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For two-way contingency tables with ordered categories, the present paper gives a theorem that the independence model holds if and only if the logit uniform association model holds and equality of concordance and discordance for all pairs of adjacent rows and all dichotomous collapsing of the columns holds. Using the theorem, we analyze the cross-classification of duodenal ulcer patients according to operation and dumping severity.

Consider the

See also Agresti ([

where

Namely this model indicates the constant of the

If the I model holds, the correlation coefficient of X and Y equals zero; but the converse does not hold. We are interested in what structure between X and Y is necessary for obtaining the I model, in addition to the correlation coefficient being to zero.

Tomizawa, Miyamoto and Sakurai [

Tomizawa et al. [

Tahata, Miyamoto and Tomizawa [

Suppose that the column variable Y is a response variable. Let

where

The logit uniform association (logit U) model (Agresti [

namely

where

Thus the logit U model indicates the constant of the odds ratios for the

The purpose of the present paper is to give the decomposition of the I model by using the logit U model (in Section 2).

Let

and

For a randomly selected pair of observations, 1)

member that ranks in row

j or below, and 2)

rather than in row i ranks in column j or below rather than in column

We shall consider the model of equality of concordance and discordance (say, CDE model) by

Then we obtain the following theorem.

Theorem 1. The I model holds if and only if both the CDE model and the logit U model hold.

Proof. If the I model holds, i.e.,

and

Thus, the CDE model holds. Also, if the I model holds, then the logit U model (with

Assuming that both the CDE model and the logit U model hold, then we shall show that the I model holds. Since the logit U model holds, we see

Thus

Since the CDE model holds, we obtain

Let

where

The data in

Operation | Dumping Severity | |||
---|---|---|---|---|

None | Slight | Moderate | Total | |

A | 61 | 28 | 7 | 96 |

B | 68 | 23 | 13 | 104 |

C | 58 | 40 | 12 | 110 |

D | 53 | 38 | 16 | 107 |

Total | 240 | 129 | 48 | 417 |

Source: Grizzle et al. [

based on

For testing the hypothesis that the I model holds assuming that the logit U model holds, the difference be- tween the

Also the CDE model fits these data poorly with

When the I model fits the data poorly, Theorem 1 may be useful for seeing the reason for the poor fit; namely, which of the lack of structure of the CDE model and that of the logit U model influences stronger.

From Theorem 1 we point out that the hypothesis that the I model holds under the assumption that the logit U model holds is equivalent to the hypothesis that the CDE model holds.

The U model indicates the constant of the

We thank the referee for comments and suggestions.

KoujiTahata,NobukoMiyamoto,SadaoTomizawa, (2015) Decomposition of Independence Using the Logit Uniform Association Model and Equality of Concordance and Discordance for Two-Way Classifications. Open Journal of Statistics,05,514-518. doi: 10.4236/ojs.2015.56054