DFFITS information

In statistics, DFFIT and DFFITS ("difference in fit(s)") are diagnostics meant to show how influential a point is in a linear regression, first proposed in 1980.^[1]

DFFIT is the change in the predicted value for a point, obtained when that point is left out of the regression:

${\displaystyle {\text{DFFIT}}={\widehat {y}}_{i}-{\widehat {y}}_{i(i)}}$

where ${\displaystyle {\widehat {y}}_{i}}$ and ${\displaystyle {\widehat {y}}_{i(i)}}$ are the prediction for point i with and without point i included in the regression.

DFFITS is the Studentized DFFIT, where Studentization is achieved by dividing by the estimated standard deviation of the fit at that point:

${\displaystyle {\text{DFFITS}}={\frac {\text{DFFIT}}{s_{(i)}{\sqrt {h_{ii}}}}}}$

where ${\displaystyle s_{(i)}}$ is the standard error estimated without the point in question, and ${\displaystyle h_{ii}}$ is the leverage for the point.

DFFITS also equals the products of the externally Studentized residual (${\displaystyle t_{i(i)}}$) and the leverage factor (${\displaystyle {\sqrt {h_{ii}/(1-h_{ii})}}}$):^[2]

${\displaystyle {\text{DFFITS}}=t_{i(i)}{\sqrt {\frac {h_{ii}}{1-h_{ii}}}}}$

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 ${\displaystyle {\sqrt {p \over n-p}}\approx {\sqrt {p \over n}}}$ times a t variate. Therefore, the authors suggest investigating those points with DFFITS greater than ${\displaystyle 2{\sqrt {p \over n}}}$.

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.^[3]