Re: R^2 and beta coefficients in multiple regression

On Sep 23, 1:29 pm, hrun...@xxxxxxxxxx wrote:
Hi,

I am trying to work out the relationship between the magnitude of the
vector of standardized regression coefficients (beta coefficients) in
a multiple linear regression framework and the coefficient of
determination (R^2) for the same model. Following Bring (1996; Amer.
Stat. Assoc.), if all variables are standardized, R^2 = ||y-hat||^2,
and ||y-hat||^2 = B1^2 + B2^2 + ... + Bk^2, where Bk are the partial
regression coefficients. This implies that the squared magnitude of
the Beta vector should equal R2. While I can confirm this for real
data in the case of simple linear regression (one independent
variable), it does not seem to work with multiple independent
variables, so I must be doing something wrong. Any suggestions would
be much appreciated.

Best,
Howard

Assisant Professor
Dept. of Biology, University of Ottawa
Ottawa, ON, Canada

When all the variables are standardized, R^2 is guaranteed to equal
B1^2 + B2^2 + ... + Bk^2 only when all the predictors are mutually
uncorrelated. In general, R^2 = r1*B1 + r2*B2 + ... + rk*Bk, where
ri is the correlation of the d.v. with predictor i.
.

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