Method Validation Edition

Linearity evaluation software for method validation Polynomial regression with nonlinearity testing — weighted models, adjustable measuring intervals and formal equivalence testing to determine the reportable range, alongside the Emancipator-Kroll nonlinearity measure. Covers the EP06 study and other linearity designs.

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Microsoft Excel with the Analyse-it tab selected, showing the Linearity section of an EP06-A Appendix C IgM report: the difference plot of the nonlinear fit minus the linear fit at each of the five dilutions, in percent, with 95% confidence intervals and the plus or minus 5% allowable nonlinearity band, and beneath it the table of linear and 2nd-order polynomial fitted values, the nonlinearity at each dilution with its 95% confidence interval and the allowable nonlinearity, four of the five dilutions flagged as not meeting the performance requirement, with the task pane open. Handwritten notes: Runs inside Excel: every analysis is on the Analyse-it tab; Nonlinearity at each dilution with its 95% CI, against the allowable nonlinearity; The linear and polynomial fits, and the difference at each dilution; Allowable nonlinearity: absolute or relative; The report is an ordinary Excel worksheet: share it, archive it, open it on any PC with Excel.
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EP06-Ed2

Determine where the assay is linear — and where it is not

I used Analyse-it for many product development, product troubleshooting, and technology evaluation activities... your product was the easiest to use, was accurate, and produced publication ready reports.
Stanley F. Cernosek, Ph.D.
Clinical Chemistry Reagent Development
Beckman Coulter, Inc.

Excel’s chart trendline fits a polynomial, reports an R² and stops. No test of whether the curvature is real, no allowable limit to judge it against, no confidence interval on the deviation at each level, no reportable range. Patient results outside the linear range of a measurement procedure cannot be reported directly — they need dilution, concentration or a qualifier. Define that range too widely and unreliable results are reported; too narrowly and the reportable interval is restricted for no reason. EP06-Ed2 provides the statistical framework. Fit polynomial models, quantify the deviation from linearity and test whether it falls within limits that are clinically acceptable.

Is the deviation from linearity real, or within the noise of the assay? Does it matter — is it inside the allowable limit at every dilution, or only at some? And where the assay is not linear across the whole range, how far must the measuring interval be narrowed before it is?

Linear, polynomial (2nd to 5th order), stepwise, best and weighted fits

The linear fit is the claim; the polynomial fits are the check. Fit linear through 5th-order polynomials, choose the model by forward stepwise selection or by hand, or take the best of the 2nd- and 3rd-order fits. The linearity plot overlays the linear and polynomial fits on the mean at each dilution, so you can see where they diverge. Where precision is not constant across the measuring interval, fit a weighted model.

  • Linear fit
  • Polynomial fit (2nd to 5th order)
  • Forward stepwise polynomial fit
  • Best (2nd or 3rd) polynomial fit
  • Weighted fits: by the variance at each level, the pooled variance or a variance function
  • Linearity plot with linear and polynomial fits
Microsoft Excel showing the Fit Model section of an EP06-A Appendix C calcium linearity report: the linear fit, the 2nd-order polynomial fit and the 3rd-order polynomial fit, each with its RMSE and a table of parameter estimates with standard errors, and t, degrees of freedom and p-value for each nonlinear term, with the footnote that the nonlinear parameter is different from zero at the 5% significance level, and the task pane open. Handwritten notes: Linear fit: RMSE and the parameter estimates; Polynomial fits, with a t-test on each nonlinear term; Fit: best polynomial, stepwise or by order.
Calcium, EP06-A Appendix C: the linear, 2nd-order and 3rd-order polynomial fits, each with its RMSE and parameter estimates, and the t-test on every nonlinear term.

Nonlinearity testing: Hsieh-Liu intervals, equality and equivalence tests

A significant nonlinear term says the curvature is real; it does not say whether it matters. The difference between the linear and the best-fitting nonlinear model is reported at each dilution with a Hsieh-Liu confidence interval. Test equality (is there significant nonlinearity?) and equivalence (is the nonlinearity within an allowable limit, such as ±5%?). The difference plot with the allowable nonlinearity band shows where the deviations occur and by how much.

  • Difference between linear and nonlinear fit
  • Hsieh-Liu confidence intervals
  • Equality test (no nonlinearity)
  • Equivalence test (nonlinearity within allowable limit)
  • Difference plot with allowable nonlinearity band
Microsoft Excel showing the Linearity section of an EP06-A Appendix C calcium report across the full measuring interval: the difference plot of the nonlinear fit minus the linear fit at each of the six dilutions, in mg/dL, with 95% confidence intervals and the plus or minus 0.2 mg/dL allowable nonlinearity band, and the table of linear and 3rd-order polynomial fitted values, the nonlinearity at each dilution with its 95% confidence interval and the allowable nonlinearity, five of the six dilutions flagged as not meeting the performance requirement, with the task pane open. Handwritten notes: Nonlinearity at each dilution with its 95% CI, against the allowable nonlinearity band; The same in a table, with dilutions outside the allowance flagged; Allowable nonlinearity.
Calcium across the full 4.65 to 16.20 mg/dL interval: the nonlinearity at each dilution with its 95% CI against the ±0.2 mg/dL allowable limit. Five of the six dilutions are flagged.

Adjustable measuring interval: the linear range and the reportable range

An assay that fails linearity across the whole range is often linear across most of it. Restrict the upper or lower end of the measuring interval and recalculate. The fits, the nonlinearity and the difference plot are re-run on the dilutions inside it. Narrow the interval until every deviation is inside the allowable limit, and the result is the analytical measuring interval — for product labelling or your laboratory’s reporting limits.

  • Adjustable measuring interval to find linear range
  • Assigned values: known, or relative by mixture, addition or dilution series
Microsoft Excel showing an EP06-A Appendix C calcium linearity report with the measuring interval reduced: the Measuring interval 4.65 to 16.20 and Reduced interval 4.65 to 15.40 rows, the precision table of mean, SD and CV at each dilution, then the Linearity section with the difference plot of the nonlinear fit minus the linear fit at the five remaining dilutions with 95% confidence intervals inside the plus or minus 0.2 mg/dL allowable band, and the table beneath with no dilution flagged, with the task pane open. Handwritten notes: Measuring interval, and the reduced interval it was cut to; Nonlinearity at each dilution with its 95% CI, against the allowable nonlinearity; Measuring interval limits, typed in.
Calcium with the measuring interval reduced to 4.65 to 15.40 mg/dL: the top dilution excluded, and every nonlinearity now inside the ±0.2 mg/dL allowable limit.

Emancipator-Kroll linearity

Emancipator and Kroll’s nonlinearity measure is a separate procedure, published independently of the EP06 polynomial approach. The measure is available alongside the polynomial approach rather than as a fallback from it. Use it where a study protocol or an established laboratory procedure specifies it.

  • Emancipator-Kroll linearity

Example analyses

See linearity evaluation results in detail — polynomial fits, difference plots and nonlinearity testing — using CLSI example datasets you can download and follow along with.

EP06 A Example 1 3 pages EP06-A — Appendix C
IgM linearity. 5 dilutions × 2 observations. Linear, second and third order polynomial fits, then nonlinearity at each dilution with 95% CIs against a ±5% allowable band, and a difference plot. Only dilution 4 meets the requirement.
EP06 A Example 2 6 pages EP06-A — Appendix C
Calcium linearity, full and reduced interval. 6 dilutions × 2 observations. Over the full interval five of the six dilutions exceed the ±0.2 mg/dL allowable nonlinearity, so a second analysis refits over a reduced interval that passes throughout. Linear and polynomial fits with nonlinearity at each dilution.

Part of measurement system analysis

Linearity is one part of measurement system analysis, alongside precision (EP05-A3), bias/trueness verification (EP15-A3), preliminary evaluation (EP10-A3-AMD) and detection capability (EP17-A2).

Related guides in the Learn section: assessing linearity, LoB, LoD and LoQ, the analytical measuring interval and reportable range, and CAP accreditation and AMR verification.

Software you can trust

Validated calculations you can defend at inspection Every calculation is performed by Analyse-it — no Excel formulas, no third-party functions. Results are validated against CLSI reference datasets, published datasets and thousands of internal test cases before every release. How Analyse-it is developed and validated →
Data stays in your facility Analyse-it runs entirely within Microsoft Excel on your PC. No cloud processing, no data transmission. Pre-submission data, data derived from patients and in-process results stay within your facility under your own data governance controls.
Standard Excel workbooks anyone can open Every analysis is an ordinary .xlsx workbook. Share with colleagues, submit to regulatory affairs, archive for audit, open on any PC with Excel. No proprietary format, no licence required to view results. Colleagues and auditors see exactly what you see.
Results that cannot be accidentally broken Analysis output contains computed values, not formulas. Nothing to accidentally overwrite, no cell references to break, no formula errors to introduce. The results you reported are exactly what you will find when you reopen the workbook months or years later for an audit.

Free trial and pricing

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Method Validation edition: US$ 475 per year or US$ 1155 for a perpetual licence. Every purchase carries a 30-day money-back guarantee. Need a quote for purchasing? Add the licence to the cart and save it as a PDF quote.