An ABA externally studentized residual calculator rescales each ordinary regression residual using an error estimate computed with that case deleted. It can expose response-side outlyingness that the full-model residual obscures, especially when leverage is substantial. It does not establish data error, case influence, model validity, treatment effect, or a clinical rule.

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

The diagnostic asks a deleted-variance question

Ordinary, standardized, internally studentized, deleted and externally studentized residuals answer different questions. An ordinary residual is ei = yi - yhati. Internal studentization uses the full-model error scale and leverage adjustment, while external studentization uses the residual variance estimated after removing the case under review.

That leave-one-out scale gives the case less control over its own denominator. The result addresses response outlyingness relative to one documented model. It is not Cook's distance, which combines residual magnitude and leverage to summarize influence on the fitted coefficient vector.

Use the ABA externally studentized residual calculator only after the full ordinary least-squares model and its row set have been fixed. Preserve the original fit even when a deletion diagnostic prompts a sensitivity review.

Record the complete source model first

Scope fieldRequired recordStop signalResponseObservable outcome, denominator, unit, and directionDefinition or scale changedFull modelFormula, coding, intercept, weights, population, phase, and versionFit cannot be reproducedRow identityAuthorized key for each included caseMissing, duplicated, or shifted rowsResidual conventionObserved minus fitted or another named conventionSign is uncertainModel sizeIncluded rows n, parameter count p, and df_resid = n - pDeleted degrees of freedom are not positiveFull-model errorSSE and residual MSESSE is negative or undefinedLeverageHat diagonal h_ii from the same designValue is outside [0, 1]Intended useDescriptive deleted-residual reviewAutomatic deletion or clinical decision

Document missing-data handling, exclusions, observation weights, transformations, software version, and calculation precision. Record phase, setting, staffing, schedule, measurement, intervention, and denominator changes beside the rows.

Expose the leave-one-out arithmetic

For a full ordinary least-squares fit with residual sum of squares SSE and residual degrees of freedom df_resid = n - p, calculate for each case:

deletedresiduali = ei / (1 - hii)

SSE(i) = SSE - ei^2 / (1 - h_ii)

si^2 = SSE(i) / (df_resid - 1)

ti = ei / (si * sqrt(1 - hii))

Here, si^2 is the leave-one-out residual variance and ti is the externally studentized residual. The statsmodels documentation describes external studentization as using leave-one-out variance and requiring an observation loop. The R core deletion-diagnostics reference distinguishes standardized residuals from Studentized residuals by their overall versus leave-one-out error estimates.

Stop when dfresid - 1 <= 0, 1 - hii is zero or unstable, or s_i^2 <= 0. Tiny negative values caused only by documented floating-point roundoff require explicit numerical review, not silent truncation.

Copy the blank deleted-residual worksheet

RowAuthorized keyy_iyhatie_ih_iiDeleted residualSSE_(i)s_i^2External t_i123...

Model summaryValueAudit noteIncluded rows nMatch full fitParameters pInclude intercept and coded columnsFull residual degrees of freedomn - pFull-model SSEPreserve precisionFull-model MSEContext onlyLargest absolute external residualDescriptive ordering onlySoftware and versionPreserve method and cross-check

Retain the full-model formula, design matrix, coefficient vector, fitted values, analyst, reviewer, date, code or spreadsheet version, and independently reproduced results.

A fictional case shows why deletion variance matters

Use the same fictional regression with x = [1, 2, 3, 4, 5, 6, 7, 8] and y = [2, 3, 5, 4, 6, 7, 8, 14]. The fitted line is y_hat = -0.1428571429 + 1.3928571429*x. With n = 8 and p = 2, full residual degrees of freedom are 6, SSE = 17.3928571429, and full-model MSE = 2.8988095238.

Rowe_ih_iiDeleted residualSSE_(i)s_i^2External t_i10.75000000000.41666666671.285714285716.42857142863.28571428570.541736338920.35714285710.27380952380.491803278717.21721311483.44344262300.225850090430.96428571430.17857142861.173913043516.26086956523.25217391300.58997546274-1.42857142860.1309523810-1.643835616415.04452054793.0089041096-0.88343784395-0.82142857140.1309523810-0.945205479516.61643835623.3232876712-0.48335258936-1.21428571430.1785714286-1.478260869615.59782608703.1195652174-0.75855827207-1.60714285710.2738095238-2.213114754113.83606557382.7672131148-1.133725472183.00000000000.41666666675.14285714291.96428571430.39285714296.2667956144

For row 8, SSE_(8) = 17.3928571429 - 3^2/(1 - 0.4166666667) = 1.9642857143. Its leave-one-out variance is 0.3928571429, and the externally studentized residual is 6.2667956144.

If only the final fictional outcome changes from 14 to 10 and the model is refitted, row 8's external residual becomes 1.3926212476. This sensitivity illustrates dependence on the fitted data. It does not show that either fictional value should be preferred.

Run invariance and failure checks

CheckExpected arithmetic behaviorWhat the check cannot proveAdd one constant to all outcomes and refitExternal residuals stay the sameResponse meaning is unchangedMultiply outcomes by a positive constant and refitExternal residuals stay the sameUnits are clinically equivalentMultiply outcomes by a negative constant and refitMagnitudes stay; signs reverseClinical direction is irrelevantPermute complete rows and refitValues follow row identityTime order never matters elsewhereh_ii approaches oneDeleted residual and scale become unstable; stopExtreme leverage can be repaired automaticallydf_resid - 1 <= 0Deleted variance is undefined; stopMore precision creates informationSSE_(i) is zerosi^2 is zero and ti is undefined; stopPerfect deletion fit validates the modelSSE_(i) is slightly negativeInvestigate precision and formula implementationClamping is always acceptable

These are implementation checks. They do not verify measurement quality, independence, sampling, coding, model form, or clinical relevance.

Read residual extremeness beside leverage and influence

The NIST regression-diagnostics guidance explains that high-leverage cases can have small ordinary residuals and describes deleted and externally studentized residuals as tools for response-side outliers that ordinary residual review may miss. The same reference separates these quantities from Cook's distance and DFFITS.

Plot ordinary residuals against fitted values and leverage. Display the original clinical observations over time and by phase. Review Cook's distance when the question concerns whole-model coefficient influence. Review coefficient-specific deletion diagnostics when that narrower question matters. A large absolute external residual alone cannot identify cause or justify removal.

Potential explanations include valid context differences, changed measurement, a denominator shift, data entry error, omitted nonlinear structure, dependence, a phase transition, or a response that the model does not represent well. Investigate provenance before interpretation.

Do not convert a reference value into an automatic rule

Formal reference distributions depend on the model assumptions, prespecified testing plan, degrees of freedom, and multiplicity across cases. This worksheet does not calculate p-values or an automatic flag. A familiar cutoff is not a universal ABA threshold and does not resolve whether a row is correct, clinically meaningful, or ethically removable.

Seek qualified methods review for formal outlier testing, repeated or clustered observations, generalized or mixed models, weights, robust methods, nonlinear fits, small samples, multiplicity, or any deletion proposal. Preserve exploratory checks as exploratory.

Clinical and data-governance review remains separate

The BACB Ethics Codes address professional conduct, competence, documentation, confidentiality, and client protection. The BCBA Test Content Outline places measurement, visual analysis, interpretation, and experimental design within professional preparation. Broader evidence and intended-use principles appear in the Standards for Educational and Psychological Testing. None validates this calculator or supplies a clinical cutoff.

Use synthetic or appropriately de-identified data for training. If an authorized workflow requires identifiable information, minimize fields and apply approved access, storage, retention, and version controls. HHS maintains separate Privacy Rule and Security Rule summaries. They do not determine applicability or certify a workflow.

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