For square contingency tables with ordered categories, the present paper gives several theorems that the symmetry model holds if and only if the generalized linear diagonals-parameter symmetry model for cell probabilities and for cumulative probabilities and the mean nonequality model of row and column variables hold. It also shows the orthogonality of statistic for testing goodness-of-fit of the symmetry model. An example is given.
umulative Probability; Global Symmetry; Linear Diagonals-Parameter Symmetry; Mean Equality; Ordinal Category; Orthogonal Test Statistic1. Introduction
Consider an square contingency table with the same row and column classifications. Let denote the probability that an observation will fall in the ith row and jth column of the table Bowker [1] considered the symmetry (S) model defined by
This model describes the structure of symmetry with respect to the cell probabilities As a model which indicates the structure of asymmetry for Agresti [2] considered the linear diagonals-parameter symmetry (LDPS) model defined by
A special case of this model obtained by putting is the S model. Yamamoto and Tomizawa [3] considered the generalized linear diagonals-parameter symmetry (LDPS(K)) model as follows; for a fixed
Especially the LDPS(0) model is equivalent to the LDPS model.
Let for
and
The S model may be expressed as
Thus the S model also has the structure of symmetry with respect to the cumulative probabilities Miyamoto et al. [4] considered the cumulative linear diagonals-parameter symmetry (CLDPS) model defined by
which indicates a structure of asymmetry for The CLDPS model is different from the LDPS model. Yamamoto and Tomizawa [3] considered the generalized cumulative linear diagonals-parameter symmetry (CLDPS(K)) model as follows; for a fixed
Especially the CLDPS(0) model is equivalent to the CLDPS model.
Let and denote the row and column variables, respectively. We consider the mean equality (ME) model as
where and and
Yamamoto et al. [5] gave Theorem 1. The S model holds if and only if both the LDPS and ME models hold.
Yamamoto and Tomizawa [6] gave Theorem 2. The S model holds if and only if both the CLDPS and ME models hold.
The present paper gives several decompositions of the S model using the LDPS(K) and CLDPS(K) models. It also proposes the mean nonequality model, and gives the orthogonal decomposition for testing goodness-of-fit of the S model. An example is given.
2. Decompositions of Symmetry Model
We shall give five kinds of decompositions of the S model using the LDPS(K) and CLDPS(K) models.
Theorem 3. For a fixed the S model holds if and only if both the LDPS(K) and ME models hold.
Proof. If the S model holds, then both the LDPS(K) and ME models hold. Conversely, assuming that the LDPS(K) and ME models hold and then we shall show that the S model holds. The ME model may be expressed as
From the LDPS(K) model, we see
Therefore we obtain. Namely the S model holds. The proof is completed.
Theorem 4. For a fixed the S model holds if and only if both the CLDPS(K) and ME models hold.
Considering the global symmetry (GS) model as
namely
we obtain Theorem 5. For a fixed the S model holds if and only if both the LDPS(K) and GS models hold.
We shall omit the proofs of Theorems 4 and 5 because these are obtained in a similar manner to the proof of Theorem 3.
For a fixed consider the mean nonequality (MNE(K)) model as follows:
which is
This model indicates that the difference between the means of and is times higher than the difference between the global symmetric probabilities. When the MNE(0) model is identical to the ME model. We obtain Theorem 6. For a fixed the S model holds if and only if both the LDPS(K) and MNE(K) models hold.
Theorem 7. For a fixed and for a fixed the S model holds if and only if both the LDPS(K) and MNE(L) models hold.
We shall omit the proofs of Theorems 6 and 7 because there are obtained in a similar manner to the proof of Theorem 3. Note that: 1) Theorem 6 is an extension of Theorem 1 because when Theorem 6 is identical to Theorem 1; 2) Theorem 7 is an extension of Theorem 3 because when Theorem 7 is identical to Theorem 3; and 3) Theorem 7 is an extension of Theorem 6 because when Theorem 7 is identical to Theorem 6.
3. Test Statistic and Orthogonality
Let denote the observed frequency in the ith row and jth column of the table with and let denote the corresponding expected frequency. Assume that has a multinomial distribution. The maximum likelihood estimates of expected frequencies under each model could be obtained, for example, using the Newton-Raphson method to the log-likelihood equations. Each model (say, model) can be tested for goodness-of-fit by the likelihood ratio chi-squared statistic with the corresponding degrees of freedom, defined by
where is the maximum likelihood estimate of under the model. The number of degrees of freedom for the S model is and that for each of the LDPS(K) and CLDPS(K) models is (being one less than that for the S model). That for each of ME, GS, and MNE(K) models is 1. Note that the number of degrees of freedom for the S model is equal to the sum of those for the decomposed models.
Lang and Agresti [7] and Lang [8] considered the simultaneous modeling of a model for the joint distribution and a model for the marginal distribution. Aitchison [9] discussed the asymptotic separability, which is equivalent to the orthogonality in Read [10] and the independence in Darroch and Silvey [11], of the test statistic for goodness-of-fit of two models (also see Tomizawa and Tahata [12], Tahata et al. [13], and Tahata and Tomizawa [14]). On the orthogonality of test statistic for models in Theorem 6, we obtain.
Theorem 8. For a fixed test statistic is asymptotically equivalent to the sum of and
Proof. The LDPS(K) model may be expressed as
where Let
where “t” denotes the transpose, and
is the vector. The LDPS(K) model is expressed as
where is the matrix with and is the vector with
where
and is matrix of 0 or 1 elements determined from (1). The matrix is full column rank which is In a similar manner to Haber [15], Lang and Agresti [7], and Tahata and Tomizawa [16], we denote the linear space spanned by columns of the matrix by with the dimension Note that where is the vector of 1 elements, and thus Let be an where full column rank matrix such that the linear space is the orthogonal component of the space Thus, where is the zero matrix. Therefore, the LDPS(K) model is expressed as
where is the zero matrix, and
The MNE(K) model may be expressed as
where
Note that From Theorem 6, the S model may be expressed as
where
Note that are the numbers of degrees of freedom for testing goodness-of-fit of the LDPS(K), MNE(K) and S models, respectively.
Let denote the matrix of partial derivatives of with respect to i.e., Let where denotes a diagonal matrix with ith component of as ith diagonal component. We see that
because and that
Thus we obtain
Therefore we obtain where
From the asymptotic equivalence of the Wald statistic and the likelihood ratio statistic (Rao [17], Darroch and Silvey [11], Aitchison [9]), we obtain Theorem 8. The proof is completed.
4. Analysis of Data
Table 1 taken directly from Agresti [18, p. 232] summarizes responses to the questions “How successful is the government in (1) providing health care for the sick? (2) Protecting the environment?”.
Table 2 gives the values of the likelihood ratio test statistic for models applied to these data. The S model does not fit these data so well. Also, each of the ME (i.e., MNE(0)), MNE(K) and the GS models does not fit these data so well. However each of the LDPS(K) models and the CLDPS(K) models fit these data very well. Using Theorems 3 through 7 (including Theorems 1 and 2), we shall consider the reason why the S model fits these data poorly. For the structure of cell probabilities we see from Theorems 3, 5, 6 and 7 that the poor fit of the S model is caused by the influence of the lack of structure of the ME model (the GS model or the MNE(K) model) rather than the LDPS(K) model For the structure of cumulative probabilities we see from Theorem 4 that the poor fit of the S model is caused by the influence of the lack of structure of the ME model rather than the CLDPS(K) model
ReferencesA. H. Bowker, “A Test for Symmetry in Contingency Tables,” Journal of the American Statistical Association, Vol. 43, No. 244, 1948, pp. 572-574.http://dx.doi.org/10.1080/01621459.1948.10483284A. Agresti, “A Simple Diagonals-Parameter Symmetry and Quasi-Symmetry Model,” Statistics and Probability Letters, Vol. 1, No. 6, 1983, pp. 313-316.http://dx.doi.org/10.1016/0167-7152(83)90051-2K. Yamamoto and S. Tomizawa, “Statistical Analysis of Case-Control Data of Endometrial Cancer Based on New Asymmetry Models,” Journal of Biometrics and Biostatistics, Vol. 3, No. 5, 2012, pp. 1-4.http://dx.doi.org/10.4172/2155-6180.1000147N. Miyamoto, W. Ohtsuka and S. Tomizawa, “Linear Diagonals-Parameter Symmetry and Quasi-Symmetry Models for Cumulative Probabilities in Square Contingency Tables with Ordered Categories,” Biometrical Journal, Vol. 46, No. 6, 2004, pp. 664-674.http://dx.doi.org/10.1002/bimj.200410066H. Yamamoto, T. Iwashita and S. Tomizawa, “Decomposition of Symmetry into Ordinal Quasi-Symmetry and Marginal Equimoment for Multi-way Tables,” Austrian Journal of Statistics, Vol. 36, No. 4, 2007, pp. 291-306.K. Yamamoto and S. Tomizawa, “Analysis of Unaided Vision Data Using New Decomposition of Symmetry,” American Medical Journal, Vol. 3, No. 1, 2012, pp. 3742. http://dx.doi.org/10.3844/amjsp.2012.37.42J. B. Lang and A. Agresti, “Simultaneously Modeling Joint and Marginal Distributions of Multivariate Categorical Responses,” Journal of the American Statistical Association, Vol. 89, No. 426, 1994, pp. 625-632.http://dx.doi.org/10.1080/01621459.1994.10476787J. B. Lang, “On the Partitioning of Goodness-of-Fit Statistics for Multivariate Categorical Response Models,” Journal of the American Statistical Association, Vol. 91, No. 435, 1996, pp. 1017-1023.http://dx.doi.org/10.1080/01621459.1996.10476972J. Aitchison, “Large-Sample Restricted Parametric Tests,” Journal of the Royal Statistical Society: Series B, Vol. 24, No. 1, 1962, pp. 234-250.C. B. Read, “Partitioning Chi-Square in Contingency Tables: A Teaching Approach,” Communications in Statistics-Theory and Methods, Vol. 6, No. 6, 1977, pp. 553562. http://dx.doi.org/10.1080/03610927708827513J. N. Darroch and S. D. Silvey, “On Testing More than One Hypothesis,” Annals of Mathematical Statistics, Vol. 34, No. 2, 1963, pp. 555-567.http://dx.doi.org/10.1214/aoms/1177704168S. Tomizawa and K. Tahata, “The Analysis of Symmetry and Asymmetry: Orthogonality of Decomposition of Symmetry into Quasi-Symmetry and Marginal Symmetry for Multi-Way Tables,” Journal de la Société Francaise de Statistique, Vol. 148, No. 3, 2007, pp. 3-36.K. Tahata, H. Yamamoto and S. Tomizawa, “Orthogonality of Decompositions of Symmetry into Extended Symmetry and Marginal Equimoment for Multi-Way Tables with Ordered Categories,” Austrian Journal of Statistics, Vol. 37, No. 2, 2008, pp. 185-194.K. Tahata and S. Tomizawa, “Orthogonal Decomposition of Point-Symmetry for Multiway Tables,” Advances in Statistical Analysis, Vol. 92, No. 3, 2008, pp. 255-269.http://dx.doi.org/10.1007/s10182-008-0070-5M. Haber, “Maximum Likelihood Methods for Linear and Log-Linear Models in Categorical Data,” Computational Statistics and Data Analysis, Vol. 3, No. 1, 1985, pp. 110. http://dx.doi.org/10.1016/0167-9473(85)90053-2K. Tahata and S. Tomizawa, “Double Linear DiagonalsParameter Symmetry and Decomposition of Double Symmetry for Square Tables,” Statistical Methods and Applications, Vol. 19, No. 3, 2010, pp. 307-318.http://dx.doi.org/10.1007/s10260-009-0127-yC. R. Rao, “Linear Statistical Inference and its Applications,” 2nd Edition, John Wiley, New York, 1973.http://dx.doi.org/10.1002/9780470316436A. Agresti, “Analysis of Ordinal Categorical Data,” 2nd Edition, John Wiley, Hoboken, 2010.http://dx.doi.org/10.1002/9780470594001
square contingency table with the same row and column classifications. Let 
denote the probability that an observation will fall in the ith row and jth column of the table 
Bowker [
As a model which indicates the structure of asymmetry for 
Agresti [
is the S model. Yamamoto and Tomizawa [
and 
Miyamoto et al. [
The CLDPS model is different from the LDPS model. Yamamoto and Tomizawa [
and 
denote the row and column variables, respectively. We consider the mean equality (ME) model as
and 
and 
the S model holds if and only if both the LDPS(K) and ME models hold.
. Namely the S model holds. The proof is completed.
the S model holds if and only if both the CLDPS(K) and ME models hold.
the S model holds if and only if both the LDPS(K) and GS models hold.
consider the mean nonequality (MNE(K)) model as follows:
and 
is 
times higher than the difference between the global symmetric probabilities. When 
the MNE(0) model is identical to the ME model. We obtain Theorem 6. For a fixed 
the S model holds if and only if both the LDPS(K) and MNE(K) models hold.
and for a fixed 
the S model holds if and only if both the LDPS(K) and MNE(L) models hold.
Theorem 6 is identical to Theorem 1; 2) Theorem 7 is an extension of Theorem 3 because when 
Theorem 7 is identical to Theorem 3; and 3) Theorem 7 is an extension of Theorem 6 because when 
Theorem 7 is identical to Theorem 6.
denote the observed frequency in the ith row and jth column of the 
table with 
and let 
denote the corresponding expected frequency. Assume that 
has a multinomial distribution. The maximum likelihood estimates of expected frequencies 
under each model could be obtained, for example, using the Newton-Raphson method to the log-likelihood equations. Each model (say, model
) can be tested for goodness-of-fit by the likelihood ratio chi-squared statistic 
with the corresponding degrees of freedom, defined by
is the maximum likelihood estimate of 
under the model. The number of degrees of freedom for the S model is 
and that for each of the LDPS(K) and CLDPS(K) models is 
(being one less than that for the S model). That for each of ME, GS, and MNE(K) models is 1. Note that the number of degrees of freedom for the S model is equal to the sum of those for the decomposed models.
test statistic 
is asymptotically equivalent to the sum of 
and 
Let
vector. The LDPS(K) model is expressed as
is the 
matrix with 
and 
is the 
vector with
is 
matrix of 0 or 1 elements determined from (1). The matrix 
is full column rank which is 
In a similar manner to Haber [
by 
with the dimension 
Note that 
where 
is the 
vector of 1 elements, and thus 
Let 
be an 
where 
full column rank matrix such that the linear space 
is the orthogonal component of the space 
Thus, 
where 
is the 
zero matrix. Therefore, the LDPS(K) model is expressed as
is the 
zero matrix, and
From Theorem 6, the S model may be expressed as
are the numbers of degrees of freedom for testing goodness-of-fit of the LDPS(K), MNE(K) and S models, respectively.
denote the 
matrix of partial derivatives of 
with respect to 
i.e., 
Let 
where 
denotes a diagonal matrix with ith component of 
as ith diagonal component. We see that
and that
where
for models applied to these data. The S model does not fit these data so well. Also, each of the ME (i.e., MNE(0)), MNE(K) 
and the GS models does not fit these data so well. However each of the LDPS(K) models 
and the CLDPS(K) models 
fit these data very well. Using Theorems 3 through 7 (including Theorems 1 and 2), we shall consider the reason why the S model fits these data poorly. For the structure of cell probabilities 
we see from Theorems 3, 5, 6 and 7 that the poor fit of the S model is caused by the influence of the lack of structure of the ME model (the GS model or the MNE(K) model
) rather than the LDPS(K) model 
For the structure of cumulative probabilities 
we see from Theorem 4 that the poor fit of the S model is caused by the influence of the lack of structure of the ME model rather than the CLDPS(K) model 