Detecting contextuality in systems with categorical variables

The Contextuality-by-Default theory describes contextual effects on random variables: how the identity of random variables changes from one context to another. Direct influences and true contextuality constitute different types of effects of contexts upon sets of random variables. Changes in the distributions of random variables across contexts define direct influences. True contextuality is defined by the impossibility of sewing all the variables of a system of random variables into a particular overall joint distribution where variables that correspond to the same property in different contexts are equal to each other as often as possible.For systems of binary random variables, the theory shows that, in cyclic systems, the two effects are in opposition. For the extension of the theory to systems with categorical random variables, I will present the nominal dominance theorem, which states a necessary condition for noncontextuality of systems where all dichotomizations of categorical variables are considered. This condition shows a case where direct effects may entail true contextuality. I will also illustrate the application of this theorem to the results of a psychophysical double-identification experiment.