Separation of symmetry for square tables with ordinal categorical data
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Separation of symmetry for square tables with ordinal categorical data Kouji Tahata1 Received: 13 October 2018 / Accepted: 10 October 2019 © Japanese Federation of Statistical Science Associations 2019
Abstract The present paper considers a model that indicates the structure of asymmetry for cell probabilities for square contingency tables with ordered categories. The model is the closest to the symmetry model in terms of the f-divergence under certain conditions and incorporates existing asymmetry models in special cases. A theorem that the symmetry model can be separated into two models which have weaker restrictions than the symmetry model is given. Also, a property between test statistics for goodness of fit is discussed. Keywords f-Divergence · Marginal homogeneity · Moment equality · Quasisymmetry Mathematics Subject Classification 62H17
1 Introduction For a square contingency table having same ordinal row and column classifications, many observations tend to concentrate on main diagonal cells or near. Thus, we are interested in considering some symmetry rather than independence between row and column classifications. The issues of symmetry have been treated in many studies, for example, Bowker (1948), Stuart (1955), Caussinus (1965), Bishop et al. (1975, Sec. 8), Agresti (1983), Kateri and Papaioannou (1997), Kateri and Agresti (2007), and Tahata and Tomizawa (2011). Let X and Y denote the row and column variables, respectively. Also, let 𝜋ij denote the probability that an observation will fall in the (i, j)th cell of an r × r square contingency table ( ). Assume that a set of known scores i = 1, … , r; j = 1, … , r u1 < u2 < ⋯ < ur can be assigned to both the rows and the columns. For a given k * Kouji Tahata [email protected] 1
Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278‑8510, Japan
13
Vol.:(0123456789)
Japanese Journal of Statistics and Data Science
( k = 1, … , r − 1 ), the asymmetry model based on the f-divergence ( ASk[f]) is defined as ) ( k ∑ h S −1 (i = 1, … , r; j = 1, … , r), ui 𝛼h + 𝛾ij 𝜋ij = 𝜋ij F h=1
where 𝛾ij = 𝛾ji , 𝜋ijS = (𝜋ij + 𝜋ji )∕2 , f is a twice-differential and strictly convex function, and F(t) = f � (t) (Tahata 2017). Note that f (0) = limt→0 f (t) , 0 ⋅ f (0∕0) = 0 , and 0 ⋅ f (a∕0) = a limt→∞ [f (t)∕t] . For the details of f-divergence, see Csiszár and Shields (2004). When 𝛼1 = ⋯ = 𝛼k = 0 , (1) the ASk[f] model is reduced to the symmetry (S) model (Bowker 1948). The ASk[f] model is the closest model to the S model in ∑ ∑ ∑ ∑ terms of the f-divergence under the condition that i j uhi 𝜋ij (or i j uhj 𝜋ij ) for h = 1, … , k as well as the sums 𝜋ij + 𝜋ji for i = 1, … , r; j = 1, … , r , are given. The present paper discusses the properties of the ASk[f] model. Section 2 introduces some special cases of the ASk[f] model. Tahata (2017) gave a theorem that the S model holds if and only if both the ASk[f] model and the marginal kth moment equality model for the scores {us } (denoted by M
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