Square contingency tables with the same categories occur frequently in applied sciences. Such tables arise from tabulating the repeated measurements of a categorical response variable. Some examples for these kind of tables are: for instance, when the subjects are measured at two different points in time (e.g., responses before and after experiments); the decisions of two experts are measured on the same set of subjects (e.g., the grading of the same cancer tumors by two specialists); two similar units in a sample are measured (e.g., the grades of vision of the left and the right eyes); matched pair experiments (e.g., social status of the fathers and sons) [1]. For square contingency tables, several models have been proposed (see, for example [2-8] but the models of symmetry (S), quasi-symmetry (QS), marginal homogeneity (MH) are classical and well known models [9,10] and the applicability of the these models is straightforward. The QS is less restrictive model than the S model [11-13].

Consider an RxR square contingency table with the same row and column classifications. Let p_{ij}-denote the probability that an observation will fall in the *i*th row and *j*th column of the table. Bowker [14] considered the symmetry (S) model for RxR tables defined by

$${p}_{ij}={p}_{ji}(i\ne j).$$The S model implies that the probability that an observation will fall in cell (*i*, *j*) of the table is equal to the probability that it falls in cell (*j*, *i*).

Multiway contingency table is obtained when a sample of n observations is cross classified with respect to T categorical variables having the same number of categories. Such tables are very popular in panel studies or matched pair examples. The symmetry model is denfied in multidimensional way.

Denote the *k*th categorical variable by *X*_{k} (*k* = 1, ..., *T*) and consider an *R*^{T} contingency table (*T* ≥ 3). Let *p*_{i1…iT} denote the probability that an observation will fall in the (*i*_{1}, ..., *i*_{T})th cell of the table.

Agresti [1] defined the S model as

$${p}_{i\mathrm{1...}iT}={p}_{j\mathrm{1...}jT},$$for any permutation (*j*_{1},…,*j*_{T}) of (*i*_{1}, …, *i*_{T}) with *i*_{t}=1,..., *r*;*t =* 1,..., *T*.

For example, when *T* = 3, let *X*, *Y* and *Z* denote the row, column and layer variables, the S model can be expressed as

$${p}_{ijk}={p}_{ikj}={p}_{jik}={p}_{jki}={p}_{kij}={p}_{kji}.$$The simplest possible model of interest is the model of complete independence, where the joint distribution of the three variables is the product of the marginals. The corresponding hypothesis is

$${H}_{0}:{p}_{ijk}={p}_{i\mathrm{..}}{p}_{\mathrm{.j}.}{p}_{\mathrm{..}k}$$Symmetry model for multiway tables is given in general as follows:

$${p}_{i\mathrm{1...}iT}=\left({\displaystyle \prod _{i=1}^{T}{\alpha}_{i}}\right)\left({\displaystyle \prod _{i=1}^{T}{\alpha}_{i}}\right){\psi}_{i\mathrm{1...}iT}$$(1.1)The common schemes for representing contingency tables are based on the row column and layer variables that are independent. In three way contingency tables, the choice of predictor and control variable is of interest to many researches. The purpose of this paper is to give some models which represent the subsymmetry and asymmetry for multiway contingency tables. We will concentrate on only three dimensional tables which are a cross-classification of observations by the levels of three categorical variables.

The models are defined in the sub symmetry and asymmetry context taking the first variable as a control variable. The models below are often used to analyze three dimensional tables.

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