Influence Diagnostics (Cook's Distance, DFFITS, Leverage)
Also known as: Cook's distance, DFFITS, leverage, influential observation detection, regression diagnostics, Etki Tanılamaları (Cook's D, DFFITS, Leverage)
Influence diagnostics are a family of post-fit measures that quantify how much each single observation affects a fitted regression. Cook's distance was introduced by R. Dennis Cook in 1977, with leverage and DFFITS formalised by Belsley, Kuh and Welsch in 1980, to flag the observations that most strongly pull the estimated coefficients.
Key highlights
- Pinpoints exactly which observations drive the regression, observation by observation.
- Distribution-free: requires no normality assumption, only a fitted model with computable residuals.
- Cook's distance, DFFITS and leverage give complementary views (overall influence, single-fit influence, predictor-space position) of the same data.
Intuition
This section is available to Pro members. Upgrade to Pro
How it works
This section is available to Pro members. Upgrade to Pro
When to use it
Use influence diagnostics after fitting any linear regression, once residuals and leverage can be computed, to check whether the conclusions rest on a handful of observations. They suit continuous outcomes and need a working regression model plus a reasonable sample of at least about 20 observations. They are a screening step rather than a final model, and in very small samples or with several clustered high-leverage points the standard measures can become unreliable.
Strengths & limitations
- Pinpoints exactly which observations drive the regression, observation by observation.
- Distribution-free: requires no normality assumption, only a fitted model with computable residuals.
- Cook's distance, DFFITS and leverage give complementary views (overall influence, single-fit influence, predictor-space position) of the same data.
- In very small samples (n < 20) Cook's distance and leverage values become unreliable, so simpler outlier detection is preferable.
- When several high-leverage points cluster together they can mask one another, hiding their joint influence from the standard one-at-a-time measures.
- Diagnostics flag influential points but do not by themselves decide whether to keep, transform, or remove them.
Common pitfalls
This section is available to Pro members. Upgrade to Pro
Applications
This section is available to Pro members. Upgrade to Pro
Frequently asked
What counts as a large Cook's distance?
A common rule of thumb flags observations with Cook's distance above 4/n, while values near or above 1 are treated as clearly influential. Thresholds are guides, not verdicts: a flagged point should be inspected, not automatically deleted.
What is the difference between leverage and influence?
Leverage measures how unusual an observation's predictor values are, regardless of its outcome. Influence (Cook's distance, DFFITS) measures how much the fitted model actually changes when the point is removed, combining leverage with the size of the residual. A high-leverage point is only influential if it also sits far from the fitted line.
Should I delete observations flagged as influential?
Not automatically. Flagging is a screening step: investigate whether the point is a data error, a genuine but rare case, or a sign of model misspecification. If influential points are pervasive, a robust regression method is usually a better response than deletion.
Do these diagnostics require normally distributed errors?
No. The measures are computed directly from the fitted residuals and the hat matrix and need no normality assumption; they only require a regression model from which residuals and leverage can be calculated.
Sources
- 1.Cook, R. D. (1977). Detection of Influential Observations in Linear Regression. Technometrics, 19(1), 15-18.
- 2.Belsley, D. A., Kuh, E., & Welsch, R. E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. Wiley.ISBN 978-0471058564
You have read it. What now?
Cite this page
ScholarGate. (2026, June 1). Influence Diagnostics. ScholarGate. https://scholargate.app/statistics/influence-diagnostics