ABA Improvement Rate Difference Calculator

An ABA improvement rate difference calculator can turn two phases into a compact number, but the number is only useful when the retained observations, removals, ties, phase lengths, and method variant stay visible. This worksheet calculates robust improvement rate difference from a least-removal strict-separation search. It displays every tied optimal solution and reruns one disputed observation as a sensitivity check.

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

What this calculator computes

Robust improvement rate difference, or robust IRD, starts with the percentage of all nonoverlapping data, commonly abbreviated PAND. When higher values are preferred, ask how many observations can remain while every retained Phase B value stays strictly greater than every retained Phase A value. If lower values are preferred, every retained Phase B value must instead be strictly lower than every retained Phase A value.

The search removes as few observations as possible. More than one removal pattern can reach the same maximum retained count. When that happens, every tied optimal solution belongs in the record. A tool that silently picks one boundary hides real information about where overlap occurred.

This page uses the robust IRD definition documented in the current SingleCaseES IRD reference and SingleCaseES mathematical definitions. The robust transformation depends on the original phase lengths as well as PAND. The result is therefore tied to this phase pair, this direction, this handling of observations, and this named method.

Robust IRD is a supplemental description. The coefficient cannot establish a functional relation, clinical importance, generality, social validity, or an intervention decision. Keep the complete chronological graph, design, replication, measurement review, treatment integrity, client priorities, stakeholder input, and qualified judgment open beside the worksheet.

Lock the question and input state first

Complete the setup before calculating a result. Record:

the outcome and its operational definition;
the unit, opportunity base, observation duration, and recording procedure;
the ordered Phase A and Phase B observations with dates or sequence positions;
whether higher or lower values are preferred, with a client-centered rationale;
the phase labels, condition change, and adjacent contrast under review;
valid, missing, invalid, corrected, late, and disputed observation states;
setting, staffing, health, access, treatment, and measurement changes;
the planned metric, strict tie rule, sensitivity rule, and rounding precision;
the graph version, analyst, reviewer, calculation date, and next review date;
client and caregiver input, assent or dissent when applicable, and unresolved questions.

Missing data remain missing. A missing observation is not a zero, and an invalid observation should not enter the phase length. If the team corrects a value or phase boundary, retain both versions and document why the active record changed.

Use the same valid-observation definition in the numerator search and the phase lengths. The original m and n in each displayed analysis must reconcile with the raw rows shown for that analysis.

Copyable ordered input table

PhaseOrderDate or sequenceRaw valueUnit and denominatorStatusContext or correction noteA1valid / missing / invalid / disputedA2A3A4B1B2B3B4

Direction selected: higher values preferred / lower values preferred

Reason the direction matters to the client: ____________________________

Leave the directional result uncalculated when the preferred direction has not been established before inspection of the answer.

Find the least-removal separation solution

For upward improvement, sort a working copy of Phase A from low to high and Phase B from low to high. Preserve the original chronological rows elsewhere. Consider boundaries that retain the lowest k Phase A values and the highest l Phase B values. A candidate is strictly separated only when the largest retained A value is less than the smallest retained B value.

When downward movement is preferred, reverse the logic: retain the highest Phase A values and the lowest Phase B values, then require the smallest retained A value to be greater than the largest retained B value.

For each valid boundary:

list the exact retained A and B values;
list the exact removed A and B values;
calculate retained total = k + l;
calculate removed total = m + n - k - l;
keep every solution with the maximum retained total.

Strict means strict. Equal values across the boundary are not separated. A tied value may require removal from A, B, or both depending on the candidate boundary. Duplicate values within the same retained phase are allowed because the separation rule concerns the boundary between phases.

The display should preserve at least one observation from each phase. If no interpretable two-phase separation record can be constructed under the locked rules, return review required and keep PAND and robust IRD uncalculated.

Calculate PAND and robust IRD

Let:

m = valid Phase A observations in the active analysis;
n = valid Phase B observations in the active analysis;
R = largest retained total across all strict-separation solutions.

Then:

PAND = R / (m + n)

The phase-length factor is:

F = (m + n)^2 / (2mn)

The robust transformation is:

robust IRD = 1 - F x (1 - PAND)

Keep PAND, m, n, R, F, and robust IRD together. Identical PAND values from different phase lengths can transform differently. A rounded standalone coefficient cannot reproduce the calculation.

The current mathematical documentation notes that robust IRD has a design-dependent feasible range, lacks a stable parameter definition, and does not have a described sampling distribution. This ABA improvement rate difference calculator therefore produces no p-value, standard error, confidence interval, or universal magnitude band. It also does not silently replace robust IRD with the original improvement rate difference introduced by Parker, Vannest, and Brown.

Blank separation and result ledger

CandidateRetained A valuesRetained B valuesRemoved A valuesRemoved B valuesStrict boundary passes?Retained total1yes / no23

OutputValueReconciliation noteValid Phase A observations, mMatches active raw rowsValid Phase B observations, nMatches active raw rowsMaximum retained total, RAll tied optima listedMinimum removalsm + n - RPANDR / (m + n)Phase-length factor, F(m + n)^2 / (2mn)Robust IRD1 - F x (1 - PAND)Display precisionFull precision retained in file

If the result changes after a correction, append a new row rather than overwriting the original calculation.

Casey's fictional worked example

Casey is fictional, and every value below is synthetic. The outcome is a count per matched observation with higher values prespecified as preferred. Phase A is 20, 20, 26, 25, 22, 23; Phase B is 28, 25, 24, 27, 30, 30, 29. Thus m = 6, n = 7, and there are 13 observations in the active analysis.

The sorted working copies are:

A: 20, 20, 22, 23, 25, 26
B: 24, 25, 27, 28, 29, 30, 30

Two solutions retain 11 observations, and no solution retains 12 or 13:

Optimal solutionRetained ARetained BRemovedRetained total120, 20, 22, 2324, 25, 27, 28, 29, 30, 30A values 25, 2611220, 20, 22, 23, 25, 2627, 28, 29, 30, 30B values 24, 2511

Both satisfy the strict increasing boundary. The first ends retained A at 23 and starts retained B at 24. The second ends retained A at 26 and starts retained B at 27. Showing both solutions makes the overlapping boundary values visible.

PAND = 11 / 13 = 0.8461538462

F = 13^2 / (2 x 6 x 7) = 169 / 84 = 2.0119047619

robust IRD = 1 - (169 / 84) x (1 - 11 / 13) = 0.6904761905

The retained calculation file keeps full precision. A presentation may show 0.6905 if its rounding rule is declared.

One-point sensitivity record

The team has a documented reason to dispute the Phase A value 26. The declared sensitivity excludes that value only. No search was conducted for a deletion that improves the answer.

The active sensitivity state has five A observations and seven B observations. A least-removal solution retains 11 of the 12 observations after removing A value 25 from the separation display.

PAND_sensitivity = 11 / 12 = 0.9166666667

F_sensitivity = 12^2 / (2 x 5 x 7) = 144 / 70 = 2.0571428571

robust IRD_sensitivity = 1 - (144 / 70) x (1 - 11 / 12) = 0.8285714286

Analysis statemnMaximum retainedPANDPhase-length factorRobust IRDLocked primary67110.84615384622.01190476190.6904761905Exclude disputed A=2657110.91666666672.05714285710.8285714286

That spread is a stability signal for review. A higher sensitivity result is neither more correct nor clinically preferable. The record should explain why the value was disputed, who reviewed that reason, and which state remains primary.

Interpret the result with phase length and metric choice visible

Robust IRD compresses overlap into retained and removed counts. It does not preserve distance between observations or chronology within phases. A value just across a boundary and a value far across it play the same separation role. Abrupt change, gradual drift, delayed response, and cyclic variation can produce similar summaries.

Short phases make each removal influential. Unequal phase lengths change the transformation factor. Ties near the separation boundary may create several equally minimal solutions. Missingness and disputed membership can move both PAND and the phase-length factor.

Reviews of single-case quantitative methods place IRD among multiple supplemental measures rather than an interchangeable universal score. See the review of single-case effect-size methods, the 25-year review of single-case effect-size reporting, and the comparative study of quantitative outcomes. Results can differ by metric. Prespecify the measure for the question, retain unfavorable or ambiguous analyses, and explain any side-by-side sensitivity rather than selecting the most favorable number.

Graph, design, and client meaning remain in the decision record

Version 5.0 of the What Works Clearinghouse Procedures and Standards Handbook centers single-case review on systematic visual analysis and repeated demonstrations. Examine level, trend, variability, immediacy, overlap, consistency, phase-change timing, measurement quality, treatment integrity, and plausible alternative explanations. One adjacent contrast cannot carry the design conclusion by itself.

Review whether the outcome and preferred direction matter to the person. Ask about benefits, burdens, adverse effects, feasibility, dignity, access, and requested changes through accessible communication. Preserve ordinary communication, mobility, sensory, and decision supports during review. The BACB Ethics Codes page is the current professional source for certificants; it does not approve this worksheet or convert a coefficient into an ethical or clinical conclusion.

Qualified professionals retain responsibility for interpretation and action within their competence, licensure, supervision, consent, organizational policy, payer requirements, and applicable law. Record disagreements and uncertainty instead of forcing consensus into the numeric field.

Privacy, security, and version control

Use synthetic or de-identified values for teaching and testing. If identifiable information is necessary for an authorized clinical workflow, use the least identifying data needed, approved systems, appropriate access controls, and retained correction history. Current HHS materials summarize the HIPAA Privacy Rule and HIPAA Security Rule. Neither summary certifies a worksheet, determines applicability, or replaces an organization's risk analysis.

Close each version with the raw-data reference, graph version, direction, formula name, all tied solutions, primary and sensitivity results, analyst, reviewer, client input, decision owner, unresolved issues, and next review date.

ABA Robust Improvement Rate Difference Sensitivity Calculator for Clinicians