An ABA Lin concordance correlation calculator asks whether two continuous measurement methods place paired values near the 45-degree line of exact equality. It combines linear precision with a correction for location and scale bias. Preserve every pair, its unit and the visual pattern because the coefficient cannot diagnose the source of disagreement.

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

Use Lin CCC for the paired continuous question

Lin's original 1989 article introduced a concordance correlation coefficient for reproducibility; the publisher DOI identifies the Biometrics paper. Lin's 2000 note is included for review of the author's later clarification. This page implements only the two-method point estimate, not its inferential extensions.

Use the worksheet for at least three complete independent pairs measured on the same continuous scale, where each row represents one prespecified unit under Method X and Method Y. A fictional exercise could compare two duration-extraction procedures on the same training records, with seconds used on both sides. The pairing key, timing window, unit, resolution and method roles must be fixed before calculation.

Do not use this version for nominal labels, ranks, unmatched samples, repeated sessions nested within people, more than two methods or a reference-standard accuracy study. Converting categories to arbitrary numbers does not make them continuous measurements.

The BACB Ethics Codes and current BCBA Test Content Outline supply professional context for competent measurement and data interpretation. They do not require Lin CCC or validate a threshold. The Standards for Educational and Psychological Testing support evaluating interpretations for a stated use and consequence.

Freeze the pairing and scale plan

Record the design before inspecting the answer.

Design fieldPrespecified entryObservable constructIndependent unit and pairing keyMethod X roleMethod Y roleShared unit of measurementTiming or observation windowResolution and rounding ruleMissing-pair ruleStop; this calculator requires complete pairsExclusions decided before analysisData version and extraction timePrimary statisticLin concordance correlation coefficientPlanned visual reviewsScatter plot and difference-versus-mean plotIntended decision and consequenceQualified reviewers

Do not switch method labels, units, transformations, pairing rules or exclusions after viewing CCC. Preserve the registered version even when the answer is inconvenient; a revised question needs a separately justified analysis.

Enter the complete pairs

Keep one row per unit. Retain more precision than the final display.

Pair IDMethod XMethod YDifference Y-XMean (X+Y)/2Included?Note12...

Confirm that both readings refer to the same unit, use the same scale and are finite numbers. Stop if either value is missing. Do not silently remove a pair because it widens disagreement.

Calculate population-form moments

For N pairs, calculate the method means:

xbar = sumi x_i / N

ybar = sumi y_i / N

Use the population-form variances and covariance for this implementation:

vx = sumi (xi - xbar)^2 / N

vy = sumi (yi - ybar)^2 / N

sxy = sumi [(xi - xbar)(yi - ybar)] / N

State the convention in the result record. Apply the same N divisor to both variances and the covariance. Mixing an N-1 variance with an N covariance changes the calculation and can break the decomposition.

Calculate Lin CCC directly

Build the denominator:

D = vx + vy + (xbar - ybar)^2

Then calculate:

rhoc = 2sxy / D

If D=0, CCC is undefined. This occurs when both methods return the same constant for every pair. Report the boundary rather than manufacturing agreement from an indeterminate ratio.

Separate precision from bias correction

When vx>0 and vy>0, calculate Pearson correlation as the precision component:

rho = sxy / sqrt(vx v_y)

Calculate the bias-correction factor:

Cb = 2sqrt(vx vy) / [vx + vy + (xbar - y_bar)^2]

The direct result should satisfy:

rhoc = rho x Cb

Pearson correlation describes how tightly pairs follow a straight line. It can remain one when that line is not the 45-degree equality line. C_b exposes the loss from unequal means or scales, but neither component identifies the operational reason for the bias.

Work the fictional exact-agreement case

Use x=[10,12,14,16,18] and y=[10,12,14,16,18]. These are fabricated unitless values.

Both means are 14. Both population variances are 8, and covariance is 8. Therefore:

D = 8 + 8 + 0 = 16

rho = 8/sqrt(8 x 8) = 1.0000000000

C_b = 16/16 = 1.0000000000

rho_c = 16/16 = 1.0000000000

This fixture checks the equality line. Real-world interpretation still depends on scale validity, sampling and consequences.

Expose a constant location shift

Keep x=[10,12,14,16,18] and use y=[11,13,15,17,19]. Every Y value is one unit higher. The means are 14 and 15; both variances and covariance remain 8.

Pearson correlation is still 1.0000000000, while:

D = 8 + 8 + (14-15)^2 = 17

C_b = 16/17 = 0.9411764706

rho_c = 16/17 = 0.9411764706

Perfect linear correlation did not equal perfect concordance. The paired table and difference plot make the systematic one-unit shift visible.

Expose scale plus location bias

Now use y=[12,15,18,21,24]. The Y mean is 18, its population variance is 18, and covariance with X is 12. Pearson again equals 1.0000000000 because the relation is exactly linear.

D = 8 + 18 + (14-18)^2 = 42

C_b = 24/42 = 0.5714285714

rho_c = 24/42 = 0.5714285714

The lower concordance reflects a line that differs from equality in both scale and location. Do not recalibrate a method from this toy result; calibration requires its own prespecified evidence and review.

Handle zero-variance boundaries explicitly

If both methods return the same constant, D=0 and CCC is undefined. If one method is constant and the other varies, the direct CCC formula has a positive denominator and returns a value, but Pearson rho and the rho x C_b decomposition are unavailable because one variance is zero.

Preserve both facts. Do not replace an unavailable precision component with zero. A constant method may signal a measurement, extraction, range or sampling problem that needs investigation before any operational decision.

Review plots and units before interpretation

Pair CCC with a scatter plot containing the equality line and a difference-versus-mean plot. Inspect nonlinear structure, changing spread, extreme pairs, restricted range, heaping and unit mismatches. The related Bland-Altman worksheet focuses directly on differences and limits of agreement; ICC depends on a selected model and intended reliability interpretation.

Treat the ABA Lin concordance correlation calculator as one documented review layer. Report N, pairing rules, units, method roles, means, variances, covariance, Pearson precision when defined, C_b, CCC, excluded or missing pairs and the visual findings. Do not apply automatic labels such as weak, moderate or excellent.

Lin CCC does not prove interchangeability, establish accuracy against a reference standard, validate either method, demonstrate clinical improvement or determine staff competence. Two datasets can share CCC while displaying different difference patterns. Practical acceptability depends on prespecified consequences and disagreement in the original units.

Stop for designs that need a fuller model

Obtain statistical or psychometric review for repeated or clustered units, more than two methods, nonlinear relationships, heteroscedastic errors, transformations, confidence intervals, hypothesis tests or sample-size planning. Pause for unmatched pairs, ambiguous pairing or a changed measurement scale.

Do not use CCC as a substitute for direct graphs, source observations, measurement-system evaluation, clinical importance, competence, supervision, client and caregiver interpretation or organizational approval. Select the analysis before viewing which output appears most favorable.

Protect paired data and provenance

Use only the minimum information needed, work inside approved systems, restrict access and preserve an audit trail. HHS explains federal privacy obligations on its Privacy Rule page and addresses safeguards on its Security Rule page. Applicable state, contractual and organizational obligations may be stricter. This is not legal advice.

Copyable result record

Result fieldValueData versionPairing key and independent unitMethod X / Method YUnit and resolutionN complete pairsxbar / ybarvx / vys_xyPearson rho or boundaryBias correction C_b or boundaryLin CCC rho_c or boundaryScatter and difference plots reviewedMissing or excluded pairsInterpretation and limitationsReviewer and review date

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