Analysis of data in square contingency tables with ordered categories using the conditional symmetry model and its decomposed models.

For the analysis of square contingency tables with ordered categories, three kinds of decompositions for the conditional symmetry model derived by Tomizawa are simply described. Using the conditional symmetry model and its decomposed models, this paper analyzes the data of unaided distance vision of women in Britain first analyzed by Stuart, the data of unaided distance vision of students in a university in Japan, and the data of unaided distance vision of pupils at elementary schools at a city in Tokyo.


Introduction
is constructed from the data of the unaided distance vision of 7477 women aged 30-39 employed in Royal Ordnance factories in Britain from 1943 to 1946. The data in Table 1 were first analyzed by Stuart (1,2). Table 2 is constructed from the data of the unaided distance vision of 4746 students aged 18 to about 25, including about 10% of the women of the Faculty of Science and Technology, Science University of Tokyo in Japan examined in April, 1982. Table 3 is constructed from the data of the unaided distance vision of 3168 pupils aged 6-12, including about half the girls at elementary schools in Tokyo, Japan examined in June 1984. In Tables 1, 2, and 3 the row variable is the right eye grade and the column variable is the left eye grade with the categories ordered from the lowest grade (1) to the highest grade (4).
To the data of Tables 1, 2, and 3 it is reasonable to   (7) proposed a point-symmetry model in J-dimensional contingency cubes. Tomizawa (8,9) proposed models of various kinds of point symmetry in two-dimensional contingency tables and gave their decompositions. In this paper we analyze the data in Tables 1, 2, and 3 using the conditional symmetry model and its decomposed models.

Models and Decompositions
Consider the square a x a contingency table with row variate denoted by X1 and column variate denoted by X2. Let pij denote the probability in the cell in row i and column j for 1i-£i a.
The models of symmetry, quasisymmetry and marginal homogeneity are defined as follows: and where the parameter y is unspecified. This model is also equivalent to the model HQ*: PijPjkPki = YPjiPkjPik for 1 i < j < ka where the parameter y is unspecified. Model H* is also expressed by a log-linear model for the pii in Tomizawa (6). A special case of model H4 obtained by putting y = 1 is the quasisymmetry model.
We introduce two kinds of modified marginal homogeneity models as follows: Hs: pij p Pji HQ: pij = aibjdij where F,ij = O>i and II0ij = 1. This model can be also expressed by a log-linear model for the pij in Tomizawa (6).
We next define the extended quasisymmetry model by for and introduce three kinds of average models as follows: where a -1 where HR*3: R = A3 a) where Xt;x, = N, and let mij denote the corresponding expected frequency under some model. We assume here that a multinomial distribution applies to thp a x a. t.ah1p The degrees of freedom for models HS, HQ, Hf,I Hj (1 = 1,2,3) and HM,(l = 1,2) are (a -2) (a + 1)/2, a(a -3)/2, a -2, 1, and a -1, respectively.
The maximum likelihood estimates of mij under model HQ can be sought by the iterative procedure in Tomizawa (6) or by the following iterative procedure: as the (k + 1)th step Here R indicates the average of ratio PijPjkPkil(PjiPkjPik) for 1 S i < j < ka in the case that the ratio parameter y in HQ changes according to each ratio, and Al indicates the average ofp iJp . i for 1 -ia -1 in the case that the ratio parameter 8 in HM1 changes according to each ratio, and also A2 iS interpreted similarly, and A3 indicates the average of two ratios in the case that the ratio p',Jp.-pis not always* equal to the ratio P.a/ p a..Therefore model HRI for 1 = 1, 2, and 3 indicates the equilibrium of two kinds of averages R and Al. We get the decompositions for model as follows. THEOREM: for = 1, 2 and 3, model HS holds if and only if all models HQ, HMI and HRI hold.
The proof of this theorem is given by Tomizawa (6). We denote special cases of models HMI(l = 1,2) obtained by putting 8 = 1 by HMI (1 = 1,2).  Table 4. Chi-square for symmetry models applied to the data in Table 1.  Table 6. Chi-square for symmetry models applied to the data in Analysis of Table 1   Table 4 presents the likelihood-ratio and the Pearson's chi-squared statistics obtained by applying the models introduced in the previous section to the data in Table 1. The value of test statistic Q in Stuart (2) for testing the goodness of fit of model HM is 11.96 with 3 degrees of freedom. The value of test statistic XM3 in Tomizawa (6) for testing the goodness of fit of the extended marginal homogeneity model HM3 is 0.005 with 2 degrees of freedom. From these values and Table 4, none of models HMt, HM2, and HM fits the data well, but all of models HMl, HM2, and HM3 fit the data very well. Moreover the maximum likelihood estimates of 8 under models HM1 and HM2 are 0.863. Since this value is less than one, we can say that the left eye is worse than the right eye. Also the values of chi-square under model HQ lie between the upper 5% and 1% tail values of the x2 distribution with 2 degrees of freedom. Under model HQ the estimated value of -y obtained by maxi-mum likelihood is 0.929. Also model HS does not fit the data well, and thus the left eye is not symmetric to the right eye. But model Hs fits adequately and the estimated values of mij/mjli for 1 i < j S 4 under model HS are 0.863. Since this value is less than one, we can say again that the left eye is worse than the right eye.
Analysis of Table 2   Table 5 presents the likelihood-ratio and the Pearson's chi-squared statistics obtained by applying various kinds of symmetry models to the data in Table 2. The value of test statistic Q in Stuart (2) for testing the goodness of fit of model HM is 11.21 with 3 degrees of freedom. This value lies between the upper 5% and 1% tail values of the x2 distribution with 3 degrees of freedom. The value of test statistic X32 in Tomizawa (6) for testing the goodness of fit of the extended marginal homogeneity model HM3 is 0.56 with 2 degrees of freedom. From this value and and HM2 are 1.228, and since this value is greater than one, this value indicates that the left eye is better than the right eye. Both models*HQ and HQ also fit adequately, and under model HQ the estimated value of y obtained by maximum likelihood is 1.208. Also, since model HS does not fit the data well, the left eye is not symmetric to the right eye. But model HS fits the data well, and the estimated values of mij/mji for 1 -i < j -4 under model Hs are 1.228. Since this value is greater than one, we can say again that the left eye is better than the right eye. Table 3   Table 6 presents the likelihood-ratio and the Pearson's chi-squared statistics obtained by applying various kinds of symmetry models to the data in Table 3. The value of test statistic Q in Stuart (2) for testing the goodness of fit of model HM is 6.85 with 3 degrees of freedom, and the value of test statistic XM32 in Tomizawa (6) for testing the goodness of fit of the extended marginal homogeneity model HM3 is 4.16 with 2 degrees of freedom. From these values and Table 6, all models fit the data well. Moreover under model HQ the estimated value of y obtained by maximum likelihood is 1.136 and the maximum likelihood estimates of 8 under models HM1 and HM2 are 0.871. We may consider these values close upon one because models HQ, HMl, and HM2 hold.

Analysis of
Also the values of statistics for the goodness of fit of models HS and HM1(1= 1, 2, 3) applied to Table 3 are greater than those applied to Table 1 and 2; namely, the goodness of fit of models HS and H;,(l = 1, 2, 3) applied to Table 3 are not so good as those applied to  Table 1 and 2 did not fit the data well, but those applied to Table 3 fit the data well. Therefore, for the data in Table 3 we can say that the left eye is symmetric to the right eye in the various senses.
The author would like to express his sincere thanks to Professor K. Kunisawa, Mr. M. Jimbo, and Dr. K. Shimizu in Science University of Tokyo for their valuable advice and encouragements.
The author is grateful to Dr. A. Nakajima and Dr. K. Fujiki in Department of Ophthalmology, Juntendo University, for their valuable advice and for giving him the data in Table 3 in this paper.
He is also deeply grateful to Dr. A. Kudo and Dr. T. Yanagawa in Kyushu University who provided him with an excellent opportunity for presenting this paper at the Environmental Risk Assessment and Statistical Methods Conference which is a satellite conference in conjunction with the 12th International Biometric Conference.