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(Ugarte, Probability and Statistics with R, 2E) In the book, it reads:
$SST$: Sum of squares of Total.
$SSE$: Sum of squares of Error.
$SSR$: Sum of squares of Regression.

Marginal increases when adding x2 to a model that already contains x1 will be denoted as

$SSR(x2|x1) = SSR(x2,x1) - SSR(x1)$

which is equivalent to

$SSR(x2|x1) = SSE(x1) - SSE(x1,x2)$.

Why is there an equivalence?

What I did:

$SSR(x2|x1) = SSR(x2,x1) - SSR(x1)$ (using $SST=SSR+SSE$)

$= SST(x2,x1) - SSE(x2,x1) - [SST(x1) - SSE(x1)] $

$= SSE(x1) - SSE(x2,x1) + [SST(x2,x1) - SST(x1)]$.

So, in fact, the question turns out to be why $SST(x2,x1) - SST(x1) = 0$!

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1 Answer 1

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Since

$$SST(X) = \sum_{i=1}^N(y_i - \bar{y})^2$$

so it doesn't depend on $X$.

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  • $\begingroup$ So, it seems the exact value of that sum is irrelevant since the cancellation happens in $SST(x2,x1)−SST(x1)$ ? $\endgroup$
    – Erdogan CEVHER
    Mar 6, 2017 at 15:56
  • $\begingroup$ @ErdoganCEVHER Exactly $\endgroup$
    – Łukasz Grad
    Mar 6, 2017 at 16:22

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