Partition of Sums of Squares - Partitioning The Sum of Squares in Linear Regression

Partitioning The Sum of Squares in Linear Regression

Theorem. Given a linear regression model including a constant based on a sample containing n observations, the total sum of squares (TSS) can be partitioned as follows into the explained sum of squares (ESS) and the residual sum of squares (RSS):

where this equation is equivalent to each of the following forms:


\begin{align} \left\| y - \bar{y} \mathbf{1} \right\|^2 &= \left\| \hat{y} - \bar{y} \mathbf{1} \right\|^2 + \left\| \hat{\varepsilon} \right\|^2, \quad \mathbf{1} = (1, 1, \ldots, 1)^T ,\\ \sum_{i = 1}^n (y_i - \bar{y})^2 &= \sum_{i = 1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i = 1}^n (y_i - \hat{y}_i)^2 ,\\ \sum_{i = 1}^n (y_i - \bar{y})^2 &= \sum_{i = 1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i = 1}^n \hat{\varepsilon}_i^2 .\\ \end{align}

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