lnOR

lnOR refers to the natural logarithm of the odds ratio and is widely used in statistics as a measure of association between a binary exposure and a binary outcome. Taking the logarithm stabilizes variance and yields quantities that are approximately normally distributed under standard conditions, which simplifies estimation, hypothesis testing, and meta-analysis. In many applied contexts, lnOR serves as the primary effect size.

For a 2x2 contingency table with a representing exposed cases, b representing unexposed cases, c representing exposed non-cases, and d representing unexposed non-cases, the odds ratio is OR = (a*d)/(b*c). The lnOR is the natural log of this quantity, lnOR = ln(a*d/(b*c)) = ln(a) + ln(d) − ln(b) − ln(c). The lnOR is zero when there is no association and positive when exposure is associated with higher odds of the outcome, negative when associated with lower odds.

Estimation and inference typically proceed through the standard error of lnOR, which for a 2x2 table is SE(lnOR) = sqrt(1/a + 1/b + 1/c + 1/d). A confidence interval for lnOR is lnOR ± z*SE, where z corresponds to the desired confidence level (for example 1.96 for 95%). Exponentiating the endpoints yields a confidence interval for the odds ratio. In meta-analysis, lnOR is favored because it is additive across independent studies and its sampling distribution is closer to normal.

lnOR also arises in logistic regression, where the regression coefficient for a binary predictor equals the log odds ratio (OR = exp(beta)). Practical cautions include handling zero cells with continuity corrections, and remembering that while lnOR and OR convey similar directional information, interpretation changes with the transformation.