1. Introduction

Consider an square contingency table with the same row and column classifications. Let denote the probability that an observation will fall in the th row and th column of the table (), and let and denote the row and column variables, respectively. The marginal homogeneity (MH) model is defined by

namely

where and (see, for example, Stuart, 1955; Bishop, Fienberg and Holland, 1975, p.293).

Let

and

for . By considering the difference between the and , the MH model also be expressed as

Namely, this states that the cumulative probability that an observation will fall in row category or below and column category or above is equal to the cumulative probability that the observation falls in column category or below and row category or above for . When the MH model does not hold, we are interested in measuring the degree of departure from MH.

For square contingency tables with ordered categories, Tomizawa, Miyamoto and Ashihara (2003) proposed the measure (denoted by in Section 2) to represent the degree of departure from MH. The measure ranges between and Also, (i) if and only if the MH model holds, and (ii) if and only if the degree of departure from MH is a maximum; that is, (then ) or (then ) for all .

However, for the analysis of square contingency tables, all cell probabilities are positive in many cases. Thus, the measure may be unsuitable for such data, because the measure cannot attain the maximum value. So, we are now interested in the measure to represent the degree of departure from MH such that it can attain the maximum value even when each of cell probabilities is not zero. Yamamoto, Masumura and Tomizawa (2011) considered such a measure for nominal square table. We are now interested in proposing such a measure for ordinal square table.

The purpose of this paper is to consider an improvement of measure for square contingency tables with ordered categories when all cell probabilities are positive.

2. Improved Measure for Marginal Homogeneity

Consider an table with ordered categories. Assume that are positive. Let

for ; and let

For a specified which satisfies and for all , consider a measure defined by

where

with

and the value at is taken to be the limit as . Thus,

where

with

Note that is the diversity index proposed by Patil and Taillie (1982), which includes the Shannon entropy when . When , then is identical to the measure given by Tomizawa et al. (2003).

Since , the minimum value of is and the maximum value of it is or (if ) when for all . So, when cannot attain the value 1. The proposed measure with is modified by using modification coefficient such that the measure can attain the value . If all are positive, then must be taken as .

Moreover, for each and a fixed , the measure has characteristics that (i) must lie between and , (ii) if and only if the MH model holds, i.e., for all and (iii) if and only if the degree of departure from MH is the largest in the sense that for all .

The measure also may be expressed as, for

where

especially

Note that is the power-divergence between and (Cressie and Read, 1984) which includes the Kullback-Leibler information when .

3. Approximate Confidence Interval for Measure

Let denote the observed frequency in the th row and th column of the table (). Assume that a multinomial distribution applies to the table. The sample version of , is given by with replaced by , where and . Using the delta method (Bishop et al., 1975, Sec. 14.6), has asymptotically (as ) a normal distribution with mean zero and variance

where for

with

and the value of variance at is taken to be the limit as .

Let denote with replaced by . Using this result, the estimated approximate confidence interval for the measure is obtained.

4. Examples

Consider the data in Table 1, taken from Andersen (1997, p.226). These data show the forecasts for production and prices for the coming three year periods given by experts in July 1956 and the actual production figures for production and prices in May 1959 given from Danish factories.

For these data, the cell probabilities are theoretically positive (not zero). Thus, it may be irrelevance to use the measure with . So we should use the measure with (for example, ) so that the measure can attain the maximum value 1.

Table 1. Results from the forecasts for production and prices and the actual production figures for production and prices (Andersen, 1997, p.226)

Table 2. When      , the estimates of      , estimated approximate standard error for      , and approximate 95% confidence interval for      , applied to Tables 1a and 1b.

If we set and , the estimated measure is for Table 1a and for Table 1b from Tables 2a and 2b. Thus, (i) for Table 1a, the degree of departure from MH is estimated to be percent of the maximum degree of departure from MH and (ii) for Table 1b, it is estimated to be percent of the maximum. Furthermore, we see from Tables 2a and 2b that the degree of departure from MH is greater for Table 1a than for Table 1b because the values in the confidence intervals for are greater for Table 1a than for Table 1b.

5. Discussion

Consider the extended MH (EMH) model defined by

also see Tahata and Tomizawa (2008). A special case of EMH model obtained by putting is the MH model. When the EMH model holds, the proposed measure is expressed as

(1)

where

For fixed and fixed, increases as increases (or as decreases). Especially, when , is identical to proposed by Tomizawa et al. (2003). When the EMH model holds, approaches 1 as approaches infinity or zero. However, when the EMH model holds, cannot attain 1 because then and , namely there is not the structure of being the condition of . The measure with can attain the maximum value 1 even if and for all . Therefore, the measure with rather than may be appropriate when the EMH model holds. Also since the probabilities are positive (not zero), the measure with rather than would be appropriate to represent the degree of departure from the MH toward the structure of maximum departure from MH which can be defined actually.

The conditional symmetry (CS) model (McCullagh, 1978) is defined by

A special case of this model obtained by putting is the symmetry model (Bowker, 1948). If the symmetry model holds, then the MH model holds. Also if the CS model holds, then the EMH model holds. Therefore when the CS model holds, the measure is expressed by with replaced by . Thus by the similar reason, when the CS model holds, the measure with rather than would be appropriate.