I cannot understand why total SS＝explained SS+unexplained SS because geometrically the sum of two small squares is not equal to a big square. I wish someone could explain that to me. Thank you.
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1Geometrically this is called the Pythagorean Theorem. There exist some very pretty geometric proofs, going back to Euclid, showing exactly how the sum of (areas) of two small squares equals the area of a big square. – whuber Mar 01 '23 at 20:40
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1Related https://stats.stackexchange.com/questions/265869/confused-with-residual-sum-of-squares-and-total-sum-of-squares and more linked from that question – Henry Mar 02 '23 at 08:52
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Thank you, guys. – J.Liu Mar 02 '23 at 15:39
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1One caveat, this decomposition holds only for models containing an intercept. If your model doesn't have an intercept, then this relationship no longer holds. – Zhanxiong Mar 02 '23 at 18:38
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1@Zhanxiong Added to my answer – Dave Mar 02 '23 at 18:38
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This question of mine has several answers that discuss the details why, but the gist is that, for a linear model (WITH AN INTERCEPT), the minimization of the residual sum of squares forces the explained and unexplained vectors to be orthogonal, and then the Pythagorean theorem applies to a right triangle. Then this post of mine gives the decomposition of the total sum of squares; in the language of that post, the orthogonality means that the $Other$ term equals zero. (That entire decomposition can be viewed as the law of cosines, where the $Other$ term corresponds to the term with the cosine in it, so zero when the vectors are orthogonal.)
Dave
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