^{1}

^{2}

For the analysis of square contingency tables with same row and column ordinal classifications, the present paper gives the decomposition of the generalized linear diagonals-parameter symmetry model using the diagonals-parameter symmetry model. Moreover, it gives the decomposition of the symmetry model using above the proposed decomposition.

Consider an

where

to the symmetry (S) (Bowker [

Yamamoto and Tomizawa [

where

Tomizawa [

For the analysis of square contingency tables with ordered categories, the purposes of this paper are (1) to give the decomposition of the LDPS(K) model using the DPS model, (2) to show that for the test statistic for the LDPS(K) model is equal to the sum of those for decomposed models, and (3) to give the decomposition of the S model using above the decomposition of the LDPS(K) model.

Tomizawa [

where

Let X and Y denote the row and column variables, respectively. The LDPMS model indicates that

Also, Tomizawa [

To consider the decomposition of the LDPS(K) model, we shall introduce a new model. For a fixed

Especially the LDPMS(0) model is equivalent to the LDPMS model.

We will denote

We obtain the following theorem.

Theorem 1. For a fixed

Proof. If the LDPS(K) model holds, then the DPS and LDPMS(K) models hold. Assuming that both the DPS and LDPMS(K) models hold, then we shall show that the LDPS(K) model holds.

From the LDPMS(K) model holds, we obtain

Also, from the DPS model holds, we see

Therefore, we obtain

Assume that a multinomial distribution applies to the

Each model can be tested for goodness-of-fit by, e.g., the likelihood ratio chi-square statistic (denoted by

where

A quick method for choosing the best-fitting model among different models is to use Akaike’s [

for each model. For more details of AIC, see Konishi and Kitagawa [

Thus, for the data, the model with the minimum AIC^{+} (i.e., the minimum AIC) is the best-fitting model.

For the analysis of contingency tables, Read [

On the orthogonality of test statistic for models in Theorem 1, we obtain the following theorem.

Theorem 2. For a fixed

The number of df for the LDPS(K) model equals the sum of number of df for the DPS and LDPMS(K) models.

Proof. First, we consider that the MLEs of expected frequencies

where

with

We can solve (3.1) for

Second, we consider that the MLEs of expected frequencies

where

Last, we consider that the MLEs of expected frequencies

where

For square contingency tables with ordered categories, Kurakami, Yamamoto and Tomizawa [

where

For a fixed

Kurakami et al. [

We will denote

Theorem 3. For a fixed

Theorem 4. For a fixed

The number of df for the S model equals the sum of the number of df for the LDPS(K) and WGS(K) models.

From the theorems given by Kurakami et al. [

Theorem 5. For a fixed

Theorem 6. For a fixed

The number of df for the S model equals the sum of the number of df for the LDPS(K) and WGS(K) models.

From Theorems 1 to 6, we obtain the following corollaries.

Corollary 1. For a fixed

Corollary 2. For a fixed

The number of df for the S model equals the sum of the number of df for the DPS, LDPMS(K) and WGS(K) models.

Consider the data in

The LDPMS(2) model is the best-fitting model among the other LDPMS(K) models because it has a mini- mum AIC^{+} value. Under the LDPMS(2) model, the MLE of

Theorem 1 would be useful for seeing the reason for its poor fit when the LDPS(K) model fits the data poorly. Thus, for the data in

Father’s status | Son’s status | Total | ||||
---|---|---|---|---|---|---|

(1) | (2) | (3) | (4) | (5) | ||

(1) | 18 | 17 | 16 | 4 | 2 | 57 |

(18.00) | (17.06) | (15.80) | (3.61) | (4.48) | ||

(2) | 24 | 105 | 109 | 59 | 21 | 318 |

(23.90) | (105.00) | (109.41) | (58.25) | (18.93) | ||

(3) | 23 | 84 | 289 | 217 | 95 | 708 |

(23.35) | (83.65) | (289.00) | (217.82) | (93.79) | ||

(4) | 8 | 49 | 175 | 348 | 198 | 778 |

(9.23) | (49.75) | (174.26) | (348.00) | (198.75) | ||

(5) | 6 | 8 | 69 | 201 | 246 | 530 |

(3.52) | (9.23) | (70.06) | (200.15) | (246.00) | ||

Total | 79 | 263 | 658 | 829 | 562 | 2391 |

Note: Status (1) is high professionals, (2) White-collar employees of higher education, (3) White-collar employees of less high education, (4) Upper working class, and (5) Unskilled workers.

Applied models | Df | AIC^{+} | |
---|---|---|---|

S | 10 | 24.80^{*} | 4.80 |

DPS | 6 | 14.84^{*} | 2.84 |

LDPS(−5) | 9 | 19.21^{*} | 1.21 |

LDPS(−4) | 9 | 19.41^{*} | 1.41 |

LDPS(−3) | 9 | 19.91^{*} | 1.91 |

LDPS(−2) | 9 | 21.94^{*} | 3.94 |

LDPS(−1) | 9 | 22.56^{*} | 4.56 |

LDPS(0) | 9 | 19.05^{*} | 1.05 |

LDPS(1) | 9 | 18.72^{*} | 0.72 |

LDPS(2) | 9 | 18.68^{*} | 0.68 |

LDPS(3) | 9 | 18.68^{*} | 0.68 |

LDPS(4) | 9 | 18.69^{*} | 0.69 |

LDPS(5) | 9 | 18.71^{*} | 0.71 |

LDPMS(−5) | 3 | 4.37 | −1.63 |

LDPMS(−4) | 3 | 4.57 | −1.43 |

LDPMS(−3) | 3 | 5.07 | −0.93 |

LDPMS(−2) | 3 | 7.11 | 1.11 |

LDPMS(−1) | 3 | 7.72 | 1.72 |

LDPMS(0) | 3 | 4.22 | −1.78 |

LDPMS(1) | 3 | 3.89 | −2.11 |

LDPMS(2) | 3 | 3.84 | −2.16 |

LDPMS(3) | 3 | 3.85 | −2.15 |

LDPMS(4) | 3 | 3.86 | −2.14 |

LDPMS(5) | 3 | 3.87 | −2.13 |

WGS(−5) | 1 | 5.59^{*} | 3.59 |

WGS(−4) | 1 | 5.39^{*} | 3.39 |

WGS(−1) | 1 | 2.22 | 0.22 |

WGS(0) | 1 | 5.73^{*} | 3.73 |

WGS(1) | 1 | 6.07^{*} | 4.07 |

WGS(2) | 1 | 6.12^{*} | 4.12 |

WGS(3) | 1 | 6.12^{*} | 4.12 |

WGS(4) | 1 | 6.11^{*} | 4.11 |

WGS(5) | 1 | 6.10^{*} | 4.10 |

^{*}Means significant at the 0.05 level.

the S model is caused by the poor fit of DPS and WGS(K) models rather than the LDPMS(K) model.

We have given the decomposition of the LDPS(K) model using the DPS model (namely, Theorem 1). Also, we have shown that the test statistic for the LDPS(K) is equal to the sum of those for the decomposed models (namely, Theorem 2). Moreover, we have given the decomposition of the S model using Theorem 1 (namely, Corollary 1), and shown that the test statistic for the S model is approximately equivalent to the sum of those for the decomposed models (namely, Corollary 2). Although details will be omitted, Yamamoto, Ohama and Tomizawa [

We thank the reviewer for the helpful comments. Also, we thank Professor S. Tomizawa and Dr. K. Tahata of Tokyo University of Science for their useful suggestions.

Shuji Ando,Hiroyuki Kurakami, (2016) Decomposition of Generalized Asymmetry Model for Square Contingency Tables. Open Journal of Statistics,06,405-411. doi: 10.4236/ojs.2016.63036