Measure of fit

Summary

Setup

• Given an OLS trendline with intercept

$$y=b_1+b_2x$$

• where $b_1,b_2$ are determined by the OLS formula and $e_i, {\hat{y}}_i$ are the OLS residuals and the OLS fitted values

TSS, ESS, RSS

• Total sum of squares (TSS)

$$TSS=\sum_{i=1}^{n}{ {\left( y_i-\bar{y} \right)}^2 }$$

• Explained sum of squares (ESS)

$$ESS=\sum_{i=1}^{n}{ {\left( {\hat{y}}_i-\bar{y} \right)}^2 }$$

• Residual sum of squares ( RSS ) (as before)

$$RSS=\sum_{i=1}^{n}{ e_i^2 }$$

• Note:
• Sample variance in $y$ -data is equal to $TSS/(n-1)$
• Sample variance in   $\hat{y}$ -data is equal to $ESS/(n-1)$
• Sample variance in $e$ -data is equal to $RSS/(n-1)$

Coefficient of determination

• Result: $TSS=ESS+RSS$
• Coefficient of determination

$$R^2= \frac{ESS}{TSS}=1- \frac{RSS}{TSS}$$

• Result: $0≤R^2≤1$