Misunderstood and Unattributed: Revisiting M. H. Doolittle's Measures of Association, With a Note on Bayes’ Theorem

Abstract

In the 1880s, American scholars developed measures of association and chance for cross-classification tables that anticipated the more widely known work of Galton, Pearson, Yule, and Fisher. Three of the measures form the historical backdrop for the earliest known use of a joint probability measure that mirrored Bayes’ theorem long before the latter gained general interest among statisticians. The joint probability measure, which served as a foundational step in M. H. Doolittle's development of the first of the two “association ratios,” has not previously been reviewed in the statistical literature. It was reintroduced as if newly developed in a subfield of experimental psychology more than a century after Doolittle's work was published. It has flourished there, but it has not seen use in other academic venues. The article describes its properties and limitations and proposes that it be disseminated and debated beyond its current narrow application. The article notes that Doolittle's first association ratio can be expressed as another joint probability and that prior treatments in the literature are inconsistent with Doolittle's understanding of its purpose. The article also demonstrates that the equivalent of Cohen's kappa (κ) was developed by Doolittle in 1887, as his second association measure.

[Received December 2014. Revised August 2015.]

• Bayes’ theorem
• Cohen's kappa
• Chi-square
• Confusion matrix
• C. S. Peirce
• Diagnostic likelihood ratios
• H. L. Wagner
• Mean square contingency.

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Correction

Notes

¹Assessment (or “frequency”) bias, B, is a measure of the extent to which a diagnostic or predictive method over- or under-estimates the occurrence of an event or the presence of a status. It is computed as (a + b)/(a + c) for the actual events (or Event 1) and as (c + d)/(b + d) for null events (or Event 2). Values of B above or below unity indicate the degree and direction of bias.

²The equations from Doolittle's era display an obsolete notation, and are occasionally misstated. Hogan and Mason (2012, table 3.3, p. 36, and p. 52) represent Doolittle's i₁ as {(ad − bc)/[(a + b)(a + c)(b + d)(c + d)]^1/2}, which yields the square root of Doolittle's i₁ instead of the true value. This appears to be a substantive misunderstanding of Doolittle's measure; the authors also incorrectly represent i₁ as being the equivalent of (χ²/n)^1/2 instead of χ²/n. Doolittle's two original formulas for i₁ in current cell identifiers appear in this article at Equations (4.1 and 4.3), and the measure's relationship with the χ² statistic appears correctly, as χ²/n in Section 3.2. (While Equation (4)(4) expresses the numerator of i₁ differently from Doolittle, it is a correct restatement of i₁.)

³Doolittle's i₁may not appear to be a joint probability, since it is the product of two factors, each comprising the difference between two conditional probabilities. It is a joint probability, nonetheless. Collapsing a factor by adding or subtracting its conditional probabilities yields a conditional probability. The first factor (TPF – FPF) is the equivalent of the conditional probability that only a tornado will be preceded by the forecast, “tornado.” The second factor (PPV – FOV) is the equivalent of the conditional probability that only a forecast of “tornado” will precede the occurrence of a tornado.

⁴Throughout the article, the term used for (TPF)(PPV) etc., is joint probability. Some may prefer a more specific term, conjoint probability, since the factors involved are conditional probabilities.

Additional information

Notes on contributors

Timothy W. Armistead

Timothy W. Armistead, Armistead Research and Investigative Services, 1564- A Fitzgerald Drive, Suite 323, Pinole, CA 94564, USA (E-mail: [email protected]). The author thanks the editor, associate editor, and peer reviewers for suggestions that greatly sharpened the arguments in this article. The author also thanks Michael Baldwin for a suggestion that led to the derivation of Equation (4.2).