An ABA Poisson rate ratio confidence interval calculator compares two event counts after accounting for their valid exposure times. This worksheet uses the exact conditional construction: condition on the combined count, calculate a Clopper-Pearson interval for the resulting binomial probability, and transform those limits to a rate ratio. It preserves both exposure clocks and all boundary decisions. The result is meaningful only when independent homogeneous Poisson models are plausible; it is not a prediction, safety threshold, or treatment-effect test.

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

A ratio needs two rates with compatible units

Suppose Record 1 contains 6 events during 3 valid hours and Record 2 contains 4 events during 5 valid hours. The rates are 2.0 per hour and 0.8 per hour, so the observed Record 1 to Record 2 rate ratio is 2.5. The comparison is multiplicative: Record 1's observed rate is two and one-half times Record 2's observed rate.

That sentence is incomplete without the event and exposure definitions. A clock that includes transition, break, sleep, unavailable materials, or time outside observation may estimate a different rate from a clock that excludes it. Counts may also change when episode boundaries, offset rules, or scorer instructions differ. This calculator cannot repair those differences.

The narrow statistical question is: assuming two independent Poisson counts with stable rates over known positive exposures, what exact conditional interval follows for lambda1 / lambda2? Deciding whether the model describes the observation record requires clinical and methods review.

Condition on the total count

Let k1 and k2 be nonnegative integer event counts, and let T1 and T2 be positive exposures in the same unit. Calculate:

rate1 = k1 / T1

rate2 = k2 / T2

R-hat = rate1 / rate2

For the conditional interval, set N = k1 + k2. When N > 0, the conditioned first-record count follows a binomial distribution with:

p = lambda1 T1 / (lambda1 T1 + lambda2 T2)

Obtain a two-sided exact Clopper-Pearson interval [pL, pU] at the prespecified confidence level. For confidence 1 - alpha:

pL = 0 when k1 = 0; otherwise BetaQuantile(alpha / 2; k1, k2 + 1)

pU = 1 when k2 = 0; otherwise BetaQuantile(1 - alpha / 2; k1 + 1, k2)

Transform the limits back to the rate-ratio scale:

RL = (pL / (1 - pL)) x (T2 / T1)

RU = (pU / (1 - pU)) x (T2 / T1)

The official R exact Poisson test documentation describes the two-sample conversion to a binomial test by conditioning on the total event count and relates the rate ratio to the binomial odds. The original Price and Bonett methods paper compares classical and Bayesian intervals for two independent Poisson rates. This worksheet names the exact conditional approach instead of presenting it as the only defensible method.

Record the model and clock before calculating

Complete the header before opening software or a spreadsheet:

FieldPrespecified entryComparison ID, record versions, and calculator versionClient-selected or client-informed questionDirection, written as Record 1 rate divided by Record 2 rateEvent definition and episode boundary ruleRecord 1 context, observer, window, and exposure clockRecord 2 context, observer, window, and exposure clockExcluded time and pause rules for each clockDisplay unit, such as events per valid hourWhy the two counts are modeled as independentWhy a stable Poisson rate is plausible within each recordConfidence level, exact method, and software versionWhether the sensitivity view is observed or hypotheticalClinical and methods reviewers

Then keep the arithmetic trail:

CalculationRecord 1Record 2Ratio resultEvent count kValid exposure TPoint rate k / TR-hat =Conditioned total NConditioned proportion k1 / NBeta shape values for lower limitBeta shape values for upper limitExact pL and pUExposure multiplier T2 / T1Full-precision RL and RUDisplay rule and displayed intervalBoundary or model concern

Software should be tested against known examples and must identify the interval method. The word exact identifies the discrete conditional construction, not perfect knowledge of a client's event process. Its achieved coverage can exceed the nominal confidence level, which is why the interval can be conservative.

Fictional calculation: 6 events in 3 hours versus 4 in 5

This example uses invented records. Record 1 has k1 = 6, T1 = 3 hours. Record 2 has k2 = 4, T2 = 5 hours.

rate1 = 6 / 3 = 2 events per hour

rate2 = 4 / 5 = 0.8 events per hour

R-hat = 2 / 0.8 = 2.5

The conditioned total is N = 10, and k1 / N = 0.6. At 95 percent confidence, the exact binomial limits are:

pL = BetaQuantile(0.025; 6, 5) = 0.2623780766

pU = BetaQuantile(0.975; 7, 4) = 0.8784477419

The exposure multiplier is T2 / T1 = 5 / 3 = 1.6666666667. Transforming gives:

RL = (0.2623780766 / 0.7376219234) x 1.6666666667 = 0.5928467965

RU = (0.8784477419 / 0.1215522581) x 1.6666666667 = 12.0448570224

With a prespecified two-decimal display, retain rate ratio 2.50; 95% exact conditional interval 0.59 to 12.04. The broad interval reflects little event-count information under this method. It neither proves equality because 1 is included nor establishes a clinically important increase from the point estimate.

Exposure sensitivity changes the estimand

Keep the same fictional counts but change Record 2's valid exposure from 5 hours to 3 hours. This is a hypothetical clock-definition view, not a rewrite of the observed record. Record 2's point rate becomes 4/3, and the ratio becomes 1.5. Because the conditioned counts are unchanged, pL and pU remain the same, while the exposure multiplier changes to 1.

The transformed interval is then 0.3557080779 to 7.2269142134. The difference from the primary result shows why an exposure rule belongs in the saved calculation. It does not show that either clock is correct or that excluded time can be chosen after seeing a preferred ratio.

Use an ABA Poisson rate ratio confidence interval calculator only after both clocks are defined. If the records use different valid-exposure meanings, report them as different estimands rather than interchangeable estimates.

Boundary cases need words as well as symbols

When k1 = 0, k2 = 4, T1 = 3, and T2 = 5, the point ratio is 0. The 95 percent interval is 0 to 2.5247780989. Zero observed events do not prove a zero underlying rate.

When k1 = 6, k2 = 0 with the same exposures, the point ratio and upper limit are infinite. The lower limit is 1.9623745429. Record infinity explicitly; do not replace it with zero, a very large arbitrary number, or a continuity correction that belongs to another method.

If k1 = 0 and k2 = 0, then N = 0. The conditional construction has no event allocation to condition on, and the rate ratio is undefined. Stop rather than displaying 1, 0, or a false finite interval.

As a check, swap the records. A finite original interval [L, U] should become [1/U, 1/L], with the point ratio replaced by its reciprocal. Boundary infinities and zeros swap accordingly.

The Poisson assumptions are substantive

The model treats each count as arising from a process with a stable rate over its exposure and suitable independence, with count variance tied to its mean. In practice, events may cluster, show refractory periods, occur only after specific antecedents, depend on schedules or support, or vary across people and settings. CDC's technical note on rate confidence intervals explicitly notes that population heterogeneity can create extra-Poisson variation and that intervals do not address systematic bias.

A methods paper on Poisson-rate tests describes the conditional binomial distribution and the conservativeness of an exact conditional test. A separate single-case methods review discusses serial dependence in repeated behavioral data. These sources help identify model limits; they do not establish that any particular ABA event stream is Poisson or independent.

Leave the output blank when exposure is not positive, a count is negative or noninteger, event definitions differ, clocks are not commensurate, or the confidence level was selected after comparing outputs. Seek qualified statistical review for overdispersion, time-varying rates, dependence, repeated measures, offset models, covariates, or more than two records.

Do not substitute the ratio for single-case analysis

A pooled ratio hides the order of events and the shape of each time series. Preserve event timestamps, direct graphs, contextual notes, integrity evidence, and phase annotations. The WWC single-case technical documentation describes visual analysis of level, trend, variability, overlap, immediacy, and consistency, with repeated demonstrations in its causal framework. One rate-ratio interval does not meet that standard.

The BCBA Test Content Outline includes operational definitions, measurement, reliability, representative data, graphing, and data-based decision making. The BACB ethics-code page provides access to the current ethics code. The Standards for Educational and Psychological Testing frame validity, reliability, fairness, and intended score use. None supplies a universal ABA rate-ratio cutoff.

Use coded identifiers and approved systems for records that include protected health information. Federal overviews of the HIPAA Privacy Rule and Security Rule are separate starting points. Consent, access, retention, safeguards, accessibility, payer, legal, regulator, and organizational controls remain local responsibilities.

Related resources

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