An ABA difference in proportions confidence interval calculator estimates an absolute difference between two binary opportunity proportions when the records can reasonably be modeled as independent binomial samples. This worksheet calculates each Wilson interval and combines the directional distances with Newcombe's square-and-add method. It also keeps the response definitions, denominators, sample labels, and independence rationale beside the result. The interval is a model-based description of uncertainty, not a mastery rule or evidence that one condition caused the difference.

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

Start with the comparison, not the subtraction

A signed difference needs a declared direction. If Sample 1 has 8 qualifying responses in 10 valid opportunities and Sample 2 has 6 in 10, the observed difference is 0.80 - 0.60 = +0.20, or 20 percentage points in the Sample 1 direction. Reversing the labels must produce -0.20 and reverse the interval limits.

The arithmetic is simple only after the two samples have comparable meanings. A clinician should be able to state what counts as a response, what creates a valid opportunity, how prompts and refusals are handled, and why the records can be treated as independent. Two adjacent phases from one client's time series usually do not become independent merely because they have different labels. Opportunities within a session may also depend on earlier trials, support, partner behavior, fatigue, or access.

This calculator therefore answers a narrow question: under two independent-binomial models, what Newcombe-Wilson interval follows for p1 - p2? It does not decide whether independence is credible or whether the difference matters clinically.

Compute two Wilson intervals first

For sample j, let xj be qualifying responses, nj be valid opportunities, and pj = xj / nj. Choose a two-sided confidence level before reviewing alternative results. At 95 percent, use z = 1.9599639845.

For each sample, calculate:

Dj = 1 + z^2 / nj

centerj = (pj + z^2 / (2nj)) / Dj

half-widthj = z x sqrt(pj(1 - pj) / nj + z^2 / (4nj^2)) / Dj

Lj = centerj - half-widthj

Uj = centerj + half-widthj

The NIST handbook's Wilson formulation supplies these single-proportion limits and explains why a simple Wald interval can produce impossible values. Wilson limits stay on the proportion scale without a manual repair.

Now set d = p1 - p2. Newcombe's independent-sample combination is:

lower = d - sqrt((p1 - L1)^2 + (U2 - p2)^2)

upper = d + sqrt((U1 - p1)^2 + (p2 - L2)^2)

If a numerical implementation drifts outside [-1, 1] because of floating-point error, cap the displayed bound at the parameter boundary and retain the uncapped working value. The original Newcombe comparison of eleven methods reports that the approach combining two Wilson intervals performed well and remained readily implementable. A later methods paper on proportion intervals reviews the poor coverage of familiar Wald calculations and places Newcombe's comparison in the broader method literature. Neither paper validates an ABA observation design.

Freeze the record before entering counts

Complete this ABA difference in proportions confidence interval calculator header before calculating:

FieldPrespecified recordComparison ID, data version, and calculator versionClient-selected or client-informed purposeSigned contrast, written as Sample 1 minus Sample 2Observable response definition shared by both samplesValid-opportunity rule shared by both samplesSample 1 window, setting, support, partner, and scorerSample 2 window, setting, support, partner, and scorerPrompt, refusal, unavailable, exclusion, and retry rulesWhy the two samples are treated as independentConfidence level and critical-value sourceWhether any comparison view is observed or hypotheticalReviewer and calculation date

Retain the intermediate values rather than saving only the final band:

CalculationSample 1Sample 2ContrastQualifying responses xValid opportunities nObserved proportion pp1 - p2 =Wilson denominator DWilson centerWilson half-widthWilson lower LWilson upper UNewcombe lower distanceNewcombe upper distanceFull-precision intervalDisplay rule and displayed intervalIndependence or comparability concern

Use a new record when either response definition, denominator rule, support condition, or sample direction changes. Do not overwrite the original comparison to make the interval appear narrower.

Fictional example: 8 of 10 versus 6 of 10

All values here are invented. Sample 1 records 8/10, so p1 = 0.8; its 95 percent Wilson limits are L1 = 0.4901624715 and U1 = 0.9433178485. Sample 2 records 6/10, so p2 = 0.6; its Wilson limits are L2 = 0.3126737697 and U2 = 0.8318196703.

The signed point difference is:

d = 0.8 - 0.6 = 0.2

The lower directional distance is:

sqrt((0.8 - 0.4901624715)^2 + (0.8318196703 - 0.6)^2) = 0.3869620829

The upper directional distance is:

sqrt((0.9433178485 - 0.8)^2 + (0.6 - 0.3126737697)^2) = 0.3210862319

The resulting interval is -0.1869620829 to 0.5210862319. With a prespecified one-decimal percentage-point display, record difference +20.0 points; 95% Newcombe-Wilson interval -18.7 to +52.1 points.

The wide interval does not erase the observed 20-point difference. The fact that it spans zero also does not prove equality for a particular person. It shows how much sampling uncertainty the procedure assigns under its independent-binomial assumptions.

Denominator sensitivity must use a separate record

For a larger fictional view, suppose separately observed samples under the same written definitions contain 40/50 and 30/50. The observed difference is again +0.20. The Wilson limits are 0.6696289407 to 0.8875624998 for Sample 1 and 0.4618143775 to 0.7239161027 for Sample 2. The Newcombe interval is 0.0201339009 to 0.3635923521.

That narrower interval answers a mathematical sensitivity question. It does not license multiplying the original counts by five, collecting repetitive trials merely to enlarge denominators, or treating clustered opportunities as new independent evidence. Label the view as separately observed or hypothetical and explain which conditions differ.

An equal-proportion check is also useful. With 8/10 in both samples, d = 0, and the interval is symmetric at -0.3413785286 to 0.3413785286. At the opposite boundary, 10/10 versus 0/10 yields d = 1 and a boundary-limited interval of 0.6075093504 to 1. A result at an endpoint still carries uncertainty.

Reverse labels as an arithmetic control

Repeat the calculation with the samples swapped. The original lower and upper limits should become the negatives of the reversed upper and lower limits. For the first fictional example, the reversal must read -0.5210862319 to 0.1869620829 around d = -0.20.

If reciprocity fails, inspect sample order, Wilson distances, copied counts, rounding, and sign labels. Perform all computations at full precision and round only the display. This control detects a common implementation mistake; it does not test whether the samples are clinically comparable.

Stop when independence or denominators are not defensible

Leave the interval blank when either denominator is zero, a count is not an integer from zero through its denominator, or the confidence level was chosen after trying several outputs. Stop as well when the response definition changed, opportunities were recoded inconsistently, invalid access was counted as failure, or unlike programs were pooled only to increase n.

Do not use this independent-sample calculator as a paired analysis. Repeated measurements on the same client, matched opportunities, alternating observations, and phase data can carry within-unit dependence that this formula does not model. The Standards for Educational and Psychological Testing provide broad guidance about validity, reliability, fairness, and intended interpretations. They do not declare a local comparison independent or clinically meaningful.

The BCBA Test Content Outline addresses operational definitions, measurement, reliability, graphing, and data-based decisions. The BACB ethics-code page directs certificants to the current code. These materials support careful practice, not a universal statistical threshold.

Keep the graph and clinical story beside the interval

An interval around a pooled difference does not show level, trend, variability, overlap, immediacy, or consistency over time. The WWC single-case technical documentation treats those features and repeated demonstrations as part of its visual-analysis framework for causal inference. This calculator supplies none of those demonstrations.

Preserve the ordered session data, opportunity rows, exclusions, integrity observations, setting changes, client and caregiver interpretation, calculation version, and reviewer comments. A band that excludes zero is not by itself proof of a treatment effect, generalization, maintenance, social validity, or clinical importance. A band that includes zero is not proof of no meaningful change.

When a record contains protected health information, use coded identifiers and approved systems. HHS maintains separate summaries of the federal Privacy Rule and Security Rule. Local consent, access, retention, safeguards, accessibility, payer, regulator, legal, and organizational requirements still need qualified review.

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