Difference between linearity and sensitivity
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Sensitivity' is a parameter that - as far as I'm aware - is out of date. It no longer appears in analytical chemical or pharmaceutical method validation protocols.
It used to be very important, and was equivalent to the slope of a calibration curve. Thus, in a long and detailed chapter on the classical analytical (beam) balance, Kolthoff & Sandell* define sensitivity as 'the amount of deflection of the beam or pointer produced by a small excess of weight on one of the pans'. They went on to state that If the balance is designed to be too sensitive, the response becomes 'irregular', presaging the use of the signal-to-noise ratio. In practice, the sensitivity of a balance usually varies as a function of total load.
Another example, from physics and electronics, is the galvanometer or pointer meter that measures current.The most sensitive workshop moving-coil meter movement gave full-scale deflection for 50µA; the sensitivity was given either as that current, or (when used for voltage readings) as '20000 ohms per volt full scale', reflecting the amount of perturbation a voltmeter introduced when testing high impedence circuitry. When sensitivity wasn't an issue, one used 1mA FSD, in the interests of robustness, stability and accuracy.
Nowadays, particularly with digital read-outs or recordings, we should think in terms of the accuracy and repeatability of the response function, which may or may not be linear. The noise level is treated as a separate parameter. In general, noise, accuracy and repeatability are functions of the quantity we are measuring, a trap for the unwary if the signal is linearised electronically or in software, as in spectrophotometers.
To conclude, 'sensitivity' is no longer considered a useful parameter. Ideally, we should report noise, accuracy and repeatability at several points covering the stated working range. Commonly, we determine several sets of parameters, for example short-term, between days, long term, between instruments (same and different models) and between labs.
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That's enough for the moment about sensitivity. To return to the question, the correct approach is to use inverse regression analysis, as presented by Miller and updated by Tellinghuisen. If you do a full method validation, you may be able to use single point calibration; I've made some remarks about the use of replicate points on RG.
Two other points: 1) serial dilutions don't give independent calibration points if the weighing and making up of the stock solution are subject to significant random variation (you may underestimate the method uncertainty), 2) I'm not convinced the usual assumption of normal distributions is valid; when weighing a powder, some may fall on the balance pan or the outside of the weighing vessel, but (apart from loss of adsorbed water) it's hard to think how the recorded weight can be too low.
*Kolthoff, I. M.; Sandell, E. B. (1951). Textbook of quantitative inorganic analysis. Macmillan.
Miller, J. N. (1991). Basic statistical methods for Analytical Chemistry. Part 2. Calibration and regression methods. A review. The Analyst, 116(1), 3. doi:10.1039/an9911600003
Joel Tellinghuisen, Simple algorithms for nonlinear calibration by the classical and standard additions methods. Analyst, 2005,130, 370-378 DOI: 10.1039/B411054D
Lee, ResearchGate (2013) Calibration uncertainty in pharmaceutical analysis: replicate single-point calibration revisited
It used to be very important, and was equivalent to the slope of a calibration curve. Thus, in a long and detailed chapter on the classical analytical (beam) balance, Kolthoff & Sandell* define sensitivity as 'the amount of deflection of the beam or pointer produced by a small excess of weight on one of the pans'. They went on to state that If the balance is designed to be too sensitive, the response becomes 'irregular', presaging the use of the signal-to-noise ratio. In practice, the sensitivity of a balance usually varies as a function of total load.
Another example, from physics and electronics, is the galvanometer or pointer meter that measures current.The most sensitive workshop moving-coil meter movement gave full-scale deflection for 50µA; the sensitivity was given either as that current, or (when used for voltage readings) as '20000 ohms per volt full scale', reflecting the amount of perturbation a voltmeter introduced when testing high impedence circuitry. When sensitivity wasn't an issue, one used 1mA FSD, in the interests of robustness, stability and accuracy.
Nowadays, particularly with digital read-outs or recordings, we should think in terms of the accuracy and repeatability of the response function, which may or may not be linear. The noise level is treated as a separate parameter. In general, noise, accuracy and repeatability are functions of the quantity we are measuring, a trap for the unwary if the signal is linearised electronically or in software, as in spectrophotometers.
To conclude, 'sensitivity' is no longer considered a useful parameter. Ideally, we should report noise, accuracy and repeatability at several points covering the stated working range. Commonly, we determine several sets of parameters, for example short-term, between days, long term, between instruments (same and different models) and between labs.
- - -
That's enough for the moment about sensitivity. To return to the question, the correct approach is to use inverse regression analysis, as presented by Miller and updated by Tellinghuisen. If you do a full method validation, you may be able to use single point calibration; I've made some remarks about the use of replicate points on RG.
Two other points: 1) serial dilutions don't give independent calibration points if the weighing and making up of the stock solution are subject to significant random variation (you may underestimate the method uncertainty), 2) I'm not convinced the usual assumption of normal distributions is valid; when weighing a powder, some may fall on the balance pan or the outside of the weighing vessel, but (apart from loss of adsorbed water) it's hard to think how the recorded weight can be too low.
*Kolthoff, I. M.; Sandell, E. B. (1951). Textbook of quantitative inorganic analysis. Macmillan.
Miller, J. N. (1991). Basic statistical methods for Analytical Chemistry. Part 2. Calibration and regression methods. A review. The Analyst, 116(1), 3. doi:10.1039/an9911600003
Joel Tellinghuisen, Simple algorithms for nonlinear calibration by the classical and standard additions methods. Analyst, 2005,130, 370-378 DOI: 10.1039/B411054D
Lee, ResearchGate (2013) Calibration uncertainty in pharmaceutical analysis: replicate single-point calibration revisited
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