Asymmetry Index on Marginal Homogeneity for Square Contingency Tables with Ordered Categories

For square contingency tables with ordered categories, the present paper considers two kinds of weak marginal homogeneity and gives measures to represent the degree of departure from weak marginal homogeneity. The proposed measures lie between –1 to 1. When the marginal cumulative logistic model or the extended marginal homogeneity model holds, the proposed measures represent the degree of departure from marginal homogeneity. Using these measures, three kinds of unaided distance vision data are analyzed.


Introduction
Consider an square contingency table with ordered categories.Let ij denote the probability that an observation will fall in the ith row and jth column of the table ( ; ).Also let X and denote the row and column variables, respectively.The marginal homogeneity (MH) model ( [1]) is defined by When the MH model does not hold, we are interested in applying the model that has weaker restriction than the MH model.As such a model, for example, there are the marginal cumulative logistic (ML) model ( [2]) and the extended marginal homogeneity (EMH) model ( [3][4][5]).We are also interested in considering the other structure of weak MH.The measures to represent the degree of departure from MH are given by, for example, [6,7].When the structure of weak MH does not hold, we are interested in measuring what degree the departure from weak MH is.
The present paper considers two kinds of structures of weak MH and proposes the measures to represent the degree of departure from weak MH.

Weak Marginal Homogeneity I and Measure
), and 3)  0 The MH model may be expressed as , and for , Note that .Assuming that  , we shall define the submeasure  as follows; , we see that 1) and   .Con- sider a measure defined by We see that 1) , 2) if and only if i (then ) and , and 3) if and only if 1, , ) and (then ) for all .Thus, indicates that 1R 1 1 p  and the other cell probabilities are zero (say, upper-rightmarginal inhomogeneity), and indicates that 1 R and the other cell probabilities are zero (say, lower-left-marginal inhomogeneity).When , we shall refer to this structure as the weak marginal homogeneity I (WMH-I).We note that if the MH model holds then the structure of WMH-I holds, but the converse does not hold.
Therefore, using the measure , we can see whether the structure of WMH-I departs toward the upper-rightmarginal inhomogeneity or toward the lower-left-marginal inhomogeneity.As the measure approaches -1, the departure from WMH-I becomes greater toward the upper-right-marginal inhomogeneity.While as the  approaches 1, it becomes greater toward the lower-leftmarginal inhomogeneity.

Weak Marginal Homogeneity II and Measure
, where for    The MH model may be expressed by for .
We shall consider the submeasure which is de- for .
The MH model may be expressed by for .
We shall consider the submeasure which is de- ) (say, conditional lower-left-marginal inhomogeneity).When c p i j  0   , we shall refer to this structure as the weak marginal homogeneity II (WMH-II).We note that if the MH model holds then the structure of WMH-II holds, but the converse does not hold.

Relationships between Measures and Models
We shall consider the relationship between the measure (or ) and the ML model.The ML model is given by where log 1 A special case of this model obtained by putting is the MH model.The ML model may also be expressed as We obtain the following theorem.
for , where


A special case of this model obtained by putting  is the MH model.Noting that ), we obtain the following theorem. Theorem 2. When the EMH model holds,  0 if and only if 1) Thus, when the ML (EMH) model holds, the measures  and  are adequate to represent the degree of departure from MH.

Approximate Confidence Interval for Measures
Let ij denote the observed frequency in the ith row and jth column of the table ( ; ).Assuming that a multinomial distribution applies to the we shall consider an approximate standard error and large-sample confidence interval for the measure  , using the delta method, as described by [8].The sample version of  , i.e., , is given by Using the delta method, we obtain the fol- lowing theorem. Theorem 3.
has asymptotically (as ) a normal distribution with mean zero and variance    , where and Also, the sample version of , i.e., , is given by with  ij replaced by   ˆij p .We obtain the following theorem.

 
has asymptotically (as ) a normal distribution with mean zero and vari-   , where ance

Examp
Example 1: Consider the unaided distance vision data in Table 1(a) taken fro [1].There are data on unaided distance vision of We see from Ta estimated value of th values in confidence interval for  are negative.Therefore, the structure of WMH-I for a woman's right and left eyes departs toward the upper-right-marginal inhomogeneity.Also we see from Table 3 that for the data in Table 1(a), the estimated value of the measure  is -0.0436 and all values in confiden e interval for c  are negative.Therefore, the structure of WMH-II for a woman's right and left eyes departs toward the conditional upper-right-marginal inhomogeneity.
Table 4 gives the values of likelihood ratio chi-squared statistic for testing goodness-of-fit of each of MH, L, and EMH models.We see from Table 4 that each of ML and EMH models fits these data well.Thus the measures  and  would indicate the degree of departure from MH.We can see from these measures that the de M gree of de inhomogeneity which indicates that the grade of right eye for arbitrary woman is "Best" and the grade of her left eye is "Worst".
Example 2: Consider the unaided vision data in Table 1(b), taken from [9].We see from  1(c), the estimated the  is 0.0125 and all values in confidence interval for  f are positive.Therefore, the structure of WMH-I for a student's right and left eyes departs toward the lower-leftmarginal inhomogeneity.Also we see from Table 3 that for the data in Table 1(c), the estimated value of the measure  is 0.0517 and all values in con idence interval for  are positive.Therefore, the structure of WMH-II for a student's right and left eyes departs toward the conditional lower-left-marginal inhomogeneity.
We see from Table 4 that each of ML and EMH models fits these data well.Thus the measures  and  would indicate the degree of departure from MH.We can see from these measures that the degree of departure from MH for the vision data in Table 1(c) is estimated to be 1.25 (5.17) percent of the maximum departure toward the (conditional) lower-left-marginal inhomogeneity which indicates t at the grade of right eye for arbitrary student is "Worst" a d the grade of his/her left eye is "Best".

Concluding Remarks h n
Fo t H r the analysis of square contingency tables with ordered categories, when the ML model, or he EM model, or other asymmetry models, for example, [11]'s conditional symmetry model (defined by ij ji p p   for i j  ) holds, the proposed measures  and  are adequate to represent the degree of departure from the MH model toward two maximum departures, i.e., toward the (conditional) lower-left-marginal inhomogeneity or toward the (conditional) upper-right-marginal inhomogeneity.ity (i.e., the r-lef he -r 8. Discussion [6,7] considered the measures to represent the degree of departure from MH.The present paper has considered two types of maximum marginal inhomogene lowe t-marginal inhomogeneity and t upper ightmarginal inhomogeneity).The measures in [6,7] take the value 1 in two types of maximum marginal inhomogeneity.The measures  and  in the present paper can distinguish these two kinds of maximum marginal inhomogeneity by the values -1 or 1 although the measures in [6,7] cannot distinguish them.Also the proposed esent the degree of departure from MH to ss measures can repr when the ML or the EMH models, or the other asymmetry models hold.Therefore for the ordinal data, the proposed measures rather than those in [6,7] may be useful to represent the degree of departure from MH.

1
the other ij are zero (

Table 2
that for the

Table 2 . Estimates of  , estimated approximate standard errors for  , and for approximate 95% confidence intervals , applied to Table
ˆ1.