The semi-symmetric intraclass contingency table is a three-dimensional (r x r x c) contingency table that is upper-triangular and a priori symmetric within layers; e.g., mijk = mjik where mijk represents the expected cell frequency. Ishii (1960) solved the maximum likelihood estimation problem of such a table by applying the maximum likelihood method directly. 

In the early 1980s, the loglinear model was modified to handle this type of data structure, and maximum likelihood estimation was carried out for the modified loglinear model (1). This approach leads to an expression of the maximum likelihood estimates exclusively in terms of the observed cell counts. This analysis is equivalent to an application of the general loglinear model to an artificially complete table obtained by splitting the off-diagonal cells in half within layers.

Applications of the semi-symmetric intraclass contingency table appear in social psychology and biology. Examples include the study of actions of pairs of individuals, the study of amino acid allele pairs in a protein, the determination of the effect of the sex-linked dwarfing gene in male chickens on resistance to E. coli infection, and the association between fingerprint patterns and genetic factors. See (1) for references, and for related work see (2), (4), (5), (9), (11), (15), and (16).

Reference

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