Extended asymmetry model based on logit transformation and decomposition of symmetry for square contingency tables with ordered categories

For the analysis of square contingency tables, this note proposes a multiplicative model imposed more restrictions on CAUSSINUS' (1965) quasi‐symmetry (QS) model. The model is not so strict as in the quasi‐double symmetry model considered by TOMIZAWA (1985b). The proposed model has the structure of the QS about the main diagonal plus an extended QS about the reverse diagonal of the square contingency table. The model is applied to the data on unaided vision of students in a university in Japan.

The symmetry model is the basic model in the analysis of square contingency tables. Multiple test statistics have been developed for the goodness of fit test. Freeman-Tukey test statistics is appropriate to be used in large samples. However, the required sample size to use the Freeman-Tukey test statistics is not clear. In this paper, the asymptotic properties of Freeman-Tukey test statistic are discussed via extensive Monte-Carlo simulation study. The Freeman-Tukey test statistic is compared with members of power-divergence family test statistic under the symmetry model. The results of simulation study are evaluated based on the Type-I error and power of a test. The results of simulation study and artificial data study show that Freeman-Tukey’s T^2 test statistic does not converge to chi-squared distribution for both sparse and non-sparse square contingency tables.

For the analysis of square contingency tables with ordered categories, Goodman (1979, Biometrika 66, 413-418) considered the diagonals-parameter symmetry (DPS) model, which has a multiplicative form for cell probabilities. This paper proposes another DPS model which has a similar multiplicative form for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (>i) or above. Special cases of the proposed model include conditional symmetry and symmetry. The relationship with a stochastic ordering for marginal distributions is also described. Moreover, the relationships between the proposed model and the (generalized) palindromnic symmetry models are described. An example is given.

For the analysis of square contingency tables with the same row and column ordinal classifications, McCullagh (1978) considered the palindromic symmetry (PS) model, which has a multiplicative form for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (> i) or above. The present paper proposes a modified PS model which indicates that (1) the symmetric odds for cumulative probabilities with distance 1 from main diagonal of the table are constant and (2) there is the structure of quasi-symmetry for them with distance k (≥ 2). Also the present paper gives the decomposition of the symmetry model using the proposed model. Examples are given.

For the analysis of square contingency tables with ordered categories, Agresti (1983) considered the linear diagonals-parameter symmetry (LDPS) model being an extension of the symmetry (S) model, which may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. This paper proposes another extension of the S model which may be appropriate for a square ordinal table if it is reasonable to assume an underlying e-contaminated bivariate normal distribution with equal marginal variances included in the class of elliptical distributions. The proposed model has a structure of weighted mixture of two LDPS models. Moreover, this paper gives a theorem that the S model holds if and only if both the proposed model and the marginal means equality model hold. The simulation studies based on the e-contaminated bivariate normal distribution are given.

For the analysis of square contingency tables with ordered categories, some models that the log odds for two symmetric cell probabilities is a linear function of the row and column values have been considered. This paper proposes a generalization of these models. This paper also proposes the model that the weighted sum of the probability that an observation will fall in one of the cells in upper right triangle of the table is equal to the weighted sum of the probability that it falls in one of the cells in lower left triangle of the table. In addition, this paper gives the theorem that the symmetry model is equivalent to both the proposed models holding simultaneously. Moreover, this paper shows that the likelihood ratio statistic for testing goodness-of-fit of the symmetry model is asymptotically equivalent to the sum of those for testing the proposed models. Examples are given.

SummaryFor the analysis of square contingency tables, it is necessary to estimate an unknown distribution with high confidence from an obtained observation. For that purpose, we need to introduce a statistical model that fits the data well and has parsimony. This study proposes asymmetry models based on cumulative probabilities for square contingency tables with the same row and column ordinal classifications. In the proposed models, the odds, for all i<j, that an observation will fall in row category i or below, and column category j or above, instead of row category j or above, and column category i or below, depend on only row category i or column category j. This is notwithstanding that the odds are constant without relying on row and column categories under the conditional symmetry (CS) model. The proposed models constantly hold when the CS model holds. However, the converse is not necessarily true. This study also shows that it is necessary to satisfy the extended marginal homogeneity model, in addition to the proposed models, to satisfy the CS model. These decomposition theorems explain why the CS model does not hold. The proposed models provide a better fit for application to a single data set of real‐world occupational data for father‐and‐son dyads.

For the analysis of square contingency tables with ordered categories, Agresti (1983) introduced the linear diagonals-parameter symmetry (LDPS) model. Tomizawa (1991) considered an extended LDPS (ELDPS) model, which has one more parameter than the LDPS model. These models are special cases of Caussinus (1965) quasi-symmetry (QS) model. Caussinus showed that the symmetry (S) model is equivalent to the QS model and the marginal homogeneity (MH) model holding simultaneously. For square tables with ordered categories, Agresti (2002, p.430) gave a decomposition for the S model into the ordinal quasi-symmetry and MH models. This paper proposes some decompositions which are different from Caussinus’ and Agresti’s decompositions. It gives (i) two kinds of decomposition theorems of the S model for two-way tables, (ii) extended models corresponding to the LDPS and ELDPS, and the generalized model further for multi-way tables, and (iii) three kinds of decomposition theorems of the S model into their models and marginal equimoment models for multi-way tables. The proposed decompositions may be useful if it is reasonable to assume the underlying multivariate normal distribution.

A class of parametric models for the analysis of square contingency tables with ordered categories Get access P. MCCULLAGH P. MCCULLAGH Department of Mathematics, Imperial CollegeLondon Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 65, Issue 2, August 1978, Pages 413–418, https://doi.org/10.1093/biomet/65.2.413 Published: 01 August 1978 Article history Received: 01 April 1977 Revision received: 01 November 1977 Published: 01 August 1978

SUMMARY A parametric model is presented for the analysis of square contingency tables where there is a one-to-one correspondence between the categories of the row variable and the categories of the column variable. The model is applicable to cross-classifications where the categories are unordered, partially ordered or completely ordered, but the interpretation of the model and associated parameters is most straightforward when the categories are at least partially ordered. Connections with the mover-stayer model and the usual model of quasisymmetry are described. Two data sets are used to illustrate the utility of the model.

SUMMARY A parametric model is developed for the analysis of square contingency tables with ordered categories. Order among the categories is a built-in feature of the new model and this means that it is unnecessary to assign arbitrary 'scores' to the row and column variables. Special cases of the proposed model include conditional symmetry and symmetry. The relationship with marginal homogeneity is also described. The model for quasisymmetry is considered and is shown to be invariant under general permutation transformations of the indices, which makes it suitable for analysing data on a nominal scale. On the other hand, the new model, called p symmetry, is invariant only under the special reverse permutation transformation. This restricted invariance property makes the latter model more suitable for analysing data on an ordinal scale.

A class of models is presented for the analysis of square contingency tables. The models fall in the class of loglinear models or models with logbilinear terms for the association. The models in this class differ in three ways: 1. the association is either assumed to be symmetric or asymmetric 2. the association is assumed to be completely different in each subtable, to have the same form but having different strength, or to be the same and having the same strength 3. for each subtable separately the association that is proposed is full, or has a logbilinear form, or is uniform. An example from research on social mobility will be discussed. The stability of the parameter estimates is studied with the jackknife.

For the analysis of square contingency tables with the same row and column ordinal classications, this article proposes a new model which indicates that the log-ratios of symmetric cell probabilities are proportional to the difference between log-row category and log-column category. The proposed model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate log-normal distribution. Also, this article gives the decomposition of the symmetry model using the proposed model with the orthogonality of test statistics. Examples are given. The simulation studies based on bivariate log-normal distribution are given.

For the analysis of square contingency tables with ordered categories, Caussinus (1965) considered the quasi‐symmetry (QS) model, Goodman (1979) considered the diagonals‐parameter symmetry (DPS) model, and Agresti (1983) considered the linear diagonals‐parameter symmetry (LDPS) model. These models show the structures of symmetry for cell probabilities. Tomizawa (1993) proposed another DPS model which has a similar multiplicative form for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (>i) or above. This paper proposes another LDPS and QS models that have the corresponding similar multiplicative forms for cumulative probabilities instead of cell probabilities. Special cases of the proposed models include symmetry. Two kinds of unaided distance vision data and endometrial cancer data are analyzed using these models. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

SUMMARY For the analysis of square contingency tables having ordered categories, this paper introduces the diagonals-parameter symmetry model and other related multiplicative models. When these models are applied to the 4 x 4 table on unaided vision first analysed by Stuart (1953, 1955), a possible explanation is obtained for (i) the 'strange residual pattern' noted by McCullagh (1978), both in his recent analysis of the data using the palindromic symmetry model and in the analysis of the data using the quasisymmetry model as fitted by Plackett (1974, p. 61) and others; (ii) the observed asymmetry in the table; (iii) the inhomogeneity observed between the row marginal, i.e. right eye vision, and the column marginal, left eye vision; and (iv) the observed association between right eye and left eye vision. The models introduced in this paper are included within the general class of multiplicative models considered by Goodman (1972), but they are different from the specific kinds of models considered in that article.