Hat Matrix Diagonal

Fit Analyses

Hat Matrix Diagonal

Data points that are far from the centroid of the X-space are potentially influential. A measure of the distance between a data point, x_i, and the centroid of the X-space is the data point's associated diagonal element h_i in the hat matrix. Belsley, Kuh, and Welsch (1980) propose a cutoff of 2 p/ n for the diagonal elements of the hat matrix, where n is the number of observations used to fit the model, and p is the number of parameters in the model. Observations with h_i values above this cutoff should be investigated.

For linear models, the hat matrix

H = X (X'X)^-1 X'

can be used as a projection matrix. The hat matrix diagonal variable contains the diagonal elements of the hat matrix

h_i = x_i (X'X)^-1 x_i'

For generalized linear models, an approximate projection matrix is given by

H = W^1/2X (X'WX)^-1 X' W^1/2

where W = W_o when the full Hessian is used and W = W_e when Fisher's scoring method is used.

The values of h_i are stored in a variable named H_yname, where yname is the response variable name.

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