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l72 - Computing R2/Evils of R2 COMPUTING R2 Here are some...

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Computing R 2 /The evils of R 2 Page 1 Computing R 2 /Evils of R 2 C OMPUTING R 2 . Here are some of the many formulas for R 2 : Our knowledge of path analysis now makes it possible to prove many of these formulas. See the optional appendix if you are interested. NOTE: I use the notation b’ k for the standardized coefficients and b k for the non- standardized, aka metric coefficients. p k is another notation often used for the standardized (aka path) coefficients. R 2 = SSR/SST Explained sum of squares over total sum of squares, i.e. the ratio of the explained variability to the total variability. R F K N K F K 2 1 * ( ) ( * ) This can be useful if F, N, and K are known R b r k yk k K 2 1 Also, R bb r i j ij j K i K 2 1 1   These formulas uses the standardized coefficients.and the zero-order correlations betwee n y and the x’s. The se (esoteric) formulas can be useful when doing path analysis. Two IV case only: R b b bb r 2 1 2 2 2 1 2 12 2       This is a special case of the last formula. One IV case only: R b 2 2 Remember that, in standardized form, correlations and covariances are the same. One IV case only: ) ( ) ( ) ( ) ( ) ˆ ( 2 2 2 u V X V X V Y V Y V R We’ll use this formula in the rest of this handout. Technically I should use rho 2 2 ) here or else put hats over the other parameters (to distinguish between population parameters and sample estimates) but since that seems to confuse some people I won’t!
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Computing R 2 /The evils of R 2 Page 2 E VILS OF R 2 (In General) While R 2 has its uses, it is far more important to have a correctly specified model than it is to have a large R 2 . For example, consider the following model: u X2 X1 X4 w X3 v You can increase R 2 by regressing a variable on variables that are causally subsequent to it, e.g. regress X1 on X2 and X3. This would be foolish, since X2 and X3 are consequence of X1, not causes of it. Remember, the data can’t tell you what the proper causal ordering is, you have to decide that for yourself. You should regress X3 on X1. If you instead regress X3 on X2, you’ll get an R 2 , but it will be meaningless (unless, perhaps, you are intentionally using X2 as a proxy for the unmeasured X1.) Such questionable modeling occurs when, say, one attitude is regressed on another, when in reality both attitudes are functions of something else.
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