{"id":6147,"date":"2026-04-08T22:30:56","date_gmt":"2026-04-09T03:30:56","guid":{"rendered":"https:\/\/ykim.synology.me\/wordpress\/?p=6147"},"modified":"2026-04-09T01:49:40","modified_gmt":"2026-04-09T06:49:40","slug":"the-impact-of-variance-components-on-the-coefficient-of-determination-r2","status":"publish","type":"post","link":"https:\/\/ykim.synology.me\/wordpress\/the-impact-of-variance-components-on-the-coefficient-of-determination-r2-6147\/","title":{"rendered":"The Impact of Variance Components on the Coefficient of Determination ($R^2$)"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

1. Executive Summary<\/h2>\n\n\n\n

The Coefficient of Determination, denoted as $R^2$, is one of the most widely used metrics for assessing the goodness-of-fit in linear regression models. However, its interpretation is often fraught with misunderstanding, particularly regarding how it fluctuates not just with the “correctness” of a model, but with the underlying distribution of the data. This report explores the mathematical and conceptual reasons why changes in variance\u2014specifically residual variance ($\\sigma^2_{\\epsilon}$) and predictor variance ($\\sigma^2_{x}$)\u2014exert a profound influence on $R^2$. By analyzing the ratio of variances, we demonstrate that $R^2$ is a relative measure of power rather than an absolute measure of model accuracy.<\/p>\n\n\n\n

2. Mathematical Definition of $R^2$<\/h2>\n\n\n\n

To understand why variance dictates the behavior of $R^2$, we must first define it through the lens of Analysis of Variance (ANOVA). In a standard linear model $Y = \\beta_0 + \\beta_1 X + \\epsilon$, the total variation in the dependent variable $Y$ can be partitioned into two distinct components:<\/p>\n\n\n\n

    \n
  1. Explained Variation (SS_{reg}):<\/strong> The variation accounted for by the relationship between $X$ and $Y$.<\/li>\n\n\n\n
  2. Unexplained Variation (SS_{res}):<\/strong> The variation resulting from the residuals or “noise” ($\\epsilon$).<\/li>\n<\/ol>\n\n\n\n

    The fundamental identity is:
    $$SS_{tot} = SS_{reg} + SS_{res}$$<\/p>\n\n\n\n

    From this, $R^2$ is defined as the proportion of the total variance in $Y$ that is explained by $X$:
    $$R^2 = \\frac{SS_{reg}}{SS_{tot}} = 1 – \\frac{SS_{res}}{SS_{tot}}$$<\/p>\n\n\n\n

    Where:<\/p>\n\n\n\n