sqrt2GMR

sqrt2GMR is a statistical metric proposed for evaluating predictive accuracy across two subpopulations in regression analysis. The name reflects a combination of a two-group geometric mean ratio with a square-root–based aggregation to yield a single score.

Definition and computation: The data are divided into two groups, A and B, with observed values y_i and predicted values ŷ_i for each observation. For nonnegative y_i and ŷ_i with ŷ_i > 0, compute r_i = y_i / ŷ_i. For each group g, compute the geometric mean ratio GMR_g = (∏ r_i)^{1/n_g} = exp( (1/n_g) ∑ log r_i ). The sqrt2GMR score is then defined as sqrt2GMR = sqrt( (GMR_A^2 + GMR_B^2) / 2 ).

Interpretation: A sqrt2GMR near 1 indicates similar predictive accuracy in both groups; values above 1 suggest underprediction and values below 1 suggest overprediction in at least one group, depending on the distribution of ratios. Using the geometric mean makes the measure multiplicatively scale-invariant and reduces sensitivity to extreme ratios compared with arithmetic averages. The square-root–based aggregation yields a root-mean-square-like combined score for the two groups.

History and usage: sqrt2GMR has appeared in a limited set of methodological discussions on cross-subgroup model evaluation and fairness in regression contexts. It is not a standard benchmark in mainstream statistics or machine learning. Practitioners typically compare it with RMSE, MAE, and group-wise GMRs, and should be mindful of limitations such as sensitivity to zero predictions, reliance on only two groups, and potential interpretive ambiguity when groups differ markedly in size or distribution.

See also: RMSE, MAE, geometric mean, group fairness, model evaluation.