Analysis of the Grip Strength Data Using Anti-Diagonal Symmetry Models

For the analysis of square contingency tables with the same row and column ordinal classifications, this article proposes new models which indicate the structures of symmetry with respect to the anti-diagonal of the table. Also, this article gives a simple decomposition in 3 × 3 contingency table using the proposed models. The proposed models are applied to grip strength data.


Introduction
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 r r × square contingency table with the same ordinal row and column classifications, let ij p denote the probability that an observation will fall in the ith row and jth column of the table ( 1, , ; 1, , i r j r = =   ).Bowker [1] proposed the symmetry model, defined by ( ) (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 . 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 1 1), the anti-diagonal cells are 13 p , 22 p and 31 p .Thus, we are interested in proposing new mod- els 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 3 3 × contingency table.Also it illustrates new models with the grip strength data in Table 1.

New Models and a Simple Decomposition
Firstly, we propose a model defined by ( ) . The symbol "*" denotes 1 i r i * = + − .This model states that the probability that an observa- tion will fall in the (i,j)th cell of the table is equal to the probability that it falls in the ( ) th j i * * , 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 1 1 . ij ij i j r i j r p p 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 3 r = , 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 1) The MLE of ij m under the AS model is The MLEs of { } ij m under the AMH model could be obtained using the Newton-Raphson method in the log-likelihood equation.Let for anti-diagonal cells (since ), thus a total of ( )( ) .Similarly, the numbers of df for testing goodness-of-fit of the AGS and AMH model are 1 and 1 r − , respectively.Note that when 4 r ≥ , 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 3 r = , it is equal to the sum of them.We shall consider the comparison between two nested models.Suppose that model

( ) 2 G
M denote the likelihood ratio chi-squared statistic for testing goodness-of-fit of model M. For the AS model, { } ij p are determined by ( ) the number of degrees of freedom (df) for testing goodness-of-fit of the AS model is ( )(

Table 1 .
Grip strength test of 805 male examinees aged 15 -18 at high schools in Japan, examined in 2011-2015.(The parenthesized values are MLEs of expected frequencies under the AMH model).