The standard method for the assessment of the precision of measurement gauges - the gauge R&R method - is not applicable for measurement systems that measure on a nominal scale. For nominal measurements, the agreement method can be used, which expresses precision in the form of an index named κ. This paper aims to provide a consistent framework for the kappa method, based on statistical modeling. The defined probability models are used to study the statistical properties of proposed estimators. Various alternative forms of κ are discussed, as well as their relationship to the so-called paradoxes of kappa.