Three dimensional contingency tables in which one variable is considered to be a factor and the other two variables have a natural relationship (such as left and right eye vision) are analyzed. Models involving symmetry and proportional symmetry between the related variables are also presented. (Author/JKS)

Using an infinite item test framework, it is argued that the usual assumption of local independence should be replaced by a weaker assumption--essential independence. The usual assumption of unidimensionality is replaced by a weaker and more appropriate statistically testable assumption of essential unidimensionality. (TJH)