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A note on the expected value of the Rand index
Author(s) -
Steinley Douglas,
Brusco Michael J.
Publication year - 2018
Publication title -
british journal of mathematical and statistical psychology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.157
H-Index - 51
eISSN - 2044-8317
pISSN - 0007-1102
DOI - 10.1111/bmsp.12116
Subject(s) - multinomial distribution , mathematics , hypergeometric distribution , null hypothesis , statistics , index (typography) , econometrics , value (mathematics) , mathematical proof , sample (material) , computer science , physics , geometry , world wide web , thermodynamics
Two expectations of the adjusted Rand index (ARI) are compared. It is shown that the expectation derived by Morey and Agresti (1984, Educational and Psychological Measurement , 44 , 33) under the multinomial distribution to approximate the exact expectation from the hypergeometric distribution (Hubert & Arabie, 1985, Journal of Classification , 2 , 193) provides a poor approximation, and, in some cases, the difference between the two expectations can increase with the sample size. Proofs concerning the minimum and maximum difference between the two expectations are provided, and it is shown through simulation that the ARI can differ significantly depending on which expectation is used. Furthermore, when compared in a hypothesis testing framework, multinomial approximation overly favours the null hypothesis.