An ABA DFBETAS calculator shows how deleting one case changes each coefficient in a documented ordinary least-squares regression. It produces a case-by-coefficient matrix, not one verdict about the row. The matrix can reveal that a case has modest influence on one parameter and much stronger influence on another, while leaving the clinical meaning of those parameters for separate review.

Clinicians & ABA Professionals / Data, Outcomes and Clinical Decision-Making.

One case can pull coefficients in different directions

DFBETA is the raw difference between a full-model coefficient and the corresponding coefficient from a fit that omits case i. DFBETAS standardizes that difference using a coefficient-specific design term and the deleted model's residual error scale. A model with an intercept and two predictors therefore has three DFBETAS values for every case.

Positive and negative signs describe the direction of the full-minus-deleted coefficient change under the stated coding. They do not label the observed response as good or bad. A value for the intercept cannot be compared as though it meant the same thing as a value for a slope, and a coefficient's statistical sensitivity is not its clinical importance.

DFBETAS differs from DFFITS, which summarizes the standardized change in a case's fitted value. It also differs from Cook's distance, which gives an overall coefficient-influence measure, and from leverage or a studentized residual, which describe other parts of the diagnostic picture.

The coefficient map must be frozen

Before using this ABA DFBETAS calculator, write down exactly what every design column means. A reordered, recoded, centered, transformed, or omitted column creates a different coefficient target even if some fitted values remain unchanged.

Audit fieldWhat belongs in the recordStop signalResponseDefinition, unit, denominator, direction, and observation windowThe measured quantity changedDesign matrixLabeled columns in exact order, intercept rule, coding, transformations, and weightsColumn identity is ambiguousPopulation and rowsStable authorized keys, inclusion and exclusion rules, missing-data handling, phaseFull and deletion fits use different row logicFull fitFormula, coefficient vector, residuals, SSE, degrees of freedom, software, and versionCoefficients cannot be reproducedDesign covariance factorDiagonal c_jj from the full-design (X'X)^-1Matrix is singular or from another designDeletion fitsbetahat(i) and deleted residual variance s_i^2 for every reviewed caseError scale is nonpositive or undefinedPurposePrespecified coefficient-influence reviewResult-driven recoding, deletion, or clinical action

Keep the original column names in every export. If software drops a reference level, aliases a rank-deficient column, changes weight handling, or uses an approximation, document that behavior rather than forcing its output into this worksheet.

Coefficient-specific standardization

Let betahatj be full-model coefficient j, betahat(i),j the same coefficient from the model fitted without case i, si^2 the leave-one-out residual mean square, and cjj diagonal element j of the full design matrix inverse C = (X'X)^-1:

DFBETAij = betahatj - betahat_(i),j

DFBETASij = DFBETAij / (si * sqrt(cjj))

The denominator changes by case through si and by coefficient through cjj. The statsmodels DFBETAS documentation identifies its ordinary least-squares result as using leave-one-observation-out fits. The library's current source computes full coefficients minus deleted-fit coefficients, divides by the square root of each deleted-fit residual variance, and then divides by the square root of the full normalized covariance diagonal.

Use the exact coefficient order returned by the full fit. Do not mix a full-model standard error with the deleted-model convention described here, and do not compare outputs from different software conventions without a documented reconciliation.

Blank coefficient-influence matrix

CaseAuthorized keyDeleted MSE s_i^2Intercept DFBETASPredictor 1 DFBETASPredictor 2 DFBETASLargest absolute valueContext note123...

Extend the table with one column for every estimated coefficient. A separate coefficient register keeps the matrix interpretable:

Column jCoefficient label and codingFull betahatjFull c_jjUnit or interpretation boundary1Intercept23...

Archive the source model, deletion-fit outputs, analyst, reviewer, calculation date, code or workbook version, displayed precision, and any later sensitivity analysis with the completed matrix.

Fictional two-predictor calculation

For the coefficient-specific illustration, use the ten synthetic rows below. No value comes from a client record.

Casex1x2y1125225733164471055396681377412881018996151010924

The full-model coefficients are beta_hat = [0.7405797101, 1.3293075684, 0.6996779388] for the intercept, x1, and x2. Full-model SSE = 23.4901771337, residual degrees of freedom are 7, and MSE = 3.3557395905. The diagonal of C = (X'X)^-1 is [0.5384057971, 0.0221417069, 0.0221417069].

CaseDeleted MSEInterceptx1x213.31383242440.7389082796-0.4132582301-0.091056898223.91244105340.0271052778-0.03609083200.021156694633.83098086290.23272107610.0353515805-0.188948102743.7075098814-0.09222708740.2526593084-0.252659308453.8653765914-0.0932690167-0.05481317210.100350268963.55237547500.02342981330.1545237470-0.282897321473.7529644269-0.0810079364-0.22192405470.221924054783.87949280460.07101854290.0228428144-0.122090904793.02188311340.1312228941-0.76281955420.4471700835100.4607502892-2.48183202212.65658831290.5853499672

Deleting case 10 gives coefficients [1.9766980664, 1.0609816559, 0.6405552801]. For the x1 slope, the raw difference is 1.3293075684 - 1.0609816559 = 0.2683259125. Its standardizer is sqrt(0.4607502892 * 0.0221417069) = 0.1010039497, producing DFBETAS_10,x1 = 2.6565883129 at full precision.

Changing only the fictional final response from 24 to 20 and refitting yields case 10 DFBETAS values [-0.7243124264, 0.7753143281, 0.1708319706]. That sensitivity does not identify which fictional response is preferable. It shows why the result must travel with the exact dataset and model version.

Reading the matrix without collapsing it

Review each coefficient column according to its prespecified meaning. A case may shift the intercept and one slope in opposite directions. Taking only the largest absolute entry loses that pattern, while averaging entries mixes parameters with different meanings.

NIST regression-diagnostics guidance describes DFBETA as influence on individual parameters rather than on the parameter vector as a whole. It mentions absolute value 1 for smaller datasets and 2 / sqrt(n) for larger datasets as common recommendations. Such values are triage heuristics. They are not universal ABA thresholds, adjusted tests across cases and coefficients, proof of data error, or authorization to delete a row.

If a formal inferential claim matters, prespecify the model and multiplicity plan and involve a qualified statistician. Do not choose the coefficient, coding, case, or comparison after finding a preferred result.

Checks that protect the implementation

Diagnostic checkExpected resultLimit of the checkAdd a constant to every response and refitAll DFBETAS stay the same under this fixed design and codingResponse meaning is unchangedMultiply the response by a positive constantDFBETAS stay the sameUnits have equal clinical meaningMultiply the response by a negative constantSigns reverse under the same codingDirection is clinically irrelevantPermute whole rowsValues follow case keysTemporal order can be ignored elsewhereCompare explicit deletion refits with a trusted libraryValues agree within declared toleranceModel assumptions are satisfiedRescale or center a predictorFitted values may agree while coefficient meanings and entries must be remappedCoefficients are interchangeableDuplicate or linearly combine a design columnMatrix becomes rank deficient; stopAutomated column dropping is acceptableDeleted MSE is zero or negativeStandardization is undefined; stopA near-perfect deletion fit is valid evidenceColumn labels and coefficient order differReconcile before calculationPositional matching is safe

Numerical agreement is necessary but not sufficient. Repeated measures, clustered cases, weights, generalized or mixed models, robust covariance estimates, penalization, nonlinear terms, small samples, or poorly conditioned designs require methods appropriate to those structures. The R core regression-deletion documentation presents DFBETAS with DFFITS, covariance ratios, Cook's distance, leverage, and studentized residuals, and explicitly notes approximation limits for generalized models.

Clinical interpretation stays outside the coefficient table

Return to direct observations and graphs after locating a sensitive coefficient. Confirm measurement quality, opportunity or denominator changes, phase and treatment integrity, setting events, staffing, schedule, and whether the model form matches the clinical question. Review leverage, residual plots, DFFITS, Cook's distance, and collinearity diagnostics when their separate questions matter.

A large DFBETAS entry does not reveal a cause. It may reflect a valid but unusual context, coding choice, data-entry problem, nonlinear relation, correlated predictors, phase transition, dependence, or a model that is too simple. Preserve the original case and fit. Any exclusion, correction, or alternative model needs documented evidence and a reviewed rationale that did not arise solely from seeking a preferred coefficient.

Ethics, evidence, and data handling

The BACB Ethics Codes anchor professional conduct and consumer protection. The BCBA Test Content Outline includes representative measurement, graphing, interpretation, experimental design, competence, confidentiality, and data-based decisions. The Standards for Educational and Psychological Testing supplies broader evidence and intended-use principles. These materials do not validate this calculator or turn a coefficient diagnostic into a clinical decision.

Prefer synthetic or appropriately de-identified data for learning and testing. For authorized identifiable data, apply organization-approved field minimization, access, storage, transmission, retention, audit, and disposal controls. HHS's Privacy Rule summary and Security Rule summary describe different regulatory safeguards. Neither summary determines a particular organization's coverage or certifies a spreadsheet, script, or review workflow.

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