1. Introduction
Contingency tables play important roles in various fields, as they highlight relationships between categorical variables. Typically, the analysis of contingency tables is interested in whether row and column variables are independent. If the independence hypothesis is rejected, then the association between the variables is of interest. Various coefficients have been proposed to measure the association, such as
gamma,
Yule’s Q,
Kendall’s tau-b,
Kendall’s tau, and
Somers’ d. See (Bishop et al. [
1], Ch. 11) and (Agresti [
2], pp. 184–192). Bishop et al. [
1], Agresti [
2], and Kateri [
3] present overviews of contingency table analysis. Additionally, Kateri [
4] reviewed the
-divergence association models related to the independence model for two-way contingency tables. Fujisawa and Tahata [
5] proposed the generalized asymmetry plus quasi-uniform association model. These works in the literature provide reviews of contingency table analysis based on association models.
Contingency tables with the same row and column classifications are often called
square contingency table. Square contingency table data arise many fields. For example, unaided distance vision data, social mobility data, father–son matched educational level data, longitudinal data in biomedical research, and so on. Typically, the analysis of square contingency tables considers the issue of symmetry rather than independence (null association) because observations tend to concentrate on or near the main diagonal. Therefore, many statisticians have considered various symmetry and asymmetry models, and modeling of symmetry is one of the important topics for the analysis of square contingency tables. For example, Bowker [
6] proposed a test for the hypothesis of symmetry (S). The S model indicates the symmetry structure of cell probabilities. Additionally, Stuart [
7] provided a large-sample test for marginal homogeneity (MH). The MH model indicates the equivalence of marginal probabilities.
Caussinus [
8] developed quasi-symmetry (QS). The QS model bridges S and MH in square tables. The QS model indicates the symmetry of the odds ratio with respect to the main diagonal of the square contingency table. If the S model holds, then the MH model holds. However, the inverse is not always true. The QS model can be used to show that the S model holds if and only if both the QS and MH models hold. This finding has influenced the methodological developments in statistical analysis of categorical data. See
Section 3 for more details.
A special issue of
Annales de la Faculté des Sciences de Toulouse, Mathématiques was published in 2002. It contained papers written by internationally distinguished authors on topics related to QS. Agresti [
9] described some generalizations of the QS model that have similar connections with generalizations of the Rasch model. Goodman [
10] commented relationship between the QS model and the quasi-independence model. McCullagh [
11] set out to list all hereditary sub-representations by real-valued square matrices, and to explain how these may be used in model construction. There are Fienberg and van der Heijden [
12], Bergsma and Rudas [
13], Dossou-Gbété and Grorud [
14], Erosheva et al. [
15], De Falguerolles and van der Heijden [
16], Stigler [
17], Thélot [
18], and Caussinus [
19] in addition to those in the special issue.
Since then, numerous studies have treated QS from a variety of perspectives. For square contingency tables, Tomizawa [
20] introduced modeling based on the cumulative probabilities. Miyamoto et al. [
21] proposed the cumulative QS model for square contingency tables with ordinal categories. Kateri et al. [
22] considered generalized QS models. To evaluate the goodness of fit, Booth et al. [
23] adapted a network algorithm to test QS in square tables. Krampe et al. [
24] proposed a method based on algebraic statistics combined with Markov chain Monte Carlo (MCMC) methods for the ordinal QS (OQS) model. Additionally, various topics related to the QS model are discussed in mathematical statistics. For example, see Rapallo [
25], Pardo and Martin [
26], and Gottard et al. [
27]. Tomizawa and Tahata [
28] reviewed some topics on various symmetry models and showed the property of test statistics of symmetry for multi-way contingency tables. Tahata and Tomizawa [
29] reviewed various models of symmetry and asymmetry and presented the relationships among models.
The topics related to the QS model have been discussed in several papers for the last 5 years. Bocci and Rapallo [
30] reviewed why the synergy between algebraic statistics and quasi-independence has been fruitful. Additionally, see Khan and Tewari [
31]. Altun [
32] considered various symmetry and asymmetry models for square contingency tables with ordinal categories. Tahata et al. [
33] proposed the model selection via the penalized likelihood approach. The symmetry models for square contingency tables are applied to the cross classified a single nucleotide polymorphism (SNP) interactions data in Karadağ et al. [
34]. Altunay and Yilmaz [
35] proposed two novel log-linear models to measure the degree of accumulation of the neutral option over the contingency tables based on Likert-type items. Additionally, Ando [
36,
37,
38] proposed models that indicate the structure of asymmetry and gave the decompositions of the models. For square contingency tables with ordinal categories, the models which have various ordered scores were compared in Ando [
39].
This review focuses on the further developments of modeling and properties of symmetry in recent years. Additionally, some comments which are not described in referenced papers are added. Herein four topics related to Tahata [
40] and Tahata et al. [
41] are reviewed: (1) modeling based on the
f-divergence, (2) the necessary and sufficient condition of symmetry, (3) partition of test statistics for symmetry, and (4) the measure of departure from symmetry.
The rest of this paper is organized as follows.
Section 2 demonstrates modeling based on the
f-divergence.
Section 3 reviews the necessary and sufficient condition of symmetry. The result given by Caussinus [
8] is included as a special case. Aitchison [
42], Darroch and Silvey [
43], Read [
44], Lang and Agresti [
45], and Lang [
46] discussed the partitioning of goodness of fit statistics.
Section 4 presents an overview of the main results presented in Tomizawa and Tahata [
28]. Caussinus’ QS model has a good property from a partitioning point of view.
Section 5 introduces the measure of departure from symmetry.
Section 6 discusses the relationships between the measure and asymmetry model. Finally,
Section 7 contains concluding remarks.
2. Modeling Based on the f-Divergence
Consider an square contingency table with the same row and column classifications. Let X and Y denote the row and column variables, respectively, and let denote the probability that an observation will fall in the cell of the table.
The S model, which is defined as
for
, indicates the symmetry structure of cell probabilities (Bowker [
6]). Additionally, the MH model indicates the equivalence of marginal probabilities and is defined as
for
where
and
(Stuart [
7]). The QS model is defined as
From (
1), the QS model can be expressed using the odds ratio as
where
and
.
This indicates the symmetry of the odds ratio with respect to the main diagonal of the square contingency table (Caussinus [
8]). For the analysis of square contingency tables, symmetry models have been proposed by using various ideas, for example, McCullagh [
47], Agresti [
48], and Tomizawa [
49].
Ireland et al. [
50], (Bishop et al. [
1], pp. 345–346) and Gilula et al. [
51] presented the method for model generation. In particular, Kateri and Papaioannou [
52] applied the method for the modeling of symmetry for square contingency tables and proposed a family of models, which includes the QS model as a special case. This family is derived by minimizing the
f-divergence under certain conditions. Additionally, Tahata [
40] extended the results to the modeling of the asymmetry structure of cell probabilities. This section reviews asymmetry models based on the
f-divergence.
Let
and
be two discrete finite bivariate probability distributions. (Csiszár and Shields [
53],
Section 4) introduced the
f-divergence between
and
. Let
f be a convex function on
with
. The
f-divergence of a distribution
from
is defined as
Here, we take , , and .
Let
f be a twice differentiable and strictly convex function, and
. Kateri and Papaioannou [
52] proposed the generalized QS (denoted by QS[
f]) model. It is defined as
where
and
. The QS[
f] model is the closest to the S model in terms of the
f-divergence under the condition where the marginals
(or
) for
and the sums
for
are given. Kateri and Papaioannou [
52] noted that if
,
, then the
f-divergence is reduced to the Kullback–Leibler divergence (Kullback and Leibler [
54]), and the QS[
f] model is equivalent to the QS model. Namely, the QS model is the closest to the S model in terms of the Kullback–Leibler divergence under this condition. Additionally, see Kateri [
4], Kateri and Agresti [
55], Kateri [
56], and Tahata [
40].
Next, consider an
square contingency table with ordered categories. Let
be a set of known scores
(with
). Replacing
by
in the QS[
f] model is the ordinal QS[
f] (OQS[
f]) model (Kateri and Agresti [
55]). The OQS[
f] model is the closest to the S model in terms of the
f-divergence under the condition where
(or
) and the sums
are given. If
,
, then (1) the OQS[
f] model with
is equivalent to the linear diagonals parameter S (LDPS) model (Agresti [
48]) and (2) the OQS[
f] model is equivalent to the OQS model (Agresti [
9], pp. 236–238).
Tahata [
40] proposed a model to fill the gap between the QS[
f] and OQS[
f] models. It should be noted that
is a set of known scores
. For a given
k (
) the asymmetry model based on the
f-divergence (AS
k[
f]) is defined as
where
and
. From the relation
, the parameters of the AS
k[
f] model must satisfy
The AS
k[
f] model is the closest to the S model in terms of the
f-divergence under the condition where
(or
) for
and the sums
for
are given. Additionally, the AS
k[
f] model can be expressed as
where
. For example,
can be set without a loss of generality. There is a one-to-one transformation between a set of
in Equation (
2) and that of
in Equation (
3) when
. When
, the AS
k[
f] model is reduced to the S model. It should be noted that (1) the AS
1[
f] model is reduced to the OQS[
f] model, (2) the AS
k[
f] model (
) is an extension of the OQS[
f] model, and (3) the AS
R−1[
f] model is reduced to the QS[
f] model.
If
for
, then the AS
k[
f] model becomes
where
. Then under Equation (
4)
where
. Equation (
5) with
is the
kth linear asymmetry (LS
k) model proposed by Tahata and Tomizawa [
57]. Therefore, the LS
k model is the closest to the S model in terms of the Kullback–Leibler divergence under the condition where
(or
) for
and the sums
are given.
Let the conditional probability that an observation will fall in cell
when the observation falls in cell
or
be denoted by
. That is,
. When
, the AS
k[
f] model becomes
where
. Then under Equation (
6)
where
with
. This model is the closest to the S model when the divergence is measured by the Pearson distance.
Equation (
5) can be expressed as
Namely, the AS
k[
f] model with
indicates that the log odds is expressed as a polynomial of
. On the other hand, the AS
k[
f] model with
(namely, Equation (
6)) indicates that the difference between symmetric conditional probabilities is expressed as a polynomial of
from Equation (
7). Let
and the composition of functions
F and
g be denoted by
H. That is,
. Then the AS
k[
f] model can be expressed as
where
. This formula indicates that a
H-transformation of conditional probability (
) has a linear combination of parameters. For
,
Therefore, the ASk[f] model is characterized as the structure with symmetric conditional probabilities . The structure depends on the function f. The function f is chosen by the user. Namely, we can apply various asymmetry models to the given dataset by changing the function f. Namely, modeling based on f-divergence enable to construct various asymmetries by using the function f.
Fujisawa and Tahata [
58] proposed the extended AS
k[
f] model for square contingency tables with ordinal categories. Additionally, Yoshimoto et al. [
59] discussed the quasi point-symmetry models based on the
f-divergence. For two-way contingency tables, the
-divergence (
f-divergence) association models are a family of models which includes the association and correlation models as special cases. Kateri [
4] presented this family of models and demonstrated the role of
-divergence in building this family. Additionally, Kateri [
56] developed the new families of
-divergence generalized QS models.
3. Necessary and Sufficient Condition of Symmetry
The S model rarely fits the given dataset due to its strong restrictions. More relaxed models, such as the QS and MH models, are often applied to a dataset. Here, we are interested in an extension of the S model, which indicates the asymmetry structure of cell probabilities. Caussinus [
8] noted that the S model holds if and only if both the QS and MH models hold. Caussinus’ result is useful to deduce the reason for a poor fit of the S model, and Bishop et al. [
1] (Section 8.2.3) derived its proof. Let
denote model M. Then
In this section, the necessary and sufficient conditions for the S model are reviewed.
The conditional S (CS) model (Read [
44] and McCullagh [
47]) is defined as
for
. The CS model with
is reduced to the S model. See also (Bishop et al. [
1], pp. 285–286).
Consider the global symmetry (GS) model, which is defined as
Namely, this model indicates that
is equal to
. If both the CS and GS models hold, then the S model holds. Since the converse holds, Read [
44] highlighted that
The LDPS model fits well when there is an underlying bivariate normal distribution (Agresti [
48]). Additionally, Tomizawa [
49] considered an extended LDPS (ELDPS) model and reported the relationship with bivariate normal distribution. Tahata and Tomizawa [
57] considered the LS
k model, which is defined by Equation (
5) with
. This model is a generalization of the LDPS and ELDPS models. That is, the LDPS and ELDPS models are the LS
1 and LS
2 models, respectively. It should be noted that
For , the difference in the degrees of freedom (df) between and is one.
On the other hand, for a given positive integer
k, the
kth moment equality (ME
k) model is defined as
where
and
. For example, the ME
1 model indicates that the mean of
X is equal to that of
Y, and the ME
2 model indicates the equality of the mean and variance for
X and
Y. Tahata and Tomizawa [
60] proved that the MH model is equivalent to the ME
R−1 model. Therefore,
For
, the difference in df between
and
is one. From the Caussinus’ result, Equations (
10) and (
11) provide various relations. For example
and
(Agresti [
2], p. 261) and Kateri and Agresti [
55] reported similar comments. However, it should be noted that
and
have more redundant conditions than the S model.
Let
denote the cell probabilities satisfying both the LS
k and ME
k models. The LS
k model is expressed as
where
. Additionally, the ME
k model is expressed as
where
and
. By considering
with
(namely, the Kullback–Leibler divergence) to measure the difference between
and
,
That is,
holds for
from the property of the Kullback–Leibler divergence. Additionally, if the S model holds, then both LS
k and ME
k models hold clearly. Therefore, for a given positive integer
k, these lead to
Tahata [
40] extended the result using the AS
k[
f] model
. Consider the generalized ME
k (denoted by GME
k) model defined as
with a set of known scores
. Tahata [
40] proved the following result:
Tahata et al. [
61] proposed the extended LS
k (ELS
k) model, which is defined as
The ELS
k model with γ = 1 is reduced to the LS
k model. It should be noted that the ELS
R−1 model is equal to the extended QS (EQS) model (Tomizawa [
62]), which indicates the asymmetry structure of the odds ratio. The QS model indicates the symmetry structure of the odds ratios. The OQS (LDPS) model indicates both the QS model and the asymmetry structure of cell probabilities and is a special case of the QS model. The EQS model is an extension of the QS model. Tahata et al. [
61] noted that
Fujisawa and Tahata [
58] proposed the model, which is a generalization of the ELS
k model using the
f-divergence and gave the generalization of Equation (
14).
4. Partition of Test Statistics for Symmetry
Consider the situation where the analyst has found hypothesis
unacceptable and has turned their attention to an examination of components
and
such that
. Let
denote the statistic for testing the goodness of fit of model M.
is asymptotically distributed as a chi-square distribution with the corresponding df. Aitchison [
42] noted that the possibility of partitioning the test statistic for
into components for testing
and
must be investigated. Namely,
. When hypotheses
and
are
separable, partitioning is possible.
Separable implies that the restricted tests of
against
and that of
against
use the same critical regions as the unrestricted test of
and that of
, respectively. It should be noted that df for
(denoted by
) is equal to the sum of df for
and
. That is,
. The acceptance of
and
separately means that
and
where
is the significance level and
c denotes the corresponding critical value. Then
. Under the separable hypotheses, hypothesis
is rejected when
. The acceptance of
and
usually results in the acceptance of
because
rarely exceeds
. Darroch and Silvey [
43] considered the separability with respect to the likelihood ratio method, which they called “independence”. Additionally, Lang and Agresti [
45] and Lang [
46] discussed the partitioning of goodness of fit statistics. This section describes a property of the statistic for testing the goodness of fit of the S model.
Let
denote the observed frequency in cell
in an
square contingency table with
. Additionally, let
and
denote the expected frequency in cell
and the corresponding maximum likelihood estimate (MLE) under a model, respectively. Assume that a multinomial distribution applies to the table. For the S model, MLEs of
are given as
See, for example, Bowker [
6] and Bishop et al. [
1] (p. 283). Additionally, MLEs of
for the CS model are given as
where
and
. MLEs of
for the GS model are given as
It should be noted that the S, CS, and GS models have closed-form estimators. Let
denote the likelihood ratio chi-square statistic for testing the goodness of fit of model M. Read [
44] noted that
From Equations (
9) and (
15),
(or
). Therefore, the CS and GS models are separable (i.e., exhibit independence).
Caussinus [
8] gave Equation (
8). The QS and MH models do not have closed-form estimators. Therefore, it is difficult to obtain a relationship similar to Equation (
15). Tomizawa and Tahata [
28] proved the relationship between test statistics for the QS and MH models. We review the result briefly.
The QS model is expressed in the log-linear form as
where
. Without the loss of generality, for example, set
. Let
Then the QS model is expressed as
where
,
,
, and
It should be noted that
is the
matrix with
, where
is the
vector of the 1 element,
and
is the
matrix of 1 or 0 elements determined from (
16). Additionally,
is the
identity matrix,
is the
zero matrix,
is the
zero vector, and ⊗ denotes the Kronecker product. The matrix
is a full column rank, which is
K. The linear space spanned by the columns of the matrix
is denoted by
. The dimension of
is
K and
is a subspace of
. Let
denote the orthogonal complement of
. Let
be an
, where
, full column rank matrix such that the columns of
span
. Since
, the QS model can be expressed as
, where
.
The MH model is defined as
Let
, where
. The MH model can be expressed as
, where
. The columns of
belong to subspace
. It should be noted that
. Since Equation (
8) holds, the S model can be expressed as
where
.
Let
denote the sample proportions, where
. From the multivariate central limit theorem,
has an asymptotic normal distribution with mean
and covariance matrix
, where
is the diagonal matrix with the elements of
on the main diagonal. In an analogous manner to Bhapkar [
63], the Wald statistic can be derived for the S model. Let
for
. Using the delta method,
has an asymptotic normal distribution with mean
and covariance matrix of
because
. Then
, where
Let
denote the Wald statistic for testing goodness of fit of model M. Since
, and
correspond to
, and
, respectively, we obtain
Additionally, from the asymptotic equivalence of the Wald statistic and the likelihood ratio statistic (Rao [
64], Section 6e. 3), we obtain
Therefore, the QS and MH models are separable and exhibit asymptotic independence. Caussinus’ result (
8) shows a good property, such as Equations (
17) and (
18).
Tahata and Tomizawa [
57] proved the asymptotic separability of the LS
k and ME
k models for Equation (
12). Tahata [
40] extended the result for Equation (
13). Additionally, Tahata et al. [
61] provided the property of the test statistics related to Equation (
14). Separable hypotheses for the S model are discussed elsewhere, for example, by Saigusa et al. [
65] and Fujisawa and Tahata [
58].
5. Measure of the Departure from Symmetry
As described in
Section 1, the analysis of contingency tables is interested in whether row and column variables are independent. If the independence hypothesis is rejected, then the association between the variables is of interest. Various coefficients have been proposed to measure the association such as
gamma,
Yule’s Q,
Kendall’s tau-b,
Kendall’s tau, and
Somers’ d. See (Bishop et al. [
1], Ch. 11) and (Agresti [
2], pp. 184–192). On the other hand, Tomizawa [
66] proposed two kinds of measures to represent the degree of departure from the S model. Tomizawa et al. [
67] gave a generalization of these measures. Their generalization is expressed as the average of the power divergence (Cressie and Read [
68]). Additionally, Tomizawa et al. [
69], Tahata et al. [
70], and Iki et al. [
71] have proposed measures to represent the degree of departure from symmetry and the marginal homogeneity. This section reviews a measure to represent the degree of departure from the QS model.
Consider an
square contingency table with
nominal categories. Caussinus [
8] proposed the QS model defined as Equation (
1). Using the conditional probability
, the QS model can be expressed as
where
and
.
Additionally, let
for
. Assuming that
for
, Tahata et al. [
41] proposed a measure defined as
When
, this measure is defined by the value taken at the limit as
. Thus
is the modification of the power divergence between
and
. Especially,
is the modification of the Kullback–Leibler divergence between them. It should be noted that the user chooses the real value
.
In addition, measure
can be expressed using the diversity index of degree
(Patil and Taillie [
72]). Let
For paraphrasing Equation (
19), please see Tahata and Kozai [
73] and Tahata et al. [
74]. Then for
Namely, measure
is a weighted sum of the diversity index, which includes the Shannon entropy (Shannon [
75]) when
and the Gini concentration (Giorgi and Gigliarano [
76]) when
.
Measure
lies between 0 and 1. For any
, the value of the measure is 0 if and only if the
contingency table has a QS structure, while its value is 1 if and only if the degree of departure from QS is the largest, in the sense that
(then
) or
(then
) for any
. When the value of the measure is 1, for any
, then
or
holds. Namely, for any
, at least one of
,
, and
is equal to 0, or at least one of
,
, and
is equal to 0. In other words, for any
, the complete asymmetry arises for at least one pair of symmetric cells. That is, it gives the partial complete asymmetry of cell probabilities. The partial complete asymmetry is a weaker condition than the complete asymmetry of cell probabilities used in Tomizawa [
66] and Tomizawa et al. [
67]. Additionally, the QS model indicates the symmetry structure of the odds ratio, while
indicates the complete asymmetry of the odds ratio because
or
∞ for any
. It is natural that the degree of departure from the QS model is the largest.
Consider an approximate standard error and large-sample confidence interval for measure
. Let
denote the sample version of
, which is given by
when
is replaced by
.
has an asymptotic normal distribution with mean
and variance
using the delta method (Bishop et al. [
1] (Section 14.6)). See Tahata et al. [
41] for details of
.
Tahata and Kozai [
73] proposed a measure to represent the degree of departure from the EQS model proposed by Tomizawa [
62]. Additionally, Tahata et al. [
74] developed a different measure to represent the degree of departure from the QS model and partitioned their measure into two components. Ando [
77] discussed the bivariate index vector to concurrently analyze both the degree and the direction of departure from the QS model.
6. Discussions
When the QS model does not fit a given dataset well, often whether an extended model is suitable is evaluated. Tomizawa [
62] proposed the EQS model for a square contingency table with ordinal categories. On the other hand, Tahata [
78] proposed the quasi-asymmetry (QA) model for a square contingency table with nominal categories. The QA model is defined as
where
and
(
) for
. Using the odds ratio, the QA model can be expressed as
This states that the odds of the symmetric odds ratio are equal to
for some
and
for other
. Using
and
in Equation (
19), the QA model can be expressed as
where
. When
(
), the QA model is reduced to the QS model. If the QA model holds, parameter
is useful to visualize the degree of departure from QS because
. Additionally, the value of
approaches 1 as the value of
increases. The parameter
may be effective to express the degree of departure from QS.
If a square table has a structure of QA, then measure
is simply expressed as
where
Namely, measure can be expressed as the weighted sum of the function of . Herein, we note that (i) when , for any . That is, the value of the measure equals 0 (i.e., the QS model holds). (ii) If we take the limit as , then (then ) or (then ) for any . That is, the value of the measure equals 1. Therefore, the maximum degree of departure from QS can be interpreted as the limitation of QA model with .
7. Concluding Remarks
Previously, Tomizawa and Tahata [
28] reviewed topics on various symmetry models to analyze square contingency tables, and Tahata and Tomizawa [
29] summarized topics related to the symmetry and asymmetry models. This paper mainly reviews four recent developments in the analysis of square contingency tables: (1) modeling based on the
f-divergence, (2) the necessary and sufficient condition of symmetry, (3) partition of test statistics for symmetry, and (4) the measure of departure from symmetry. These are based on the results in Tahata [
40] and Tahata et al. [
41].
This paper focuses on the analysis of square contingency tables. Previous papers have examined contingency tables with other structures. For example, Tahata and Tomizawa [
79] discussed the double symmetry, which is characterized as both symmetry and point symmetry. In this review, the problem of estimation of cell probabilities and goodness of fit test are omitted. For such issue, Kateri [
3], Lawal [
80] and Tan [
81] would be useful to study computational aspects using R or SAS. Additionally, Lang [
82,
83] provided the methodology and corresponding R code.
In this paper, the models related to the QS model are described. Especially, the characterization of models and its properties are discussed. The models described this paper is useful for analyzing real dataset. For example, unaided distance vision data, social mobility data, father–son matched educational level data, longitudinal data in biomedical research, and so on. Such applications can be seen in each of references. For details of the estimation of cell probabilities and goodness of fit test, please see corresponding papers.
Various topics are generalized for multi-way contingency tables with the same classifications. Bhapkar and Darroch [
84] proposed the QS and MH models for a general order and gave the generalization of Equation (
8). Tomizawa and Tahata [
28] proved the generalization of Equation (
18). For multi-way contingency tables, the structures of asymmetries are considered in Tahata and Tomizawa [
57], Tahata et al. [
85], Tahata et al. [
86] and Shinoda et al. [
87]. Moreover, Tahata and Tomizawa [
88] and Yoshimoto et al. [
59] treated the problem of point symmetry for multi-way contingency tables.
In future work, the generalization of the results in described in this review will be considered for the multi-way contingency tables. Additionally, the goodness of fit test for symmetry and asymmetry models should be considered for sparse contingency tables.