General Classes of Influence Measures

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Description

These functions implement the general classes of influence measures for multivariate regression models defined in Barrett and Ling (1992), Eqn 2.3, 2.4, as shown in their Table 1.

They are defined in terms of the submatrices for a deleted index subset I

H_I = X_I (X^T X)^{-1} X_I

Q_I = E_I (E^T E)^{-1} E_I

corresponding to the hat and residual matrices in univariate models.

For subset size m = 1 these evaluate to scalar equivalents of hat values and studentized residuals.

For subset size m > 1 these are m \times m matrices and functions in the J^{det} class use |H_I| and |Q_I|, while those in the J^{tr} class use tr(H_I) and tr(Q_I).

The functions COOKD, COVRATIO, and DFFITS implement some of the standard influence measures in these terms for the general cases of multivariate linear models and deletion of subsets of size m>1, but they are only included here for experimental purposes.

Usage

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Jdet(H, Q, a, b, f)

Jtr(H, Q, a, b, f)

COOKD(H, Q, n, p, r, m)

COVRATIO(H, Q, n, p, r, m)

DFFITS(H, Q, n, p, r, m)

Arguments

H

a scalar or m \times m matrix giving the hat values for subset I

Q

a scalar or m \times m matrix giving the residual values for subset I

a

the a parameter for the J^{det} and J^{tr} classes

b

the b parameter for the J^{det} and J^{tr} classes

f

scaling factor for the J^{det} and J^{tr} classes

n

sample size

p

number of predictor variables

r

number of response variables

m

deletion subset size

Details

These functions are purely experimental and not intended to be used directly. However, they may be useful to define other influence measures than are currently implemented here.

Value

The scalar result of the computation.

Author(s)

Michael Friendly

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.