An ABA Deming regression calculator fits a straight method-comparison line when both paired continuous variables may contain measurement error. Its answer depends on the direction of the fit and a positive, prespecified error-variance ratio. Keep the paired data, ratio evidence, scatterplot, residuals, and difference view beside the slope and intercept.

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

Confirm that an errors-in-variables line matches the question

Deming's 1936 paper addresses least-squares estimation when both coordinates have error. The official NIST Deming Fit documentation describes Deming regression as a paired-measurement errors-in-variables model and makes the assumed error-variance ratio explicit. CLSI EP09 covers quantitative measurement-procedure comparison, including Deming techniques, experiment design, plots, bias estimates and uncertainty. These sources do not make the method an ABA standard or approve a clinical decision.

Use this worksheet only for complete paired continuous measurements on the same independent units, expressed on a common scale, when a straight-line comparison is defensible and measurement error may affect both axes. A statistician or measurement specialist should review whether unweighted Deming assumptions fit the design.

Ordinary least squares treats X as fixed or measured without relevant error in the standard setup. Deming regression assigns error to both coordinates through a variance ratio. Lin CCC summarizes paired concordance, and Bland-Altman analysis displays differences. None of these answers can substitute for the others.

Professional duties still govern whether this analysis is competent and useful. The current BACB Ethics Codes and BCBA Test Content Outline supply relevant context, but neither mandates Deming regression. The Standards for Educational and Psychological Testing emphasize linking each interpretation to its evidence and intended use.

Define lambda before fitting

This worksheet defines:

lambda = error variance in Y / error variance in X

Here, error variance means variance in the measurement error around the unobserved value, not variance of the observed X or Y scores. That orientation must be written down because software and references can define a ratio in the reciprocal direction. A value of lambda=1 says the two measurement-error variances are assumed equal. It does not say the observed variances of X and Y are equal.

Use replicate-measurement or other defensible precision evidence to set lambda. Do not estimate it from the same unreplicated comparison pairs merely to obtain a preferred slope. If no defensible ratio is available, stop or present a prospectively chosen range as a labeled sensitivity analysis rather than disguising one value as known.

Register the method-comparison plan

Design fieldPrespecified entryObservable constructIndependent unit of analysisCommon measurement unitX method and roleY method and rolePairing key and timing windowInclusion and exclusion rulesMissing-pair ruleStop; complete pairs required.Lambda definitionError variance Y / error variance X.Lambda value and evidenceReplicate or precision evidence versionSensitivity values chosen before fittingData version and extraction timeIntended consequenceQualified reviewers

Preserve this record with the result. Swapping axes, changing lambda, excluding observations, or transforming a scale changes the analysis.

Enter one row per independent unit

Preserve acquisition order in the row sequence.

Pairing keyX measurementY measurementUnit/version noteUnit 1Unit 2Unit 3...

Stop for unmatched pairs, mixed units, duplicate units, qualitative categories, or a time window that allows the underlying construct to change. Inspect a scatterplot and a Y-minus-X difference plot before relying on a straight line. Curvature, changing spread, clusters, time drift, ceiling effects, or isolated influential points can invalidate a simple fit.

Calculate the centered sums

For n pairs, calculate the means xbar and ybar, then use unrounded values:

SSx = sumi (xi-xbar)^2.

SSy = sumi (yi-ybar)^2.

SSxy = sumi (xi-xbar)(yi-ybar).

The common divisor used for variances and covariance would cancel from the slope formula. This worksheet keeps the centered sums so the arithmetic is easy to audit.

Calculate the Deming line

With this worksheet's lambda orientation, calculate:

beta = [SSy - lambda SSx + sqrt((SSy-lambda SSx)^2 + 4 lambda SSxy^2)] / (2 SSxy)

alpha = ybar - beta xbar

The fitted line is Yhat = alpha + beta X. Stop if lambda<=0, SSxy=0, or the result is nonfinite. Keep full precision through every step and round only displayed values. Record the exact lambda convention with the calculation rather than relying on a software default.

For each unit, preserve Yhati and the vertical raw residual ei=Yi-Yhati. These vertical residuals are a convenient diagnostic; they are not the error components minimized by Deming regression and should not be interpreted as ordinary least-squares residuals. A qualified reviewer may need orthogonal or ratio-scaled residuals, weighted fitting, replicate models or bootstrap uncertainty beyond this worksheet.

Pairing keyXYFitted Y_hatVertical diagnostic Y-Y_hatReview note

Use additional rows as needed.

Work a fictional equal-error example

Five fabricated paired values use X=[1,2,3,4,5] and Y=[1,2,4,4,7]. No person, assessment or clinical record is represented. Set lambda=1 prospectively for the arithmetic demonstration.

The means are xbar=3 and ybar=3.6. The centered sums are SSx=10, SSy=21.2, and SS_xy=14.

beta = [21.2-10 + sqrt((21.2-10)^2 + 4(14^2))] / 28

beta = 1.4770329614

alpha = 3.6 - 1.4770329614(3) = -0.8310988843

The fictional fitted line is Y_hat=-0.8310988843 + 1.4770329614X. Do not convert its slope and intercept into a pass/fail conclusion.

Run only a prespecified lambda sensitivity

Using the same fictional pairs:

The table records the prespecified range.

LambdaDeming slopeIntercept0.51.4922151764-0.876645529211.4770329614-0.831098884321.4577199400-0.7731598200

Keep every row in the final record.

For comparison, ordinary least squares of Y on X gives slope SSxy/SSx=1.4 and intercept -0.6. The difference is not evidence that Deming is automatically better. It shows that the error model changes the fitted line.

Do not widen, narrow or relocate the sensitivity range after seeing which result looks favorable. Preserve the rationale for every value and report the full table.

Check axis orientation and boundaries

Swapping X and Y also requires replacing lambda with 1/lambda. Recalculate from the swapped centered sums; for a valid finite nonzero slope, verify beta(X on Y) x beta(Y on X) = 1 within numerical tolerance. Merely swapping the column labels while retaining lambda does not perform this check and can conceal an inverted ratio.

Exact collinear pairs on Y=a+bX, with a finite nonzero slope and at least two distinct X values, recover that line for any positive lambda. If all X values are identical, all Y values are identical, or SS_xy=0, this simple worksheet stops rather than forcing a slope. Near-zero covariance can produce unstable slopes and demands review.

Interpret slope, intercept, and residual structure together

Use the ABA Deming regression calculator as an auditable method-comparison record. Report the axis roles, common unit, pair count, lambda definition and evidence, centered sums, slope, intercept, fitted values, residual review, sensitivity table, exclusions, and intended consequence.

Slope one and intercept zero are reference values, not proof of interchangeability. A line near those values can coexist with wide differences, clinically important residuals, range restriction or poor precision. A line away from them does not identify which method is correct. Review the scatterplot, Y-minus-X differences, fitted-value residual pattern, clinical decision points and measurement process.

This point-estimate worksheet provides no confidence intervals or hypothesis tests. It does not validate a method, establish treatment response, demonstrate experimental control, prove causation or authorize replacing one measurement process with another.

Stop when a different model is needed

Pause for changing error variance across the range, nonlinear structure, repeated or clustered units, replicate-specific error modeling, outliers requiring a prespecified robust process, or censored values. Unequal units, weights, more than two methods, and uncertain pair identity also exceed this worksheet. Obtain statistical or metrology review for weighted Deming, confidence intervals, bootstrap procedures, Passing-Bablok or other robust regression, sample-size planning and decisions at clinical thresholds.

Preserve raw values and plots. Do not repair data, remove inconvenient points, swap axes, redefine lambda or choose a method after comparing which result appears most favorable.

Protect paired measurements and provenance

Limit the worksheet to information required for review. Identifiable unit-level measurements belong only in approved systems with controlled access and an audit trail. Federal duties for regulated entities are summarized in the HHS Privacy Rule and HHS Security Rule overviews. Additional state and organizational requirements may apply. This page is not legal advice.

Copyable result record

Result fieldValueData versionRecord the preserved version.X and Y methods/rolesCommon unit and pairing keyState both.Complete pairs nLambda definition, value and evidenceName the ratio orientation and evidence.SSx, SSy, SS_xyDeming slope betaIntercept alphaKeep full precision.Prespecified sensitivity resultsAxis-swap checkRecord the numerical product.Scatter/difference/residual reviewMissing or excluded entriesExplain any change.Intended interpretation and limitsReviewer and review dateAdd both.

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