Consider the data in Table 1. Table 1 is the data of grip strength of 805 male examinees aged 15 - 18 at high schools in Japan, which visited Tokyo University of Science, Open Campus, August, examined in 2011-2015. In Table 1 the row variable is the right hand muscle strength level and the column variable is the left hand muscle strength level. The category in Table 1 means muscle strength level compared with other people of one’s age and sex. Generally, for such data with similar classifications, many observations tend to fall (or near) the main diagonal cells. For the data in Table 1, 73% of observations concentrate in the main diagonal. Thus, the independence between classifications is unlikely to hold. Therefore, we are interested in whether or not there is a structure of symmetry with respect to the main diagonal in the table.

For the analysis of an square contingency table with the same ordinal 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] proposed the symmetry model, defined by

where (see also Martin and Pardo [2] ; Kolassa and Bhagavatula [3] ; Tahata and Tomizawa [4] ). This

model states that the probability that an observation will fall in the (i,j)th cell of the table is equal to the probability that it falls in the (j,i)th cell. Namely, this model describes a structure of symmetry with respect to the main diagonal of the table. Stuart [5] proposed the marginal homogeneity model, defined by

where and. This model states that the row marginal distribution is identical to the

column marginal distribution. Read [6] considered the global symmetry model, defined by

This model states that the probability that an observation will fall in one of the upper-right triangle cells above the main diagonal of the table is equal to the probability that it falls in one of the lower-left triangle cells below the main diagonal.

For the data in Table 1, we see that many observations fall in the upper-right triangle cells above the main diagonal. Thus, the models for symmetry between classifications are unlikely to hold. Then, the symmetry with respect to the anti-diagonal may hold for the data in Table 1. Note that the probabilities for the anti-diagonal cells are for the table. When the number of the categories is 3, i.e., , (such as the data in Table 1), the anti-diagonal cells are, and. Thus, we are interested in proposing new models for symmetry with respect to the anti-diagonal, which would hold for the data in Table 1.

The present paper proposes three models and gives a simple decomposition using the proposed models in contingency table. Also it illustrates new models with the grip strength data in Table 1.

Firstly, we propose a model defined by

where. The symbol “*” denotes. This model states that the probability that an observation will fall in the (i,j)th cell of the table is equal to the probability that it falls in the cell. Namely, this model indicates the structure of symmetry with respect to the anti-diagonal of the table. We shall refer to this model as the anti-diagonal symmetry (AS) model. Note that the AS model is a special case of the reverse conditional symmetry model, proposed by Tomizawa [7] .

Secondly, we propose a model defined by

Let X and Y denote the row and column variables, respectively. Then, this model is also expressed as

We shall refer to this model as the anti-diagonal global symmetry (AGS) model.

Finally, we propose a model defined by

This model states that the row marginal distribution is identical to the column marginal distribution in reverse order. We shall refer to this model as the anti-diagonal marginal homogeneity (AMH) model.

We obtain the following theorem.

Theorem 1. When, the AS model holds if and only if both the AGS and AMH models hold.

Proof. If the AS model holds, then the AGS and AMH models hold. Assuming that both the AGS and AMH models hold, then we shall show that the AS model holds. If the AMH and AGS models hold, then we have (i.e.,), (i.e.,) and . Thus, we see, and. Namely, the AS model holds. The proof is completed.

Note that this theorem does not hold when. Let denote the observed frequency in the (i,j)th cell of

the table () with, and let denote the corresponding expected frequency.

Assume that have a multinomial distribution. The maximum likelihood estimates (MLEs) of under the AS and AGS model, are expressed as the closed-forms as follows:

1) The MLE of under the AS model is

2) The MLE of under the AGS model is

where

The MLEs of under the AMH model could be obtained using the Newton-Raphson method in the log-likelihood equation. Let denote the likelihood ratio chi-squared statistic for testing goodness-of-fit of model M. For the AS model, are determined by of, of

for anti-diagonal cells (since), thus a total of. Therefore, the number of degrees

of freedom (df) for testing goodness-of-fit of the AS model is. Similarly, the numbers of df for testing goodness-of-fit of the AGS and AMH model are 1 and, respectively. Note that when, the number of df for the AS model is greater than the sum of numbers of df for the AGS and AMH models, and when, it is equal to the sum of them.

We shall consider the comparison between two nested models. Suppose that model is a special case of model; that is, is simpler than, so when holds, necessarily also holds. For testing that model holds assuming that model holds, we can use the likelihood ratio statistic which is the difference between the and. When model holds, this statistic has an asymptotic chi-squared distribution with df being equal to the difference between the df for and for.

Consider the data in Table 1 again. All the AS, AGS and AMH models fit these data well, yielding the likelihood ratio statistic with 3 df, with 1 df, and with 2 df, respectively. Since the AS model is a special case of the AGS model, we shall test the hypothesis that the AS model holds assuming that the AGS model holds. Since with 2 df being the difference between the numbers of df for the AS and AGS models, this hypothesis is accepted at the 0.05 significance level. Thus, the AS model would be preferable to the AGS model. Similarly, since the AS model is a special case of the AMH model, we shall test the hypothesis that the AS model holds assuming that the AMH model holds. Since with 1 df being the difference between the numbers of df for the AS and AMH models, this hypothesis is rejected at the 0.05 significance level. Therefore, the AMH model would be preferable to the AS model.

Under the AMH model, the probability that an examinee’s right hand grip strength level is “Excellent (1)”, is estimated to be equal to the probability that an another examinee’s left hand grip strength level is “Poor (3)”. Also, the probability that an examinee’s right hand grip strength level is “Poor (3)”, is estimated to be equal to the probability that an examinee’s left hand grip strength level is “Excellent (1)”.

The decomposition of the AS model into the AGS and AMH models, given by Theorem 1, would be useful for seeing the reason for its poor fit when the AS model fits the data poorly, and it should be considered for ordinal categorical data because all the AS, AGS and AMH models are not invariant under arbitrary same permutations of row and column categories.

The authors would like to thank the referee for their helpful comments.

Kiyotaka Iki, (2016) Analysis of the Grip Strength Data Using Anti-Diagonal Symmetry Models. Open Journal of Statistics,06,590-593. doi: 10.4236/ojs.2016.64049