DFFITS

DFFITS is a diagnostic meant to show how influential a point is in a statistical regression. It was proposed in 1980. It is defined as the Studentized DFFIT, where the latter is the change in the predicted value for a point, obtained when that point is left out of the regression; Studentization is achieved by dividing by the estimated standard deviation of the fit at that point:

where y i ^ and y i ( i ) ^ are the prediction for point i with and without point i included in the regression, s ( i ) is the standard error estimated without the point in question, and h i i is the leverage for the point.

DFFITS is very similar to the externally Studentized residual, and is in fact equal to the latter times h i i / ( 1 − h i i ) .

As when the errors are Gaussian the externally Studentized residual is distributed as Student's t (with a number of degrees of freedom equal to the number of residual degrees of freedom minus one), DFFITS for a particular point will be distributed according to this same Student's t distribution multiplied by the leverage factor h i i / ( 1 − h i i ) for that particular point. Thus, for low leverage points, DFFITS is expected to be small, whereas as the leverage goes to 1 the distribution of the DFFITS value widens infinitely.

For a perfectly balanced experimental design (such as a factorial design or balanced partial factorial design), the leverage for each point is p/n, the number of parameters divided by the number of points. This means that the DFFITS values will be distributed (in the Gaussian case) as p n − p ≈ p n times a t variate. Therefore, the authors suggest investigating those points with DFFITS greater than 2 p n .

Although the raw values resulting from the equations are different, Cook's distance and DFFITS are conceptually identical and there is a closed-form formula to convert one value to the other.