SSresSStot

SSres and SStot are two fundamental sums of squares used in linear regression to quantify model fit. SSres, the residual sum of squares, measures the discrepancy between observed values y_i and the model’s predictions ŷ_i: SSres = sum_i (y_i - ŷ_i)^2. SStot, the total sum of squares, captures the overall variability of the observed values around their mean ȳ: SStot = sum_i (y_i - ȳ)^2. In ordinary least squares regression, the total variability can be decomposed as SStot = SSres + SSreg, where SSreg = sum_i (ŷ_i - ȳ)^2 is the explained sum of squares.

The ratio SSres/SStot represents the proportion of total variability the model fails to explain. The coefficient of determination, R^2, is defined as R^2 = 1 - SSres/SStot = SSreg/SStot and expresses the fraction of variance explained by the model. With an intercept included, R^2 typically lies between 0 and 1, while without an intercept negative values of R^2 (and thus ratios exceeding 1) are possible. In practice, SSres and SStot are used to assess fit, compare models, and perform variance decomposition in regression analysis.

The term SSresSStot may be encountered as a shorthand in some contexts for the ratio SSres/SStot or as a symbolic label for the pair of statistics used in the same calculation, but it is not a separate standard metric beyond its role in deriving R^2. Related concepts include adjusted R^2, which accounts for model complexity, and analysis of variance (ANOVA), which uses these sums of squares to test hypotheses about model components.