lnR2

lnR2 is the natural logarithm of the coefficient of determination, R^2, used in statistical analysis to transform the scale of a model’s explanatory power. R^2 measures the proportion of variance in the dependent variable explained by a regression model, with 0 ≤ R^2 ≤ 1. The quantity lnR2 is defined for models with R^2 > 0 and yields values in (-∞, 0], with lnR2 = 0 only when R^2 = 1. In practice, R^2 can be exactly 0 for some models, which makes lnR2 undefined, so researchers may apply a small positive offset in such cases.

Computation involves fitting the regression, obtaining R^2, and then taking the natural log of this value. Because the natural logarithm is monotone on (0, 1], the ranking of models by R^2 is preserved when transforming to lnR2, but the scale and interpretability change. lnR2 can be used to stabilize variance in certain resampling or meta-analytic contexts, though it is not a standard model-selection criterion on its own.

Limitations include the fact that lnR2 emphasizes very small improvements in R^2 more than larger ones when R^2 is already near zero, and it does not address issues of model adequacy beyond explained variance. R^2 itself is sensitive to data range, variable inclusion, and model form, and lnR2 inherits these caveats. It is also not defined for R^2 = 0 and is not applicable to many generalized linear models that report alternative goodness-of-fit measures, such as pseudo R-squared values.

Example: if R^2 = 0.64, then lnR2 ≈ -0.446; if R^2 = 0.25, lnR2 ≈ -1.386. See also R-squared, adjusted R-squared, pseudo R-squared, and log transformations.