Digby's H

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.

Formula

For a 2×2 table of binary variables U and V with cell frequencies or proportions

the odds ratio is

${\displaystyle \omega ={\frac {ad}{bc}}.}$

Digby's H is defined as

${\displaystyle H={\frac {(ad)^{3/4}-(bc)^{3/4}}{(ad)^{3/4}+(bc)^{3/4}}}.}$

Equivalently, in terms of the odds ratio,

${\displaystyle H={\frac {\omega ^{3/4}-1}{\omega ^{3/4}+1}}.}$

Properties

Digby's H ranges from −1 to 1. It equals zero under statistical independence of U and V, since independence implies ${\displaystyle ad=bc}$ and hence ${\displaystyle \omega =1}$. 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.

Digby's H belongs to a one-parameter family of odds-ratio transformations sometimes called generalized Yule coefficients,

${\displaystyle S_{q}={\frac {(ad)^{q}-(bc)^{q}}{(ad)^{q}+(bc)^{q}}},}$

where ${\displaystyle q=1}$ gives Yule's Q,

${\displaystyle Q={\frac {ad-bc}{ad+bc}},}$

${\displaystyle q=1/2}$ gives Yule's Y,

${\displaystyle Y={\frac {{\sqrt {ad}}-{\sqrt {bc}}}{{\sqrt {ad}}+{\sqrt {bc}}}},}$

and ${\displaystyle q=3/4}$ gives Digby's H. All members of this family are zero under independence and share the same sign as ${\displaystyle ad-bc}$.

See also

• Contingency table
• Phi coefficient

References