Range and Linearity: Where a Calibration Curve Stops Meaning Anything
A sample reads above the top standard. The software returns a value anyway, extrapolating the line. Nothing flags it, the number looks like every other number, and it is the one result in the run with no evidence behind it.
Quick answer
What is range? ICH Q2(R2) defines the range of an analytical procedure as the interval between the lowest and highest results in which the procedure has a suitable level of precision, accuracy and response. Note what that definition contains: range is not a property of the instrument or the curve. It is the interval over which the other validation characteristics were actually demonstrated.
The guideline adds a second, more practical term. The reportable range includes all values from the lowest to the highest reportable result for which there is a suitable level of precision and accuracy, typically given in the same unit as the specification acceptance criterion — and it should be inclusive of the upper and lower specification or reporting limits.
That last requirement is the one with teeth: a method whose validated range does not bracket the limit it is being judged against cannot decide whether the limit was met.
What linearity actually requires
ICH Q2(R2) asks for a linear relationship between analyte concentration and response to be evaluated across the range, demonstrated on the product, suitable reference materials, separate weighings of analyte, or predefined mixtures of components.
The specific requirements are worth listing, because most of them are routinely skipped:
| Requirement | What the guideline says |
|---|---|
| Number of levels | A minimum of five concentrations appropriately distributed across the range is recommended |
| Statistical treatment | Results evaluated by an appropriate statistical method, e.g. a regression line by least squares |
| What to report | A plot of the data, the correlation coefficient or coefficient of determination, the y-intercept and the slope of the regression line |
| Residuals | Analysis of deviation of actual data points from the line; the impact of any non-random pattern in the residuals plot should be assessed |
Why the residuals matter more than the r-squared
This is the part that gets lost. A coefficient of determination close to 1 is compatible with a curve that is systematically wrong at both ends and systematically wrong in the middle, in opposite directions. The number summarises total scatter; it does not detect shape.
A residuals plot does. If the deviations from the line form a pattern — a smile, a frown, a drift — the relationship is not linear, however high the r-squared. That is precisely why ICH Q2(R2) asks for the residuals to be assessed in addition to the correlation coefficient rather than instead of it.
If you take one practical habit from this page: look at the plot. A correlation coefficient is a one-number summary of a picture, and the picture contains information the number cannot carry.
What a validated range looks like in practice
A 2022 method developed to quantify residual HEPES buffer in gallium-68 radiopharmaceuticals reported linearity between 3 and 100 µg/mL, with a correlation coefficient for the calibration curves equal to 0.999, coefficients of variation under 2%, and a quantitation limit of 3 µg/mL. The range and the quantitation limit coincide at the bottom, which is the normal arrangement: the lower end of a reportable range is usually set by where quantitation becomes reliable.
A 2022 stability-indicating method for a three-drug combination reported ranges of 125–750 µg/mL, 1.25–7.5 µg/mL and 0.625–3.75 µg/mL for its three analytes, each with a correlation coefficient above 0.999. Each range is a six-fold span centred on the expected concentration — not an open-ended line.
The lower end: detection and quantitation limits
Where the required range approaches the bottom of what a procedure can do, ICH Q2(R2) sets out how the detection limit (DL) and quantitation limit (QL) may be estimated. The approaches include visual evaluation, signal-to-noise, and calculation from the standard deviation of the response and the slope of the calibration curve:
- Signal-to-noise: a ratio of 3:1 is generally considered acceptable for estimating the DL; for the QL, a ratio of at least 10:1 is considered acceptable.
- From the calibration curve: DL = 3.3σ/S and QL = 10σ/S, where σ is the standard deviation of the response and S the slope.
- Confirmation: an estimated QL should subsequently be validated by analysing a suitable number of samples known to be near or at the QL.
The guideline also sets a rule that explains a lot of certificate formatting: for impurity tests, the QL for the analytical procedure should be equal to or below the reporting threshold. What a limit does and does not establish is covered in what not detected actually means.
When the relationship is not a line
ICH Q2(R2) is explicit that some procedures show non-linear responses, and that this is not a defect. Immunoassays and cell-based assays may show an S-shaped response, which occurs when the concentration range is wide enough that responses are constrained by upper and lower asymptotes; four- or five-parameter logistic functions are the common models.
For those procedures, the guideline says evaluation of linearity is separate from consideration of the shape of the curve, and linearity of the concentration-response relationship is not required. What is required is that performance be evaluated across the range to obtain values proportional to the true sample values.
What this means at the bench
- Know your two numbers. The concentration below which you cannot quantify, and the concentration above which you have not calibrated. Results outside them are extrapolations.
- Dilute into the range rather than extrapolating out of it. A dilution adds a small, known uncertainty. Extrapolation adds an unknown one.
- Bracket your samples. If your expected result sits near the top standard, add one above it.
- Keep five levels, not three. It is the guideline's recommended minimum and it is what makes a residuals plot readable at all.
- Record the range with the result. A concentration without the calibration range it came from cannot be re-checked later — the same record discipline described in measuring peptide concentration.
Key takeaways
- Range is the interval over which precision, accuracy and response were demonstrated — not wherever the curve can be drawn.
- The reportable range must be inclusive of the upper and lower specification or reporting limits.
- ICH Q2(R2) recommends a minimum of five concentrations distributed across the range for linearity.
- Reporting should include the plot, the correlation coefficient or coefficient of determination, the y-intercept and the slope.
- Any non-random pattern in the residuals plot must be assessed; a high r-squared does not rule one out.
- Signal-to-noise of 3:1 is acceptable for estimating the detection limit, and at least 10:1 for the quantitation limit.
- Non-linear responses are permitted; for S-shaped assays, linearity of the concentration-response relationship is not required.
Frequently asked questions
Is an r-squared of 0.999 enough?
It is necessary but not sufficient. It summarises scatter around the fitted line without revealing whether the line is the right shape, which is why the guideline asks for the residuals to be examined as well.
Can I just dilute everything into the middle of the curve?
That is the correct instinct. The cost is the dilution's own uncertainty, which is small, well understood and recordable — all things extrapolation is not.
Why must the range bracket the specification limit?
Because a decision about whether a limit was met requires evidence at that concentration. If the method was never validated there, the result at that point has no demonstrated accuracy or precision behind it.
What if my sample reads below the quantitation limit?
Then it is reportable as below the QL, not as the number the software printed. Between the detection limit and the quantitation limit, analyte presence can be detected but not reliably quantified — that is the definition of the gap.
Does a wider range make a method better?
No. A wider range means more work to validate and usually worse performance at the extremes. Methods are built around the concentrations actually used, which is why published ranges tend to be a few-fold span rather than orders of magnitude.
Sources
- International Council for Harmonisation. ICH Harmonised Guideline Q2(R2): Validation of Analytical Procedures. Final version adopted 1 November 2023. ICH guideline
- Migliari S, Scarlattei M, Baldari G, Silva C, Ruffini L. A specific HPLC method to determine residual HEPES in [68Ga]Ga-radiopharmaceuticals: development and validation. Molecules. 2022;27(14):4477. DOI
- Vankalapati KR, Alegete P, Boodida S. Stability-indicating HPLC method development and validation for simultaneous estimation of metformin, dapagliflozin, and saxagliptin in bulk drug and pharmaceutical dosage form. Biomed Chromatogr. 2022;36(7):e5384. DOI
Journal citations retrieved via PubMed. The cited methods concern a buffer residue and small-molecule drugs; the validation principles transfer, the specific ranges belong to those analytes.
This article describes analytical methodology only. It contains no usage instructions, no dosing information and no protocols of any kind.
For laboratory and in-vitro research use only. Not for human or veterinary use.
For laboratory and in-vitro research use only. Not for human or veterinary use. See our research-use disclaimer.
