Gaj Vidmar <[EMAIL PROTECTED]> wrote:
>This is about rewriting the basic formula for the hat elements
>("leverages") in multiple linear regression (see note at the end
>of the message for context).
>
>- I am familiar with the expression for the matrix in the diagonal of which
>lie the hat elements (Stevens, 1996, p. 109):
>
>H = X (X' X)^-1 X'
>
>where X is the raw-scores matrix (with 1's in the 1st column, corresponding
>to the constant term).
>
>Knowing the simplification for the bivariate case (Myers & Well, 1991)
>
>h_sub_ii = 1/N + (x_sub_i - x_bar)^2 / SS_sub_x ,
>
>it is this kind of non-matrix expression that I am looking for for the
>general (p-variate) case. In other words, I don't want to
>"carry arround" the entire data matrix which the model has been
>derived from, when predicting Y (with CI) for a single <new> case.
You don't have to "carry around" X, just (X'X)^-1 , or your
S/sigma_2 below.
>... What I wonder is whether I can use this formula if the
>predictors are correlated to a non-negligible extent, in which case, of
>course, I would use (Stevens, 1996, p. 111)
>
>MD_sub_i^2 = (x_sub_i - x_bar)' S^-1 (x_sub_i - x_bar)
>
>whereby x_sub_i and x_bar are case-vector and centroid, respectively, and S
>is the covariance matrix).
Don't bother with x_bar, etc. Just leave the constant term
in x_sub_i, and use:
h_sub_ii = x_sub_i'(X'X)^(-1)*x_sub_i
or
h_sub_ii = x_sub_i'(P/sigma_2)*x_sub_i
Clint Cummins
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