Generalized Linear Asymmetry Model and Decomposition of Symmetry for Multiway Contingency Tables

A k-order generalization of the linear diagonals-parameter symmetry model is proposed, and related orthogonal decompositions of the generalization are inspected. Applications to randomized clinical trials are given. J o ur na l o f B iometrics & Bistatis t i c s ISSN: 2155-6180 Journal of Biometrics & Biostatistics Citation: Tahata K, Tomizawa S (2011) Generalized Linear Asymmetry Model and Decomposition of Symmetry for Multiway Contingency Tables. J Biomet Biostat 2:120. doi:10.4172/2155-6180.1000120 Volume 2 • Issue 4 • 1000120 J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journal Page 2 of 6 of generality, we may set (1) 1 l α = (l = 1,...,k). We shall refer to this model as the kth linear asymmetry model (denoted by T k LS ). A special case of the T k LS model with { } 1( ) ( ) 1 s k s α α = = =  , s =1,..,T, is the ST model. Special cases of the T k LS model with k=1 and with k=2 are LST (i.e., 1 T LQ ) and ELST models, respectively. The 1 T Q model can be expressed as


Introduction
Consider an r 2 square contingency table with the same row and column classifications. Let p ij denote the probability that an observation will fall in the (i,j) th cell of the table (i=1,…,r;j=1,…,r). For the analysis of square contingency tables, one of our interests is whether or not there is a structure of symmetry (or asymmetry) rather than independence in the table. The symmetry (S 2 ) model is defined by where ψ ij =ψ ji . This indicates that the probability that an observation will fall in the (i,j) th cell is equal to the probability that the observation falls in the (j,i) th cell. As a model that has the weaker restrictions than the S 2 model, Caussinus [1] considered the quasi-symmetry ( 2 1 Q ) model defined by P ij = µα i β j ψ ij where ψ ij =ψ ji . A special case of this model with {α i = β i } is the S 2 model. Also Caussinus [1] showed a theorem that the S 2 model holds if and only if both the 2 1 Q and the marginal homogeneity models hold. For the analysis of data, the theorem (say decomposition of the S 2 model) may be useful for seeing the reason for the poor fit when the S 2 model fits the data poorly.
The S 2 and 2 1 Q models indicate the structure of symmetry of cell probabilities and odds-ratios, respectively. As a model that indicates the structure of asymmetry of cell probabilities, Agresti [2] considered the linear diagonals-parameter symmetry (LS 2 ) model defined by where ψ ij =ψ ji . This model is a special case of 2 1 Q model. In this way various symmetry and asymmetry models have been proposed by many statisticians (also see Agresti [3]; Tomizawa and Tahata [4]).
Consider an r T contingency table with ordered categories. Let i=(i 1 ,…,i T ) for i k =1,…,r (k =1,…,T), and let P i denote the probability that an observation will fall in the i th cell of the table. Let X k (k =1,…,T), denote the k th variable. Tahata et al. [5] considered the linear diagonalsparameter symmetry (LS T ), and extended LS T (ELS T ) models. The ELS T model is defined by   For the analysis of data, when the S T model does not hold, one may be interested in applying various asymmetry models; for example, the LS T , ELS T and T h LQ models. If these models do not hold, we are interested in applying a more generalized asymmetry model. In addition we are interested in seeing the reason for the poor fit of the S T model by using the decomposition of the S T model. Thus the present paper proposes the generalization of the ELS T model, and gives the orthogonal decomposition of the S T model.

Generalized linear asymmetry model
Consider a new model defined by, for a fixed k (k=1,..,r-1), . This model can also be expressed as Consider the case of T=2. For a fixed k (<r), the 2 k LS model can be expressed as where ψ ij =ψ ji . Under this model, the ratio of P ij to P ji is . Namely, this model indicates that the log ratio of symmetric cells is expressed as the polynomial. Note that the

Decomposition of symmetry model
For a fixed k (k=1,..,r-1), consider a model defined by We shall refer to this model as the marginal k th moment equality ( T k MME ) model. Then we obtain the following theorem. We give the proof in the Appendix 1. Note that special cases of Theorem 1 with k=1 and k=2 are given by Tahata et al. [5].
Also although the detail is omitted, we can see that the  [6].
By the way, the 1 T MME model is expressed as where C is unknown constant. We shall refer to this model as the covariance equality (CE T ) model. Then, in a similar manner to Theorem 1 and Tahata et al. [7], we can obtain the following theorem. The relationships among models are given in (Figure 1). and CE T models could be obtained using the iterative procedure, for example, the general iterative procedure for log-linear models of Darroch and Ratcliff [8] or using the Newton-Raphson method to the log-likelihood equations. where φ i = φ j for any permutation j=(j 1 ,…,j T ) of i=(i 1 ,…,i T ) with Each model can be tested for goodness-of-fit by, e.g., the likelihood ratio chi-squared statistic with the corresponding degrees of freedom (df). The numbers of df for the S T ,

Orthogonal decomposition of test statistic
Also the number of df for the  The proof of Theorem 3 is given in the Appendix 2. In a similar manner to Tahata et al. [5,7], we can obtain the following theorem. Note that special cases of Theorem 3 with k =1 and k =2 are given by Tahata et al. [5].

Analysis of data
Analysis of table 1: The data in (Table 1), taken from Stuart [9], are constructed from unaided distance vision of 7477 women aged 30-39 employed in Royal Ordnance factories in Britain from 1943 to 1946 (see, e.g., Caussinus [1]; Tomizawa and Tahata [4]). The S 2 model fits the data in (Table 1) poorly (see Table 3). By using the decompositions for the S 2 model, we shall consider the reason why the S 2 model fits these data poorly. The  LS model may be expressed as Therefore, under this model the probability that a woman's right eye grade is i and her left eye grade is j(>i) is estimated to be  , and 3(2) 0 998 α = .
. Note that ( LS models are obtained although the details are omitted.
According to the test based on the difference between the G 2 values for the 2 2 LS and LS model.

Analysis of table 2:
Consider the data in (Table 2) taken from Tahata et al. [5]. These are the results of the treatment group only in randomized clinical trials conducted by a pharmaceutical company in anemic patients with cancer receiving chemotherapy. The response is the patient's hemoglobin (Hb) concentration at baseline (before treatment) and following 4 weeks and 8 weeks of treatment. (Table 2) shows the 3×3×3 array of counts of Hb response that is classified as (1) ≥10 g/dl, (2) 8-10 g/dl, and (3) < 8 g/dl. It is reasonable to explore this array for various asymmetries involving time. Namely, we are interested in considering the transition of patient's Hb concentration rather than the interchangeability of evenly spaced points in time with respect to those concentrations. For example, we want to see whether there is an asymmetric transition of those concentrations or not, when the value of those concentration at baseline was given.
We see from ( Table 3) that (1) each of the S 3 , 3 k MME (k=1,2,) , and CE 3 models fits the data in (Table 2) poorly, however, (2) the 3 2 LQ model fits them well. The S 3 model fits the data in (Table 2) poorly (see Table 3). By using the decompositions for the S 3 model, we shall consider the reason why the S 3 model fits these data poorly. The 3 2 LQ model fits them well, but the other models fit them poorly. So, we see from Theorem 2 (i.e., decomposition of the S 3 model into the 3 2 LQ and CE 3 models) that the poor fit of the S 3 model is caused by the influence of the lack of The response categories are (1) ≥10 g/dl, (2) 8 − 10 g/dl, (3) < 8 g/dl. LQ model).
For Table 1 For The symbol "*" means significant at 5% level.  and 3 0 21 θ = . , respectively. Therefore, under the 3 2 LQ model, (i) the conditional probability that the state of the Hb concentration is j at 4 weeks and that is k (>j) at 8 weeks, is estimated to be 1.82 k-j times higher than the conditional probability that the state of the Hb concentration is k at 4 weeks and that is j at 8 weeks on condition that the patient's Hb concentration is (1) ≥10g/dl at baseline, (ii) those conditional probability is estimated to be 0.63 k-j times higher than the corresponding conditional probability on condition that the patient's Hb concentration is (2) 8-10 g/dl at baseline, and (iii) those conditional probability is estimated to be 0.21 k-j times higher than the corresponding conditional probability on condition that the patient's Hb concentration is (3) <8 g/dl at baseline. Therefore we could infer that (i) when a patient's Hb concentration is (1) ≥10 g/dl at baseline, those concentration tend to decrease from 4 weeks to 8 weeks since the maximum likelihood estimates of θ 1 is greater than 1, (ii) when a patient's Hb concentration is (2) 8-10 or (3) <8 g/dl at baseline, those concentration tend to increase from 4 weeks to 8 weeks since the maximum likelihood estimates of θ 2 and θ 3 are less than 1.

Concluding Remarks
In this paper, we have proposed the k-order generalization of the linear diagonals-parameter symmetry model that is including the first order quasi-symmetry model, and have given the decomposition of the symmetry model. When the S T model fits the data poorly, the decomposition of the S T model (i.e., Theorems 1 and 2) would be useful for seeing the reason for its poor fit. As seen in analysis of (Tables 1, 2), we can see that (1) for the data in (Table 1), the poor fit of the S 2 model is caused by the poor fit of the 2 k MME models rather than the 2 k LS (k=1,2,3) models, and (2) for the data in (Table 2), the poor fit of the S 3 model is caused by the CE 3 model rather than the  From above equations, we see