Extension of Marginal Complementary Log–Log Model and Separations of Marginal Homogeneity for Ordinal Categorical Data

The present paper considers a model that indicates the structure of inhomogeneity for marginal distributions for ordinal categorical data. The model is based on the complementary log–log transformation of marginal cumulative probability. A theorem that the marginal homogeneity model holds if and only if the proposed model and the marginal mean and variance equality model holds is given. Also, a model based on the conditional marginal distribution is considered under certain condition.

between two variables does not hold in the contingency table which is constructed from matched-pairs data. Then, we are interested in considering various symmetry (or asymmetry) instead of independence. This paper treats methods for analyzing square tables.
For a r × r square contingency table with ordered categories, let X and Y be the row and column random variables, respectively. Also, let p ij denotes the probability that an observation will fall in the ith row and jth column of the table for i = 1, … , r;j = 1, … , r . We assume p ij > 0 for all i and j. Bowker [4] considered the symmetry model defined by p ij = p ji for i < j . The marginal homogeneity (MH) model is defined by where p i⋅ = ∑ r t=1 p it and p ⋅i = ∑ r s=1 p si (Stuart [15]). This model indicates the structure that satisfies the identity of marginal distributions of row and column. Also, Caussinus [5] showed the theorem that the symmetry model holds if and only if both the quasi-symmetry model, which indicates a symmetric structure for odds ratios, and the MH model holds. We shall refer this theorem as separation of symmetry. For the details of method for joint distribution, please see Bishop et al. [3] and Agresti [2].
Let F X i and F Y i denote the marginal cumulative probabilities of X and Y, respectively, namely F X i = ∑ i s=1 p s⋅ and F Y i = ∑ i t=1 p ⋅t for i = 1, … , r − 1 . The MH model may also be expressed as p i⋅ = p ⋅i (i = 1, … , r), When the MH model fits poorly for a real data set, we are interested in applying some extension of the MH model. Indeed, the MH model fits the data in Table 1 poorly (see Sect. 6.1). McCullagh [11] considered the marginal cumulative logistic (ML) model defined as follows: where L X i and L Y i denote the logit transformations of F X i and F Y i , respectively. That is, Also, see Agresti [1, p. 205]. The ML model with = 0 is the MH model, namely the ML model is the extension of the MH model. Moreover, Miyamoto et al. [12] proposed the conditional ML (CML) model, and Saigusa et al. [13] proposed the marginal complementary log-log (MCLL) model. The ML (CML) model states that one (conditional) marginal distribution is a location shift of the other (conditional) marginal distribution on a logistic scale. The MCLL model states that one marginal distribution is a location shift of the other marginal distribution in terms of a function 1 − exp(− exp(x)) . In this paper, we consider a model that is more relaxed than these models. The extensions of the ML and CML models exist (Kurakami et al. [9]). Therefore, an extension of the MCLL model is proposed in Sect. 2. Caussinus [5] gave the separation of symmetry. The separation may be useful to see a reason for the poor fit of the symmetry model when the symmetry model fits the data poorly. Thus, we are interested in considering a necessary and sufficient condition of the MH model. We give the separaton of marginal homogeneity in Sect. 3. This paper is organized as follows. Section 2 proposes an extension of the MCLL model. Section 3 shows two theorems. Section 4 extends the proposed model into multi-way contingency table. Section 5 discusses the goodness-of-fit test. Section 6 gives examples. Section 7 concludes this paper.

Extension of MCLL
(1) that the probability that X is i + 1 or above, is equal to the probability that Y is i + 1 or above to the power of i 1 2 , for i = 1, … , r − 1 , namely, Also, in Eq. (1), i log( 1 ) + log( 2 ) is the difference between the two random variables on complementary log-log scale, namely (i) the MCLL model (i.e., 1 = 1 ) indicates that one marginal distribution is a location shift of the other marginal distribution, and (ii) the EMCLL model indicates that the difference between two marginal distributions depends on the value of category i. The Gumbel (minimum) distribution function is where is the location parameter and ( > 0 ) is the scale parameter. Let GX(x) be the Gumbel distribution function with parameter ( 1 , 1 ) and GỸ (x) be the Gumbel distribution function with parameter ( 2 , 2 ) . Then, the difference between two complementary log-log transforms of GX(x) and GỸ (x) is expressed as This structure is similar to the EMCLL model. If we set 1 = 2 = 1 without loss of generality, then This is similar to the Eq. (1) with 1 = 1 , that is, the location shift model. Also, if we set 1 = 2 = 0 without loss of generality, then The difference depends only on the scale parameter. This is similar to the Eq. (1) with 2 = 1 . For ordinal categorical data, if it is reasonable to assume an underlying Gumbel distribution, then the proposed model may be appropriate for the square contingency tables. Miyamoto et al. [12], Tahata et al. [18], and Shinoda et al. [14] considered the models depending on the conditional marginal cumulative probability. These models indicate the structure of marginal inhomogeneity on the condition that an observation will fall in one of off-diagonal cells of the table.
Let F X(c) i and F Y(c) i denote the conditional marginal cumulative probabilities of X and Y, respectively, given that X ≠ Y . That is, for i = 1, … , r − 1, where with = ∑ ∑ s≠t p st = Pr(X ≠ Y). We shall consider the conditional EMCLL (CEMCLL) model which is defined by where Similarly, (i) the CEMCLL model with * 1 = 1 reduces to the conditional MCLL (CMCLL) model proposed by Shinoda et al. [14]. The CMCLL model indicates that one conditional marginal distribution is a location shift of the other conditional marginal distribution. The CEMCLL model indicates that the difference between two conditional marginal distributions depends on the value of category i. In a similar manner to the EMCLL model, the CEMCLL model may be appropriate if it is reasonable to assume underlying Gumbel distribution for the conditional marginal distribution.
Both the EMCLL and CEMCLL models show the inhomogeneity of two marginal distributions. The EMCLL model shows the difference between the complementary log-log transformation of the row and column marginal distribution function is linear with respect to category i. On the other hand, the CEMCLL model shows that of row and column conditional marginal distribution function is linear with respect to category i.

A Necessary and Sufficient Condition for the MH
The MH model does not fit for a given dataset because the MH model is very restrictive. Marginal inhomogeneity models such as the ML and MCLL models were proposed. We are interested in detecting the probabilistic structure for the poor fit of the MH model. When the MH model can be separated into two or more models, analyzing these models may be helpful to elucidate the reason for the poor fit of the MH model. Consider a model defined by We shall refer to the model as the mean equality (ME) model. Also, we consider a model defined by Thus, we shall refer to the Eq. (3) as the mean and variance equality (MVE) model.
We give the following lemma.

Lemma 1 The MVE model is equivalent to the following restrictions:
Proof We can see that We obtain the following theorem.  (7) and (8)  In a similar manner to proof of Theorem 1, this leads to the equation .

Thus, we can obtain
This is the MH model. Therefore, we can obtain the following theorem.

Theorem 2 The MH model holds if and only if both the CEMCLL and MVE models hold.
From Theorems 1 and 2, we can obtain the following corollary.

Corollary 1 If the MVE model holds, then the EMCLL model holds if and only if the CEMCLL model holds.
Saigusa et al. [13] and Shinoda et al. [14] gave the separation of marginal homogeneity. We would like to note that Theorems 1 and 2 include these results.

Extension for Multi-Way Table
Consider a multi-way r T contingency table of same classification having ordered categories. Let X t denotes the t-th random variable for t = 1, … , T , and let Pr( We obtain the following theorem.

Theorem 3 The MH[T] model holds if and only if both the EMCLL[T] and the MVE[T] models hold.
Proof denotes the conditional marginal cumulative probability of X t , given that there is at least one set of unequal random variables. That is, for We would like to note that the MVE[T] model can be expressed as Then, we can obtain the following theorem.

Theorem 4 The MH[T] model holds if and only if both the CEMCLL[T] and MVE[T] models hold.
From Theorems 3 and 4, we can obtain the following corollary.

Goodness-of-Fit Test
Let n i 1 ⋯i T denote the observed frequency in the ( i 1 , … , i T ) cell of the r T table with n = ∑ ⋯ ∑ n i 1 ⋯i T , and let m i 1 ⋯i T denote the corresponding expected frequency, that is, m i 1 ⋯i T = np i 1 ⋯i T . We assume that a multinomial distribution applies to the table.
Consider two models, say M 1 and M 2 , such that if model M 1 holds, then model M 2 holds. For testing the goodness-of-fit of model M 1 assuming that model M 2 holds, the conditional likelihood ratio statistic is given by . The number of df for the conditional test is the difference between the numbers of df for models M 1 and M 2 . The conditional test statistics are more powerful because they are based on fewer degrees of freedom. As

Occupational Status for Japaneses
Consider the data in Table 1 taken from Tominaga [19, p. 131] again. Table 2 gives the values of likelihood ratio chi-square statistic G 2 for testing the goodness-of-fit of models. We analyze the data using the new model and the properties of the MH model. First, we want to see whether the marginal distribution of father's status is equal to that of his son. Since the MH model fits poorly from Table 2, we can infer that the marginal distribution of father's status is different from that of his son. Then, the extended models (i.e., the MCLL and EMCLL models) are applied to the data, and neither fit well.
Second, we want to see whether the conditional marginal distribution of father's status is equal to that of his son on the condition that his status is different from that of his son. Then, we shall apply the extended models (i.e., the CMCLL and CEM-CLL models) based on the conditional marginal cumulative distributions. The CEM-CLL model only fits well. Under the CEMCLL model, the MLEs of * 1 and * 2 are ̂ * 1 = 1.59 and ̂ * 2 = 0.75 with the standard errors 0.099 and 0.127, respectively. We want to see whether * 1 = 1 and * 2 = 1 . Consider the hypothesis that the MH model holds under the assumption that the CEMCLL model holds. According to the test based on the difference between G 2 values for the MH and CEMCLL models, this hypothesis is rejected at the 0.05 significance level because 203.55 − 1.67 = 201.88 with 2 df. Therefore, the CEMCLL model may be preferable to the MH model. Hence, under the CEMCLL model, the probability that the son's status in a pair is i + 1 or above, is estimated to be equal to the probability that his father's status in a pair is i + 1 or above to the power of 0.75 × 1.59 i , for i = 1, 2, 3 , on condition that the father's status is different from his son's status.
and that difference tends to be greater as i increases, where F X(c) i and F Y(c) i are MLEs of the conditional marginal cumulative probabilities of X and Y for i = 1, 2, 3 . Therefore, the occupational distribution for son is stochastically lower than the occupational distribution for father on the condition that father's status is different from his son's status. The difference becomes greater as the status category number increases. Please see Fig. 1.
Last, according to Theorem 2, we can see that the poor fit of the MH model is caused by the influence of the lack of structure of the MVE model rather than the CEMCLL model.   [20], Lang and Scott [10], Sobel et al. [16], and Tahata [17]. These articles mainly focused on the structure of joint distributions. On the other hand, we focus on the structure of marginal distributions. A cursory glance at the data reveals that the MH model is inappropriate. Indeed, G 2 (MH) = 32.80 for testing its fit, with df = 4 . For the population represented by this sample, we analyze whether the (conditional) occupational distribution for sons differs from the (conditional) occupational distribution for fathers.

Occupational Status for British
First, we apply the MCLL and EMCLL models to compare two marginal distributions. These models fit well, having G 2 (MCLL) = 4.26 ( df = 3 ) and G 2 (EMCLL) = 3.04 ( df = 2 ), respectively. Their parameter estimates are ̂2 = 0.88 with the standard error 0.020 under the MCLL model and ̂1 = 0.98 and ̂2 = 0.96 with the standard errors 0.022 and 0.079, respectively, under the EMCLL model. Consider the hypothesis that the MCLL model holds under the assumption that the EMCLL model holds, that is, 1 = 1 . According to the test based on the difference between G 2 values for the MCLL and EMCLL models, this hypothesis is accepted at the 0.05 significance level because 4.26 − 3.04 = 1.22 with 1 df. Therefore, the MCLL model may be preferable to the EMCLL model. The occupational distribution for son is stochastically higher than the occupational distribution for father. As the reference, we give the observed and estimated marginal distribution functions in Fig. 2.
Last, we apply the CMCLL and CEMCLL models to compare two conditional marginal distributions on the condition that father's status is different from his son's status. These models fit well, having G 2 (CMCLL) = 5.35 ( df = 3 ) and G 2 (CEMCLL) = 3.53 ( df = 2 ), respectively. Their parameter estimates are ̂ * 2 = 0.82 with the standard error 0.031 under the CMCLL model and ̂ * 1 = 0.95 and ̂ * 2 = 0.97 with the standard errors 0.035 and 0.130, respectively, under the CEMCLL model. Consider the hypothesis that the CMCLL model holds under the assumption that the CEMCLL model holds, that is, * 1 = 1 . According to the test based on the difference between G 2 values for the CMCLL and CEM-CLL models, this hypothesis is accepted at the 0.05 significance level because 5.35 − 3.53 = 1.82 with 1 df. Therefore, the CMCLL model may be preferable to the CEMCLL model. The occupational distribution for son is stochastically higher than the occupational distribution for father on the condition that father's status is different from his son's status. As the reference, we give the observed and estimated conditional marginal distribution functions in Fig. 3.  tables with ordinal categories. We would like to note that these models should not apply to the data with nominal categories.
In comparison with the MCLL[T] (CMCLL[T]) model, additional parameters 1t ( * 1t ) for t = 2, … , T allow us to consider degrees of inhomogeneity proportional to category number as it shown in Sect. 6. The conditional test statistics are more powerful because they are based on fewer degrees of freedom. Thus, the proposed models enable more detailed analysis for the contingency tables with the same ordinal categories.