1. INTRODUCTION

Consider an r×r square contingency table with the same row and column ordinal classifications. Let pij denote the probability that an observation will fall in the ith row and jth column of the table (i=1,…,r;j=1,…,r), and let X and Y denote the row and column variables, respectively. The marginal homogeneity (MH) model is defined by

namely where pi⋅=∑t=1rpit and p⋅i=∑t=1rpti ([1], [2, p. 282]). This model is also expressed as namely where

This indicates that the conditional row marginal distribution is identical to the conditional column marginal distribution under the condition that an observation will fall in one of the off-diagonal cells of the table.

We consider the (r−1)(r−2)∕2 (being r−12) ways of collapsing the r×r original table with ordered categories into a 3×3 table by choosing cut points after the sth and tth rows and after the sth and tth columns for 1≤s<t≤r−1. We define each collapsed 3×3 table as the Tst table (1≤s<t≤r−1). This process means that r categories in the r×r original table get into 3 categories (for example, A, B and C) by dividing at the cut points s and t and then a Tst table (1≤s<t≤r−1) has the new 3 categories. In a collapsed Tst table (1≤s<t≤r−1), let Gij (s,t) indicate the corresponding probability for row value i (i=1,2,3) and column value j (j=1,2,3); that is,

Then, the MH model is expressed as

where for all s and t (1≤s<t≤r−1); see [3].

When the MH model does not hold, we are interested in measuring the degree of departure from MH. For square contingency tables with nominal categories, Tomizawa and Makii [4] proposed a measure which expresses the degree of departure from MH. In addition, for square contingency tables with ordered categories, Tomizawa, Miyamoto and Ashihara [5] proposed the measure Γ(λ), which represents the degree of departure from MH. See Appendix A for the measure Γ(λ).

Moreover, when the MH does not hold, we are interested in measuring the degree of departure from MH for every collapsed table Tst (1≤s<t≤r−1). The purpose of this paper is to propose a new measure which expresses the degree of departure from MH by adopting collapsed 3 ×3 tables. Section 2 considers such a measure and Section 3 gives an approximate variance and a confidence interval for the measure. Section 4 shows examples using the proposed measure. Section 5 gives the concluding remarks.

2. MEASURE OF DEPARTURE FROM MH FOR COLLAPSED 3 × 3 TABLES

Let for 1≤s<t≤r−1,

and

Assume that Gi⋅ c(s,t)+G⋅i c(s,t) are all positive. The MH model is also represented as

for 1≤s<t≤r−1.

Consider a measure defined by

where and the value at λ=0 is taken to be continuous limit as λ→0. Namely where

The submeasure Ωst (λ) (1≤s<t≤r−1) describes the degree of departure from the MH model for the collapsed Tst table. Note that Ist (λ)Gi⋅ c(s,t),G⋅i c(s,t);πst(i)c∗,πst(i)c∗ is the power-divergence between Gi⋅ c(s,t),G⋅i c(s,t) and πst(i)c∗,πst(i)c∗, and especially, Ist (0)(⋅) is the Kullback–Leibler information. Also, note that a real value λ (>−1) is chosen by users. See [6] for the power-divergence.

Let for 1≤s<t≤r−1,

Note that G1(i) c(s,t)+G2(i) c(s,t)=1. The MH model may be expressed as

for 1≤s<t≤r−1.

Then, Ωst (λ) (1≤s<t≤r−1) may also be defined by

where and the value at λ=0 is taken to be continuous limit as λ→0. Namely,

Therefore, Ωst (λ) represents the weighted sum of the power-divergence Ist(i) (λ)Gk(i) c(s,t);12. Moreover, Ωst (λ) can be expressed as

where and the value at λ=0 is taken to be continuous limit as λ→0. Namely where

Note that Hst(i) (λ)Gk(i) (s,t) is Patil and Taillie’s [7] diversity index of degree λ for G1(i) c(s,t),G2(i) c(s,t), which includes the Shannon entropy (when λ = 0). Therefore, Ωst (λ) represents the weighted sum of the diversity index Hst(i)(λ)({Gk(i)c(s,t)}). Note that for each λ>−1,

1. 0≤Hst(i) (λ)(⋅)≤(2λ−1)∕(λ2λ),

2. Hst(i) (λ)(⋅)=0 if and only if G1(i) c(s,t)=1 (then G2(i) c(s,t)=0) or G2(i) c(s,t)=1 (then G1(i) c(s,t)=0),

3. Hst(i) (λ)(⋅) = (2λ−1)∕(λ2λ) if and only if G1(i) c(s,t)=G2(i) c(s,t) =12, that is Gi⋅ c(s,t) = G⋅i c(s,t).

Thus, we conclude that the measure Ω(λ) lies between 0 and 1, and the submeasure Ωst (λ) also lies between 0 and 1. For each λ>−1,

• d.

  there is a structure of MH in the r×r table if and only if Ω(λ) = 0,

• e.

  the degree of departure from MH in the r×r table is the largest, in the sense that G1(i) c(s,t)=1 (then G2(i) c(s,t)=0) or G2(i) c(s,t)=1 (then G1(i) c(s,t)=0) for i=1,2,3 and 1≤s<t≤r−1 if and only if Ω(λ)=1,

and for fixed s and t (1≤s<t≤r−1),

1. f.

   there is a structure of MH in a collapsed 3×3 table Tst if and only if Ωst (λ)=0,

2. g.

   the degree of departure from MH in a collapsed 3×3 table Tst is the largest, in the sense that G1(i) c(s,t)=1 (then G2(i) c(s,t)=0) or G2(i) c(s,t)=1 (then G1(i) c(s,t)=0) for i=1,2,3 if and only if Ωst (λ)=1.

3. APPROXIMATE CONFIDENCE INTERVAL FOR MEASURE

Let nij denote the observed frequency in the ith row and jth column of the r×r table (i=1,…,r;j=1,…,r). Assuming that a multinomial distribution applies to the table, we shall consider an approximate standard error and a large sample confidence interval for the measure Ω(λ), using the delta method. The sample version of Ω(λ), that is, Ω̂(λ), is given by Ω(λ) with {pij} replaced by {p̂ij}, where p̂ij=nij∕n and n=∑∑nij. Using the delta method, n(Ω̂(λ)−Ω(λ)) has asymptotically (as n→∞) a normal distribution with mean 0 and variance σ2[Ω(λ)]. See Appendix B for the details of σ2[Ω(λ)] and Appendix C for the details of the submeasure Ωst (λ).

Let σ̂2[Ω(λ)] denote σ2[Ω(λ)] with {pij} replaced by {p̂ij}. Then σ̂[Ω(λ)]∕n is an estimated approximate standard error for Ω̂(λ), and Ω̂(λ)±zp∕2σ̂[Ω(λ)]∕n is an appriximate 100(1−p) percent confidence interval for Ω(λ), where zp∕2 is the percentage point from the standard normal distribution corresponding to a two-tail probability equal to p.

4. EXAMPLES

Consider the data in Table 1. These data describe the cross-classification of father’s and son’s occupational status categories in Japan and British.

Since the confidence intervals for Ω(λ) applied to the data in Tables 1a and 1b do not include zero for all λ (see Table 2), these would indicate that there is not the structure of MH in both tables.

When the degrees of departure from MH in Tables 1a and 1b are compared using the confidence intervals for Ω(λ), it is greater in Table 1a than in Table 1b. Then, we can see that the degree of departure from MH is greater for Table 1a than for Table 1b. Therefore, the difference between the father’s classification distribution and his son’s distribution is greater in Japan than in British.

We shall further analyze the data in Tables 1a and 1b using the submeasure Ωst (λ) (1≤s<t≤r−1). We see from Table 3 that for Table 1a, (i) the degree of departure from MH in the collapsed table T12 is the smallest and (ii) those in the other collapsed tables are greater than that in the T12 table. Thus, it is seen that (i) when we combine the categories (3) to (5) in Table 1a, the degree of departure from MH for the collapsed table is slightest, and (ii) when we combine the categories in the other patterns in Table 1a, those for the collapsed tables are greater than that for case (i).

However, there is no possibility to decide in which collapsed table the degree of departure from MH is largest, because the values in the confidence intervals for them overlap each other except for Ω12 (λ). So, we can get a conclusion, for Table 1a, that even if we get the original 5 categories into the 3 categories in any pattern to make us interpret the original table more easily, the marginal distributions for each collapsed table are different between father and his son’s social classes. Moreover, if we consider the original categories (1) as the high class, (2) as the middle class and (3) to (5) as the low class, social mobility between father and his son’s social classes is less than that for the other classification patterns. We also see from Table 3 that for Table 1b, the confidence intervals for the submeasures Ω12 (λ), Ω13 (λ) and Ω23 (λ) include zero, on the other hand, those for the other submeasures do not include zero. These results mean that when we combine the categories (3) to (5) in Table 1b, the marginal distribution for father’s social class is equal to that for his son’s one. Note that we can also obtain similar interpretations when we combine the categories (2) to (3) and (4) to (5), and (1) to (2) and (4) to (5) in Table 1b. Thus, if we consider the original categories (1) as the high class, (2) as the middle class and (3) to (5) as the low class, we could not see that social mobility between father’s social class and his son’s one happened. Also note that we are able to get a similar interpretations when we consider combining the categories (2) to (3) and (4) to (5), and (1) to (2) and (4) to (5) in Table 1b. As a result, for Table 1b, if we regard only the category (5) as the low class, when we consider the same specified category between father and his son, the probability that fathers whose son’s social class is lower than him is different from the probability that sons whose father’s social class is higher than him.

We can also see from Table 4 that the degree of departure from MH is greater for Table 1a than for Table 1b by using the measure Γ(λ) proposed by Tomizawa et al. [5]. However, when we would like to analyze these data in more detail, for example, by using collapsed tables as described above, it is impossible to do it by using the measure Γ(λ). So, in such a situation, the proposed measure Ω(λ) would be useful.

Furthermore, we can know more about the degree of departure from MH in Tables 1a and 1b. From Ω̂(λ), for example, with λ=1, (i) for Table 1a, the degree of departure from MH is estimated to be 42.26 percent of the maximum degree of departure from MH, and (ii) for Table 1b, the degree of departure from MH is estimated to be 1.04 percent of the maximum degree of departure from MH.

5. CONCLUDING REMARKS

The measure Ω(λ) always ranges from 0 to 1 independent of the dimension r and sample size n. Thus, it may be useful for comparing the degrees of departure from MH in several tables.

When the MH model does not hold, we are interested in (i) seeing what degree the departure from MH is for the original r×r table, (ii) seeing what degree the departure from MH is for Tst tables (1≤s<t≤r−1) and (iii) seeing in which Tst table (1≤s<t≤r−1) the degree of departure from MH is the largest. We recommend to use the proposed measure Ω(λ) for (i) and the proposed submeasure Ωst (λ) for (ii) and (iii). The submeasure may also be used in the case that you would like to just analyze one collapsed table. Note that the measure Ω(λ) is not invarianrt under the arbitrary permutations of row and column categories except the reverse order, so this measure should be applied only for ordinal data.

Consider the artificial data in Table 5. Let G2 indicate the likelihood ratio statistic for goodness-of-fit of MH model. Table 6a gives the values of G2 applied to these data. We shall compare the values of G2 for Tables 5a and 5b. We see that the value of G2 for Table 5a is greater than that for Table 5b. In contrast, for any fixed λ(>−1), the value of Ω̂(λ) is greater for Table 5b than for Table 5a (see Table 6b). In terms of G^i⋅c(s,t)/G^⋅ic(s,t), (i=1,2,3, 1≤s<t≤r−1) (see Table 5), it seems natural to conclude that the degree of departure from MH is less for Table 5a than for Table 5b. Therefore we recommend using Ω̂(λ) for comparing the degrees of departure from MH among several tables. To many readers, it might seem that G2∕n is also a reasonable measure for representing the degree of departure from MH. However, G2∕n is not such a measure for us. For instance, consider the artificial data in Tables 5b and 5c. The values of G2∕n are 0.0261 for Table 5b and 0.1284 for Table 5c. Therefore, the value of G2∕n is less for Table 5b than for Table 5c. On the other hand, for any fixed λ(>−1), the value of Ω̂(λ) for Table 5b is equal to that for Table 5c (see Table 6b). Moreover, G^i⋅c/G^⋅ic, i=1,2,3 for Table 5b is identical to that for Table 5c (see Table 5). Therefore, it seems natural to get a conclusion that there are no differences between Tables 5b and 5c for the degree of departure from MH. As a result, Ω̂(λ) may also be more desirable to measure the degree of departure from MH than G2∕n.

Finally, the readers may be interested in considering a measure based on unconditional marginal probabilities (say, Γ(λ)), instead of the proposed measure Ω(λ) based on conditional marginal probabilities. The measure Γ(λ) takes the minimum value 0, when there is the structure of MH in the original table. On the other hand, we cannot define the maximum value, because it is impossible to define the structure of the furthest departure from MH. Thus, when the readers wants to consider the degree of departure from MH in addition to define the maximum departure from MH, we recommend using the measure Ω(λ). If not, each measure, Γ(λ) and Ω(λ), is enough to use.

CONFLICT OF INTEREST

Authors have no conflict of interest to declare.