DFFITS

A regression influence diagnostic measuring the studentized change in an observation's fitted value when that observation is omitted from model estimation.

Version

v2 · 2026-09-19 · History

Domain-specific #

4140

Domain group

Formal Sciences

Origin domain

Experimental Design & Statistics

Subdomains

Regression Diagnostics, Influence Analysis → Experimental Design & Statistics

Core Idea

DFFITS for observation i is its full-fit prediction minus its deletion-fit prediction divided by an estimated standard deviation of that fitted difference. Deleting an observation changes coefficients according to its residual and leverage; studentization makes the induced prediction change comparable across points. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of regression diagnostics. It is case-specific standardized influence on its own fitted value rather than residual extremeness alone.

How would you explain it like I'm…

The Bossy Dot Detector

Imagine drawing a best-fit line through a bunch of dots. Now take away just one dot, draw the line again, and see how much the line's guess at that dot's spot moved. If it moved a lot, that one dot was pushing the whole line around. DFFITS is a number that measures that push in a fair way, so you can compare every dot.

How Much One Point Moves Its Fit

When statisticians fit a line or model to data, they sometimes worry that one data point is bossing the whole result. DFFITS checks this one point at a time. For a point, you compare the model's prediction for that point when the point is included to the prediction when the point is left out. Then you divide by how much that difference would normally vary, so the numbers can be compared across points. A big DFFITS means that point has a lot of influence on its own predicted value — which is different from simply being far from the line.

Standardized Leave-One-Out Influence

DFFITS is a regression diagnostic that measures how influential one observation is on its own fitted value. For observation i, you fit the model with all the data and again with just observation i deleted, then take the difference between the two predictions at point i. You divide that difference by an estimate of its standard deviation, so the result is on a comparable scale across points. How much a point matters depends on both its residual (how far it is from the fit) and its leverage (how unusual its predictor values are). A point with a large residual isn't necessarily influential, and DFFITS captures the combination rather than residual size alone. You also need to say which variance estimate you used — one computed with or without the deleted point.

 

DFFITS for observation i is (ŷ_i − ŷ_i(i)) divided by an estimated standard deviation of that difference, where ŷ_i is the full-data fitted value and ŷ_i(i) is the prediction for case i from the fit with exactly case i deleted, using the same response and design matrix. Deleting a case changes the coefficient estimates by an amount determined by its residual and its leverage h_ii, so the numerator reflects both. The denominator studentizes the change so it is comparable across cases; one must state whether the variance estimate is internal (from the full fit) or external (from the deletion fit), since conventions differ. The diagnostic targets case-specific standardized influence on the case's own fitted value, which is distinct from residual extremeness alone and from global influence measures on all coefficients.

Scope of Application

DFFITS belongs to regression diagnostics and is useful where the analyst can specify a fitted linear model, an observation i, full-data and leave-one-out predictions, leverage, residual variance, a studentization convention, and reference thresholds, then evaluate the diagnostic uses the same response and design with exactly observation i deleted and reports the selected internal or external variance convention. The scope is broad within that domain but bounded by the need for the diagnostic uses the same response and design with exactly observation i deleted and reports the selected internal or external variance convention. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making the diagnostic uses the same response and design with exactly observation i deleted and reports the selected internal or external variance convention the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name DFFITS can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to DFFITS. DFFITS compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: a fitted linear model, an observation i, full-data and leave-one-out predictions, leverage, residual variance, a studentization convention, and reference thresholds. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express the diagnostic uses the same response and design with exactly observation i deleted and reports the selected internal or external variance convention independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of regression diagnostics because they reuse a fitted linear model, an observation i, full-data and leave-one-out predictions, leverage, residual variance, a studentization convention, and reference thresholds, Deleting an observation changes coefficients according to its residual and leverage; studentization makes the induced prediction change comparable across points., and type the carrier, state every parameter and convention in the definition, test that the diagnostic uses the same response and design with exactly observation i deleted and reports the selected internal or external variance convention, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction DFFITS Domain-specific

Parents (1) — outgoing typed relations

• DFFITS is a kind of Statistical Inference Prime

  The diagnostic quantifies sensitivity of a fitted inference to one case.

Hierarchy paths (3) — routes to 3 parentless roots

• DFFITS → is a kind of Statistical Inference → is a kind of Inductive Reasoning

Show alternative paths (2)

Neighborhood in Abstraction Space

DFFITS sits in a crowded region of the domain-specific corpus (17^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Regression Diagnostics & Linear Models (11 abstractions)

Nearest neighbors

• Regression diagnostic — 0.92
• Regression analysis — 0.92
• Working–Hotelling procedure — 0.91
• Partial residual plot — 0.90
• Best linear unbiased prediction — 0.90

Computed from structural-signature embeddings · 2026-10-25