Residualit

Residualit is a proposed statistical measure intended to quantify how large a model’s residuals are relative to the inherent variation in the data. It is defined as a dimensionless ratio that objects to the absolute scale of residuals by comparing the model’s prediction error to the observed spread of the response variable.

Calculation proceeds from a fitted model. For data (x_i, y_i) with predictions ŷ_i, residuals are e_i = y_i − ŷ_i. The root-mean-square error (RMSE) is sqrt((1/n) sum e_i^2). Let s_y be the sample standard deviation of the observed responses y. The residualit is then defined as R = RMSE / s_y. If s_y is zero, the metric is undefined. Values near zero indicate a fit that is small relative to data variation; values near or above one suggest residuals on the order of, or larger than, the data’s natural dispersion.

Interpretation and use are still evolving. Residualit can be used to compare models across datasets with different scales, since it normalizes error by data spread. It complements other model-evaluation metrics such as RMSE, R^2, and AIC. However, it has limitations: it depends on the scale of y, can be affected by outliers, and there is no universally standardized threshold for interpretation. It should be used alongside other diagnostics, especially when heteroscedasticity or nonlinearity is a concern.

History and status: residualit is not part of the formal statistical canon and appears only in limited methodological discussions. There is no broadly accepted standard definition or usage, and practitioners may define variants tailored to their applications.

See also: residual, RMSE, standard deviation, model evaluation.