Order‐based statistics and monotonicity: A family of ordinal measures of association*

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Abstract
This paper reviews standards for measures of association. In particular, standards for order‐based measures are examined. The concept of monotonicity is shown to be ambiguous as it has been applied in this area, and it is clarified. The result is the specification of three — instead of the usual two — kinds of monotonic relations. These three monotonic models provide the basis for defining three non‐arbitrary measures of ordinal association. Eleven order‐based measures are reviewed in the light of measurement standards as extended by the clarification of monotonicity. Eight are shown either to embody non‐order‐based elements or to contain ad hoc characteristics. The remaining three are shown to be members of a single family of monotone based ordinal measures that can be applied where ever their particular monotone models are appropriate.

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