An ABA DFFITS calculator asks a narrow regression question: how much would one case's own fitted value change if that case were omitted, after putting the change on a deleted-model error scale? The signed result helps a reviewer locate cases that deserve closer model and provenance review. It does not say that a case is wrong, dispensable, clinically unimportant, or responsible for the pattern.

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

What a DFFITS value represents

DFFITS is short for difference in fits. For case i, it compares the prediction at that case's predictor values from the full ordinary least-squares model with the prediction from a model fitted without case i. The change is standardized so cases within the same documented fit can be reviewed on a common scale.

The sign records direction. A positive DFFITS means the full-model fitted value at the case is higher than the deleted-fit value under the convention used here. A negative result means it is lower. Absolute magnitude describes fitted-value sensitivity, not the size or importance of a client's observed behavior change.

This is different from leverage, which describes predictor position; an externally studentized residual, which emphasizes response-side outlyingness; Cook's distance, which summarizes joint coefficient influence; and DFBETAS, which separates influence coefficient by coefficient.

A reproducible model record comes first

Use this ABA DFFITS calculator only when every input belongs to one fixed fit. Record the analysis contract before inspecting influence values.

Model elementRequired recordPause conditionResponseObservable definition, unit, denominator, and directionMeaning or scale changedDesignEvery predictor column, coding rule, transformation, weight, and intercept conventionA column or row cannot be reproducedCasesAuthorized stable key, inclusion rule, exclusion rule, and missing-data handlingCases are missing, duplicated, or shiftedModel identityFormula, population, phase, software, version, and calculation dateOutputs came from different fitsError termsOrdinary residuals and leave-one-out residual variance for each caseDeleted error scale is zero or undefinedLeverageHat diagonal h_ii from the same full designh_ii is outside [0, 1]Intended useDescriptive influence reviewAutomatic deletion or clinical verdict

Keep the source observations, fitted values, residual plots, leverage plot, phase labels, setting, staffing, schedule, measurement changes, and intervention changes with this record. A compact influence table is not a substitute for the clinical graph.

Two calculation routes should agree

Let p be the number of estimated parameters, including the intercept. Let ti be the externally studentized residual from the same ordinary least-squares fit and let hii be the case's leverage:

DFFITSi = ti * sqrt(hii / (1 - hii))

The direct deletion form uses the full-model fitted value yhati, the fitted value at the same predictor row from the model without case i, yhat(i),i, and the deleted-model residual standard error s_i:

DFFITSi = (yhati - yhat(i),i) / (si * sqrt(h_ii))

For a full-rank ordinary least-squares model, the two routes are algebraically equivalent when the same conventions and precision are used. The current statsmodels DFFITS documentation describes the diagnostic as single-case influence on a predicted value and returns 2 * sqrt(p / n) as an investigation heuristic. Its published source implementation multiplies the externally studentized residual by sqrt(hii / (1 - h_ii)) and obtains the deleted error scale through leave-one-out fits.

Do not silently substitute an internally studentized residual. Stop if 1 - hii is effectively zero, the deleted residual degrees of freedom are not positive, or si is zero or undefined. Keep full precision through the calculation and round only for display.

Blank fitted-value influence worksheet

CaseAuthorized keyy_iyhatih_iiDeleted MSEExternal t_iDFFITS_iContext note123...

Fit-level audit fieldValueReproduction noteIncluded cases nMatch the fixed fitParameters pInclude intercept and coded columnsFull residual degrees of freedomn - pFull-model SSE and MSEPreserve source precisionLargest absolute DFFITSDescriptive ordering onlyFormula and software versionName external versus internal conventionIndependent cross-checkCompare formula and deletion refit

Retain the worksheet with the model formula, full coefficient vector, design-matrix column order, analyst, reviewer, code or spreadsheet version, and any sensitivity-fit decision.

Fictional ten-case example

Consider a synthetic model with an intercept and two predictors. The rows use x1 = [1,2,3,4,5,6,7,8,9,10], x2 = [2,5,1,7,3,8,4,10,6,9], and y = [5,7,6,10,9,13,12,18,15,24]. These numbers are invented and have no client or treatment meaning.

The full fit is:

y_hat = 0.7405797101 + 1.3293075684x1 + 0.6996779388x2

There are n = 10 cases, p = 3 parameters, 7 residual degrees of freedom, SSE = 23.4901771337, and MSE = 3.3557395905.

Casey_iyhatiResidualh_iiDeleted MSEExternal t_iDFFITS153.46924315621.53075684380.35040257653.31383242441.04332302390.7662672953276.89758454110.10241545890.32463768123.91244105340.06300473640.0436821745365.42818035430.57181964570.35161030603.83098086290.36281566440.267176819541010.9555555556-0.95555555560.26666666673.7075098814-0.5795141841-0.3494602017599.4861513688-0.48615136880.20668276973.8653765914-0.2776209035-0.141703591061314.3138486312-1.31384863120.20668276973.5523754750-0.7826407522-0.399476421571212.8444444444-0.84444444440.26666666673.7529644269-0.5090180066-0.306949407281818.3718196457-0.37181964570.35161030603.8794928046-0.2344373512-0.172639254691516.9024154589-1.90241545890.32463768123.0218831134-1.3316746192-0.9232709548102420.33075684383.66924315620.35040257650.46075028926.70689547084.9258710239

For case 10, 6.7068954708 * sqrt(0.3504025765 / (1 - 0.3504025765)) = 4.9258710239. It is the largest absolute result in this fictional fit. That ordering is an invitation to inspect the row, context, model, and deletion fit, not permission to remove the case.

Change only the fictional case 10 response from 24 to 20 and refit. Its DFFITS becomes 1.4375951158. The difference shows that DFFITS belongs to the complete fitted dataset, not to a case in isolation.

Sign, magnitude, and the investigation heuristic

The sign tells which way the case pulls its own fitted value. Use absolute values only when ranking magnitude, and keep the sign visible for interpretation. A near-zero DFFITS can occur when the residual is small even if leverage is notable. A large residual at low leverage and a high-leverage row with a tiny residual can also yield quite different results.

Statsmodels reports 2 * sqrt(p / n) as a threshold for investigation. NIST regression diagnostics likewise presents DFFITS as fitted-value influence and describes common size-dependent rules of thumb. These are screening conventions, not hypothesis tests, data-quality verdicts, or universal ABA cutoffs. Small samples, many simultaneous case reviews, model misspecification, dependence, selection after looking at results, and software conventions all affect what a qualified reviewer may do next.

Implementation checks and failure boundaries

CheckExpected arithmetic behaviorWhat it cannot establishAdd a constant to every response and refitDFFITS values remain unchangedResponse meaning stayed constantMultiply the response by a positive constant and refitValues remain unchangedNew units are clinically interchangeableMultiply the response by a negative constantMagnitudes remain and signs reverseDirection has no clinical meaningPermute complete rows and refitValues follow their stable case keysOrder is irrelevant to every ABA analysisRefit each one-case deletionDirect differences agree with the residual-leverage formulaThe model is correctly specifiedRecode predictors with an equivalent full-rank basisFitted values and DFFITS remainCoefficients retain the same interpretationLet h_ii approach oneThe calculation becomes unstable; stopExtreme leverage can be repaired automaticallyDeleted MSE is zero or undefinedDFFITS is undefined; stopAn exact fit validates the dataDesign matrix loses rankCoefficients and deletion results may be nonunique; stopDropping a column is acceptable

For repeated or clustered observations, mixed models, generalized models, nonlinear fits, robust estimators, survey weights, or formal case testing, obtain qualified statistical review. The R core deletion-diagnostics reference documents DFFITS alongside DFBETAS, covariance ratios, Cook's distance, leverage, and studentized residuals, and notes that generalized-model results can be approximations.

The diagnostic belongs beside direct clinical evidence

Plot the observed measure across time and phase. Review measurement integrity, opportunities and denominators, implementation conditions, setting events, staffing, schedule changes, and the plausibility of the model form. Examine residual-versus-fitted and residual-versus-leverage plots. Use Cook's distance for a joint coefficient-influence view and DFBETAS when a named coefficient is the question.

DFFITS does not identify why a case is influential. Possible explanations include valid context, data entry error, an omitted predictor, nonlinearity, changed measurement, dependence, a phase transition, or an observation the model represents poorly. Investigate provenance first and preserve the unaltered fit plus any reviewed sensitivity fit.

Professional and information-governance limits

The BACB Ethics Codes provide current professional-conduct and consumer-protection requirements. The BCBA Test Content Outline includes measurement, graphing, interpretation, experimental design, competence, confidentiality, and data-based decisions. Broader evidence and intended-use principles appear in the Standards for Educational and Psychological Testing. None of these sources validates this worksheet or supplies a DFFITS clinical cutoff.

Use synthetic or appropriately de-identified records for training. When an authorized workflow needs identifiable information, minimize the fields and use approved access, retention, transmission, audit, and disposal controls. HHS publishes separate summaries of the Privacy Rule and the Security Rule. Those summaries do not decide organization-specific applicability or certify this implementation.

Related resources

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