DFBETAS

A standardised version of the DFBETA diagnostic, scaled by its standard error, allowing influence of observations to be compared consistently across models.

Source: Belsley, Kuh & Welsch 1980

DFBETAS reveals how much an individual observation influences each estimated regression coefficient, measuring the change in a coefficient when that one point is removed, scaled by its standard error. Because it is standardised, its values can be compared across coefficients and models, and it pinpoints influence on specific predictors rather than on the fit as a whole. A large value flags a case that noticeably shifts a particular coefficient, prompting scrutiny. Isolating a point's effect on each coefficient is what it does. Kirkwood and Sterne (2003) describe this diagnostic.

Source: Kirkwood & Sterne 2003

DFBETAS are calculated for each observation and each coefficient as the difference between the coefficient estimated with all the data and the coefficient estimated with that observation removed, divided by an estimate of the coefficient's standard error. This gives a standardised measure of the change attributable to the observation. So DFBETAS are calculated by refitting the model without each observation, taking the change in each coefficient, and standardising it by its standard error, which yields, for every observation and coefficient, a scaled value indicating how strongly that observation influences that particular coefficient, comparable across coefficients because of the standardisation.

Source: Belsley, Kuh & Welsch 1980

DFBETAS are interpreted by their magnitude, with larger absolute values indicating greater influence of an observation on a coefficient; a common rough threshold flags values exceeding two divided by the square root of the sample size, though such cut-offs are guides rather than strict rules. So DFBETAS are interpreted by identifying observations whose standardised influence on a coefficient is large relative to a threshold or to the other observations, marking them as influential for that particular coefficient, which prompts investigation of those cases, since a large DFBETAS value signals that a single observation substantially affects the estimate of a specific regression parameter.

Source: Belsley, Kuh & Welsch 1980

DFBETAS measure the influence of an observation on each individual coefficient separately, giving a value for every observation and coefficient, while Cook's distance summarises an observation's overall influence on all the fitted values, or the whole set of coefficients, in a single measure. So DFBETAS and Cook's distance differ in granularity, with DFBETAS pinpointing which coefficients an observation affects and Cook's distance giving an aggregate measure of its total influence, and they are complementary, since Cook's distance flags generally influential observations and DFBETAS reveal which specific coefficients those or other observations most affect.

Source: Cook 1977

DFBETAS are useful because they reveal not just whether an observation is influential but which specific coefficients it affects, allowing targeted investigation of how individual data points drive particular estimates. This is valuable when the interest is in specific coefficients. So DFBETAS are useful for diagnosing the influence of observations at the level of individual coefficients, complementing overall influence measures, which helps the analyst understand how particular cases affect the estimates of interest and decide whether the results depend unduly on a few observations, supporting robust and transparent regression modelling.

Source: Belsley, Kuh & Welsch 1980