Measuring agreement among several raters classifying subjects into one-or-more (hierarchical) nominal categories. A generalisation of Fleiss' kappa

Cohen's and Fleiss' kappa are well-known measures for inter-rater reliability. However, they only allow a rater to select exactly one category for each subject. This is a severe limitation in some research contexts: for example, measuring the inter-rater reliability of a group of psychiatrists diagnosing patients into multiple disorders is impossible with these measures. This paper proposes a generalisation of the Fleiss' kappa coefficient that lifts this limitation. Specifically, the proposed $\kappa$ statistic measures inter-rater reliability between multiple raters classifying subjects into one-or-more nominal categories. These categories can be weighted according to their importance, and the measure can take into account the category hierarchy (e.g., categories consisting of subcategories that are only available when choosing the main category like a primary psychiatric disorder and sub-disorders; but much more complex dependencies between categories are possible as well). The proposed $\kappa$ statistic can handle missing data and a varying number of raters for subjects or categories. The paper briefly overviews existing methods allowing raters to classify subjects into multiple categories. Next, we derive our proposed measure step-by-step and prove that the proposed measure equals Fleiss' kappa when a fixed number of raters chose one category for each subject. The measure was developed to investigate the reliability of a new mathematics assessment method, of which an example is elaborated. The paper concludes with the worked-out example of psychiatrists diagnosing patients into multiple disorders.