Coefficients of determination measured on the same scale as the outcome: Alternatives to R² that use standard deviations instead of explained variance.

Abstract

The coefficient of determination, R², also called the explained variance, is often taken as a proportional measure of the relative determination of model on outcome. However, while R² has some attractive statistical properties, its reliance on squared variations (variances) may limit its use as an easily interpretable descriptive statistic of that determination. Here, the properties of this coefficient on the squared scale are discussed and generalized to three relative measures on the original scale. These generalizations can all be expressed as transformations of R², and alternatives can therefore also be calculated by plugging in related estimates, such as the adjusted R². The third coefficient, new for this article, and here termed the CoDSD (the coefficient of determination in terms of standard deviations), or Rπ (R-pi), equals R²/(R²+1-R²). It is argued that this coefficient most usefully captures the relative determination of the model. When the contribution of the error is c times that of the model, the CoDSD equals 1/(1 + c), while R² equals 1/(1 + c²). (PsycInfo Database Record (c) 2024 APA, all rights reserved).