// Workers AI · dad joke modeWhat did Digby's H say to the other letter? "I'm hooked on you.
Digby's H is a measure of association for a 2 × 2 contingency table. It is a nonlinear transformation of the odds ratio, proposed by P. G. N. Digby in 1983 as a computationally simple approximation to the tetrachoric correlation coefficient, which otherwise requires iterative numerical estimation.[1][2]
Formula
[edit]For a 2×2 table of binary variables U and V with cell frequencies or proportions
| V = 0 | V = 1 | |
|---|---|---|
| U = 0 | a | b |
| U = 1 | c | d |
the odds ratio is
Digby's H is defined as
Equivalently, in terms of the odds ratio,
Properties
[edit]Digby's H ranges from −1 to 1. It equals zero under statistical independence of U and V, since independence implies and hence . Positive values indicate positive association; negative values indicate negative association.
Because H depends on the table only through the ratio ad/bc, it is invariant under multiplication of all four cell frequencies by the same positive constant, and — like any function of the odds ratio alone — its value is determined by ω independently of the row and column marginal totals.[2]
Digby's H belongs to a one-parameter family of odds-ratio transformations sometimes called generalized Yule coefficients,[3]
where gives Yule's Q,
gives Yule's Y,
and gives Digby's H. All members of this family are zero under independence and share the same sign as .
See also
[edit]References
[edit]- ↑ Digby, P. G. N. (1983). "Approximating the tetrachoric correlation coefficient". Biometrics. 39 (3): 753–757. doi:10.2307/2531104.
- 1 2 Warrens, Mathieu J. (2008). "On Association Coefficients for 2×2 Tables and Properties That Do Not Depend on the Marginal Distributions". Psychometrika. 73 (4): 777–789. doi:10.1007/s11336-008-9070-3.
- ↑ Bonett, D. G.; Price, R. M. (2007). "Statistical Inference for Generalized Yule Coefficients in 2 × 2 Contingency Tables". Sociological Methods & Research. 35 (3): 429–446. doi:10.1177/0049124106292358.