Calibration Model Selection in Forensic Toxicology

Quick answer: Choosing between linear and quadratic calibration is a model-order decision, not a contest for the highest R². In quantitative forensic toxicology, the laboratory should show that the selected model adequately describes analytical response across the validated range using residual behavior, calibrator back-calculations, bias and precision, independent quality controls, and predefined acceptance criteria. Weighting is a separate regression decision and is addressed in the dedicated calibration-weighting guide linked below.

This page focuses specifically on calibration-model selection. For the broader role of calibrators in forensic testing, see Calibration Evidence Review.

Why calibration-model selection matters

Quantitative LC-MS/MS and GC-MS methods convert instrument response into an estimated concentration. That conversion depends on a mathematical relationship established from calibrators of known concentration. If the chosen model does not reflect how variance and response behave across the analytical range, the calculated concentration can be biased even when the chromatographic peaks look clean and the run otherwise appears acceptable.

The problem is especially important when a method spans a wide concentration range. Variance often increases with concentration, a condition called heteroscedasticity. In that setting, unweighted least-squares regression can allow high-concentration calibrators to dominate the fit and may degrade accuracy at the lower end of the range. Weighting can reduce that imbalance, but the weighting scheme itself should be supported by validation data rather than selected by habit.

Linear, quadratic, and weighted models answer different statistical problems

Linear regression

A linear model assumes that the relationship between concentration and measured response is adequately represented by a straight line over the validated range. Linear regression can be entirely appropriate when residuals are acceptably distributed and calibrator and QC performance support the model.

Quadratic regression

A quadratic model introduces curvature. It may be appropriate when the analytical response is not adequately described by a straight line across the full working range. More parameters, however, do not automatically make a model better. Added flexibility can also fit noise, so a quadratic model should be justified through validation rather than chosen simply because it produces a larger R² value.

Weighting is a separate regression decision

Model order and weighting should not be collapsed into one question. A laboratory can use a linear or quadratic model with an appropriate weighting scheme, depending on the validated variance structure. This page focuses on whether a straight-line or curved model is justified. For the separate statistical question of heteroscedasticity and 1/x versus 1/x² weighting, see Calibration Curve Weighting in Forensic Toxicology.

Why R² alone is not enough

A high coefficient of determination can coexist with poor performance at individual calibration levels. This is particularly true across broad analytical ranges, where the highest concentration points can drive an apparently excellent correlation while low-level residuals remain unacceptable.

A stronger review considers the full calibration evidence:

Back-calculated calibrator concentrations: How closely does the fitted model reproduce the known concentrations of the calibrators?
Residuals: Are errors randomly distributed, or do they show systematic curvature, concentration-dependent bias, or widening variance?
Independent quality controls: Do QCs that were not used to build the curve perform acceptably at low, middle, and high concentrations?
Bias and precision: Does the validated method remain accurate and reproducible across the intended range?
Run-specific acceptance: Did the actual casework batch satisfy the laboratory’s pre-established calibration and QC criteria?

Back-calculation is central to evaluating the curve

Back-calculation applies the fitted equation to each calibrator’s observed response and compares the calculated result with the calibrator’s assigned concentration. A model that generates an attractive R² but repeatedly miscalculates one portion of the range should not be accepted solely because the overall correlation appears strong.

Review should also identify whether calibrators were excluded. Exclusion can be scientifically justified under a validated procedure, but repeated removal of a particular level can reveal a range or model problem. The case record should allow a reviewer to determine which calibrators were initially present, which were accepted or rejected, why any were excluded, and whether the remaining curve still met the laboratory’s criteria.

Residual plots can reveal what a correlation coefficient hides

A residual is the difference between an observed or back-calculated value and the value expected under the fitted model. Residual plots are useful because they can reveal patterns that a single summary statistic conceals. Systematic curvature may suggest that a linear model is inadequate. Increasing residual spread at higher concentrations may show heteroscedasticity. Concentration-dependent positive or negative residuals may indicate that the model favors one part of the range.

Peer-reviewed toxicology work on calibration-model selection has emphasized model order, weighting, normalized variance, and residual assessment rather than reliance on correlation alone. In one published evaluation, mis-selection of the calibration model produced substantial QC error, illustrating why model selection is a validation question rather than a cosmetic curve-fitting choice.

Calibration range defines where quantitative claims are supported

A validated calibration range is not merely the set of numbers printed on a curve. It is the concentration interval over which the laboratory has demonstrated acceptable performance for the intended method. A result above the upper calibrator generally should not be converted into a precise concentration by simple extrapolation unless the laboratory has validated a procedure that supports that use.

Common approaches for specimens above the validated range include validated dilution and reanalysis or reporting the result as greater than the upper reporting limit. See Dilution Integrity for the separate question of whether dilution preserves acceptable quantitative performance.

Model selection should be analyte- and method-specific

A laboratory may use different models or weighting schemes for different analytes within the same multi-analyte method. That is not inherently problematic. Ionization behavior, detector response, analytical range, matrix effects, and variance structure can differ by compound. What matters is that the selected model for each reported analyte was evaluated and shown to be fit for purpose.

Recent forensic methods illustrate this point. Published nitazene assays have used weighted quadratic calibration for many analytes while selecting different weighting for particular compounds. Other validated blood methods have used weighted linear models after evaluation of residuals and back-calculated concentrations. The existence of multiple defensible approaches is precisely why the laboratory’s own validation data matter.

Calibration is connected to matrix effects, internal standards, and carryover

Curve behavior should not be reviewed in isolation from the rest of the method. Matrix-dependent ion suppression or enhancement can alter response, particularly when calibrators and case specimens are not appropriately matched. Stable-isotope internal standards can compensate for some variation but do not guarantee correction under every chromatographic or matrix condition. Sequence effects and carryover can also distort low-level responses.

Case-review checklist for calibration evidence

For a questioned quantitative result, useful records commonly include:

the method-validation report and the study supporting model selection;
the number and concentrations of calibrators used during validation;
the tested model orders and weighting schemes;
residual plots or other model-fit evaluations;
back-calculated calibrator results and acceptance criteria;
low-, mid-, and high-level QC results;
the calibration curve from the actual casework batch;
documentation of any excluded calibrator and the reason for exclusion;
the laboratory’s rules for results above or below the validated range;
records of dilution or reanalysis when the initial response exceeded the range.

Those records permit a reviewer to distinguish a curve that merely looks acceptable from a calibration process that was actually validated and accepted under the laboratory’s procedure.

Frequently asked questions

Does an R² of 0.99 or 0.999 prove a calibration curve is valid?

No. R² describes one aspect of fit but does not by itself establish acceptable accuracy across the range. Back-calculated calibrators, residuals, QC performance, bias, precision, and pre-established acceptance criteria are more informative when evaluating quantitative reliability.

Is 1/x² weighting always better than 1/x?

No. The appropriate weighting depends on the variance structure and performance of the specific analytical method. A laboratory should support its choice with validation data.

Can a laboratory use a quadratic calibration curve?

Yes, when validation demonstrates that a quadratic model appropriately represents the analytical response over the intended range and produces acceptable quantitative performance. The mere fact that a quadratic equation fits more flexibly is not, by itself, sufficient justification.

Can a result above the highest calibrator simply be extrapolated?

Usually the more defensible approach is to follow the laboratory’s validated procedure for above-range specimens, such as dilution and reanalysis or an above-range report. Precise extrapolation beyond the demonstrated calibration range requires specific support.

What is the most important document for reviewing calibration-model selection?

The method-validation report is usually the starting point because it should show how the model, weighting, range, and acceptance criteria were established. The actual case batch then shows whether those validated requirements were satisfied for the reported result.

Scientific foundation

AAFS Standards Board, ANSI/ASB Standard 036, Standard Practices for Method Validation in Forensic Toxicology.
Desharnais B, Camirand-Lemyre F, Mireault P, Skinner CD. Procedure for the Selection and Validation of a Calibration Model I—Description and Application. Journal of Analytical Toxicology. 2017;41(4):261-268. Journal article.
Recent forensic LC-MS/MS validation studies applying analyte-specific calibration models, residual assessment, and weighted regression include work on nitazenes, cannabinoids, and multi-class blood testing in the Journal of Analytical Toxicology.

Technical note: This page addresses analytical calibration and quantitative method review. It does not determine whether a measured drug concentration establishes dose, timing, impairment, toxicity, intent, or causation; those are separate interpretive questions.

Calibration Model Order in Forensic Toxicology: Linear vs Quadratic Regression