NOTES ON STATISTICS, PROBABILITY and MATHEMATICS

R-Squared: SST, SSE & SSR:

See also this post.

From these Wikipedia definitions:

\[\begin{align} \text{SST}_{\text{otal}} &= \color{red}{\text{SSE}_{\text{xplained}}}+\color{blue}{\text{SSR}_{\text{esidual}}}\\ \end{align}\]

or, equivalently,

\[\begin{align} \sum(y_i-\bar y)^2 &=\color{red}{\sum(\hat y_i-\bar y)^2}+\color{blue}{\sum(y_i-\hat y_i)^2} \end{align}\]

and

\(\large \text{R}^2 = 1 - \frac{\text{SSR}_{\text{esidual}}}{\text{SST}_{\text{otal}}}\)

So if the model explained all the variation, \(\text{SSR}_{\text{esidual}}=\sum(y_i-\hat y_i)^2=0\), and \(\bf R^2=1.\)

From Wikipedia:

Suppose \(r = 0.7\) then \(R^2 = 0.49\) and it implies that \(49\%\) of the variability between the two variables have been accounted for and the remaining \(51\%\) of the variability is still unaccounted for.

Proof by example:

fit = lm(mpg ~ wt, mtcars)
summary(fit)$r.square
[1] 0.7528328
> sse = sum((fitted(fit) - mean(mtcars$mpg))^2)
> ssr = sum((fitted(fit) - mtcars$mpg)^2)
> 1 - (ssr/(sse + ssr))
[1] 0.7528328

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NOTE: These are tentative notes on different topics for personal use - expect mistakes and misunderstandings.