nominal scale. Examples are given.     

A class of exact UMP unbiased tests for conditional symmetry in small-sample square contingency tables

Testing conditional symmetry against various alternative diagonals-parameter symmetry models often provides a point of departure in studies of square contingency tables with ordered categories. Typically, chi-square or likelihood-ratio tests are used for such purposes. Since these tests depend on the validity of asymptotic approximation, they may be inappropriate in small-sample situations where exact tests are required. In this paper, we apply the theory of UMP unbiased tests to develop a class of exact tests for conditional symmetry in small samples. Oesophageal cancer and longitudinal income data are used to illustrate the approach.     

Marginal asymmetry model for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, this paper proposes a model which indicates the structure of marginal asymmetry. The model states that the absolute values of logarithm of ratio of the cumulative probability that an observation will fall in row category i or below and column category i+1 or above to the corresponding cumulative probability that the observation falls in column category i or below and row category i+1 or above are constant for every i. We deal with the estimation problem for the model parameter and goodness-of-fit tests. Also we discuss the relationships between the model and a measure which represents the degree of departure from marginal homogeneity. Examples are given.     

Quasi local odds symmetry model for square contingency table with ordinal categories

This paper proposes a new model for square contingency tables. The proposed model tests the equality of local odds ratios between the one side of the main diagonal and corresponding other side and it represents the non-symmetric structure of the square contingency table. The proposed model is compared with twenty-five models introduced for analysing the square contingency tables for both symmetric and non-symmetric structures. The results show that the proposed model provides best fit performance than other existing models for square contingency tables.     

Double linear diagonals-parameter symmetry and decomposition of double symmetry for square tables

For square contingency tables with ordered category, the present paper proposes the double linear diagonals-parameter symmetry (D-LDPS) model which implies the structure of both asymmetry with respect to the main diagonal and with respect to the reverse diagonal in the table. The D-LDPS model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. The present paper also gives the orthogonal decomposition of the double symmetry model into the D-LDPS model and the double marginal mean equality model. An example is given.     

A model of symmetry with exponents along every subdiagonal and its application to data on unaided vision of pupils at Japanese elementary schools

For the analysis of square contingency tables with ordered categories, this paper introduces a simple symmetry model in which the expected frequency has an exponential form along every subdiagonal of the table. The model is applied to the data on unaided distance vision of 3168 pupils at elementary schools in Tokyo which were reported earlier by Tomizawa, and a result of interest is obtained.     

DECOMPOSITIONS FOR CONDITIONAL SYMMETRY MODEL INTO PALINDROMIC SYMMETRY AND MODIFIED MARGINAL HOMOGENEITY MODELS

For square contingency tables with ordered categories, the conditional symmetry model is decomposed into the palindromic symmetry and the modified marginal homogeneity models, and moreover into the generalized palindromic symmetry model and two other models. The data on unaided vision first analysed by Stuart (1953, 1955) is analysed again by using the decompositions.     

Diagonal,weighted marginal homogeneity model and decompositions for linear diagonals-parameter symmetry model

For square contingency tables rith ordered categories, this paper proposes two kinds of extensions of marginal homogeneity model and gives decompositions for the Liseer diagonals-parameter symmetry model considered by Agresti (1983a) using the proposed models- The proposed models are also applied to an unaided vision data.     

Partitioning chi-squape in contingency tables: A teaching approach

A representation of sums and differences of the form 2n log n, the lnn function, is introduced to express likelihood-ratio chi-square test statistics in contingency table analysis. This is a concise explicit form to display when partitioning chi-square statistics in accordance with hierarchical models. The lnn representation gives students insights into the construction of test statistics, and assists in relating identical forms under differing model sets. Hierarchies are presented for independence and equi-probability in two-way tables, for symmetry in correlated square tables, for independence-and-homogeneity of two-way responses across levels of a factor, and for mutual independence in three-way tables, along with relevant partitions of chi-square.     

A new C(α) test for 2×2 tables

Two by two contingency tables which arise from experiments in which each subject serves as a self control are examined, Two types of experimental design are discussed: the confounded design and the counterbalanced design. In the case of the counterbalanced design, the locally asymptotically optimal C(α) test is compared with the binomial test for symmetry in a 2×2 contingency table. The binomial test is found to be less efficient than the C(α) test.     

The iterative proportional fitting algorithm for adjusted agreement in a non‐inferiority diagnostic clinical trial

In this paper, we describe a method of comparing agreement between two diagnostic contingency tables after adjustment to more clinically relevant marginal distributions using the iterative proportional fitting algorithm. When the categories of a contingency table represent mild, moderate, and severe outcomes, the majority of patients often are in the mild category. Because it is often of more interest to evaluate agreement when patients are uniformly distributed among categories, we present the primary results of two clinical trials with adjustment to this structure. We also describe the relationship between the sponsor's pre‐specified agreement measure for the observed contingency table and kappa for the adjusted table; and by either criterion, we then show that the agreement of the new diagnostic tool with the standard diagnostic tool is comparably non‐inferior to the agreement of the standard diagnostic tool with itself. Copyright © 2014 John Wiley & Sons, Ltd.     

Tests of symmetry with one-sided alternatives in three-way contingency tables

For a two-dimensional contingency table of probabilities, the concept of symmetry around the main diagonal is well defined. Statistical hypothesis test of symmetry versus positive bias have also been explored. For tables of higher (three or more) dimensions, however, different concepts of symmetry are available. In this study, we consider statistical inference procedures of symmetry in partial tables versus various biases in three-dimensional tables. We find the maximum likelihood estimates of the cell probabilities and the asymptotic distribution of the likelihood ratio test statistic in each case. Simulation studies are used to investigate the sizes and powers of the tests. The methodologies developed are applied on real data sets.     

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.