Selectivity, sensitivity, detection limits and method validation

Validation asks whether the procedure suits the intended purpose

Method validation evaluates performance against a defined intended use, including relevant analytes, matrices, concentration interval and uncertainty or decision requirements. Verification checks that a specified procedure can be performed adequately under the actual laboratory conditions. An established method still needs evidence that it works in the setting where it is used.

Selectivity, precision, trueness, range, lower limits, robustness and uncertainty may matter differently for different objectives. Validation of the analytical stage alone may leave sampling and preparation questions unresolved. State what was assessed, under which conditions and with what acceptance criteria; a single attractive number cannot establish overall fitness.

A steeper response need not detect a smaller concentration

Sensitivity of a measuring system is change in indication per corresponding change in quantity; in a linear response model it is the slope. Detection performance also depends on noise, background, selectivity and the chosen decision probabilities. Sensitivity should not be used as a synonym for detection limit.

Two invented systems use the same response unit U and concentration basis. A has slope 2.00 U L mg⁻¹ and response standard deviation 0.100 U; B has slope 4.00 U L mg⁻¹ and response SD 0.400 U. Under a fixed linear additive-noise model, concentration-equivalent noise σ_C=σ_R/m is 0.0500 and 0.100 mg L⁻¹. B has the steeper slope but twice the concentration-equivalent noise.

This calculation is not complete detection-limit validation. Different response units cannot be compared by raw slope alone. Changing variance, interferents and uncertainty of the calibration parameters can require a different model.

A decision threshold and a detection concentration are different quantities

For an optional deeper original teaching model, one indication R at true concentration C follows a normal distribution with mean b+mC and known constant SD σ. Choose b=1.00 U, m=2.00 U L mg⁻¹ and σ=0.100 U. Calibration values and noise are treated as known; these are invented inputs, not method data.

Choose false-presence probability α=0.0500 and false-absence probability β=0.0500. The one-sided standard-normal 95th percentile is z=1.64485362695. A decision rule declares a positive indication when R exceeds R_crit=b+zσ≈1.16448536 U. A blank with C=0 has 5% probability of exceeding it.

The concentration corresponding to that threshold by inverse calibration is C_crit=(R_crit−b)/m≈0.0822426813 mg L⁻¹. At that true concentration, the mean indication equals the threshold, so only 50% of indications exceed it. It is not the true concentration giving 95% detection.

For the latter level under this model, C_D=(z_(1−α)+z_(1−β))σ/m≈0.164485363 mg L⁻¹. At C_D, 95% of indications exceed R_crit and 5% do not. R_crit is an observed-indication cutoff; C_D is a true-concentration performance level. Neither labels an unobserved sample as exactly zero.

The plotted probability is Φ((b+mC−R_crit)/σ), where Φ is the standard-normal cumulative distribution function. Estimated noise or calibration, changing variance, multiple observations and matrix interference require appropriate new treatment. The probability of a positive signal given a blank is also different from the probability that an observed positive sample is a blank.

A chosen lower quantification level needs confirmation

In the same simple response model, choose a teaching concentration C_10=10σ/m=0.500 mg L⁻¹. Its net response above blank is 1.00 U, ten times the assigned response SD. The response-noise-only relative SD is 10%.

That result explains one commonly used scale, but it does not establish an actual quantification limit or acceptable total uncertainty. Required precision, trueness, selectivity and matrix behavior depend on the intended use. Lower-limit estimates need confirmation with appropriate evidence rather than automatic acceptance of a multiplier.

A second response can supply information only if its model is valid

Consider two invented exact response equations with no intercept or noise: R₁=2C_A+C_I and R₂=C_A+3C_I, with concentrations numerically in mg L⁻¹ and response coefficients in U L mg⁻¹. A is the intended analyte and I an interferent. At C_A=2.00 and C_I=1.00 mg L⁻¹, both indications are 5.00 U.

The coefficient matrix has determinant 2×3−1×1=5. It is invertible: C_A=(3R₁−R₂)/5=2.00 and C_I=(2R₂−R₁)/5=1.00 on the stated numerical basis. If only R₁ were used while falsely setting C_I=0, the result would be C_A=2.50 mg L⁻¹.

Two channels do not automatically prove selectivity, orthogonality or identity. The actual coefficients, interfering forms, conditioning and noise need validation. This nonsingular exact toy illustrates added information, without claiming experimental performance.

A performance claim should retain its tested boundaries

Report the entities, matrix, interval, conditions and procedure for each validation result. Robustness examines the effect of specified variations, while reproducibility comparisons state what locations, operators or systems changed. Neither term means every conceivable condition was tested.

If the target, preparation or matrix changes beyond that scope, assess whether further validation is needed. Fitness is a supported relationship between a procedure and its intended use, rather than a permanent label granted by one successful calibration.

Worked examples

Try it yourself

1. A new fixed linear toy has slope 5.00 U L mg⁻¹ and response SD 0.100 U. Find concentration-equivalent noise and the chosen ten-noise concentration.

   Show the answer and reasoning

   0.0200 and 0.200 mg L⁻¹.

   Divide noise by slope, then multiply that concentration scale by ten.

2. In the known-normal model, keep slope and α/β fixed but double σ to 0.200 U. Find C_D.

   Show the answer and reasoning

   Approximately 0.328970725 mg L⁻¹.

   C_D is proportional to the specified constant noise SD; the decision threshold also changes.

3. Does an indication failing the chosen positive rule prove zero analyte?

   Show the answer and reasoning

   No.

   The model permits false absence and below-threshold indications from positive true concentrations.

4. Does merely adding a second channel prove experimental selectivity?

   Show the answer and reasoning

   No.

   The coefficients, identities, interference scope, noise and model conditioning need evidence.

What to carry forward

• Validate for a stated purpose and scope.
• Separate response slope from concentration noise.
• Distinguish observed cutoff and true detection level.
• A chosen quantification multiplier needs confirmation.
• Multiple responses need valid models and selectivity evidence.