The F Test
3.2 Methodology, Terms and Definitions
3.2.5 Analytical Figures of Merit
In order to describe the performance qualities of an analytical method, some meas- urable quantities are used; these are called analytical figures of merit. These num- bers reveal the capabilities that can be reached by using an analytical method, such as the lowest concentration or amount measurable and the concentration values for which the method is applicable.
3.2.5.1 Detection Limit and Limit of Quantitation
One of the most important analytical figures of merit is detection limit (DL) or limit of detection (LOD). DL may be defined in terms of the analytical signal or concen- tration (or absolute amount) corresponding to this analytical signal. The conversion from signal to concentration units is simply done by using the calibration plot, which relates these two quantities. In either case, DL is the smallest quantity, which is statistically different than 0, the absence of the signal or analyte. Certainly, most analytical signals are meaningful only when compared to a blank value.Therefore, very often the DL refers to the smallest quantity, which is statistically different from the blank value. It should be remembered that several approaches are used to define and compute DL; therefore, the procedure used to calculate it should be clearly defined. It is not possible or practical to carry out meaningful analytical measurements at DL, because this value is inherently imprecise and involves an uncertainty of about 50% RSD. Useful measurements start at about 10 times the DL, a value that is commonly termed as quantitation limit (QL) or limit of quanti- tation (LOQ).
The following definitions are often used:
DL3s⫽3s(1/m) (3.4)
where sis the standard deviation of blank measurements and mthe slope of the cal- ibration plot.
When blank concentration value,Cblank, is significantly large, this definition may have the following form:
DL3s⫽3s(1/m)⫹Cblank (3.5)
In this case, the concentration value of blank is a limiting factor for DL.
It must be stressed that particular analytical techniques may have different defini- tions of DL. Therefore, it is best to mention the approach and the formula used while reporting a DL, as well as all the conditions present in its determination.
3.2.5.2 Analytical Range
Analytical range is the part of the calibration plot, which is used for analysis, in order to obtain a required performance. For example, if the required precision is 2%
RSD or better, a part of the non-linear portion of a calibration plot may also be included. On the other hand, the calibration plot may be totally curved and still may fulfil the requirements. Alternatively, a linear calibration plot may totally fail in pro- viding the required precision, resulting in no usable range. Some of these cases are shown in Figure 3.4. Rapid and accurate computation facilities are present in most of the analytical instruments. However, even in our times, there seems to be a men- tal fixation for the absolute necessity of linearity in calibration.
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Figure 3.4 Examples for analytical range and %RSD. Range for %RSD⫽2 or better (a) is given for a partly linear calibration plot; (b) does not exist for a linear calibra- tion plot
The analytical rangestarts at about LOQ and will end at the high concentration extreme where the required precision is not met anymore. The analytical range, some- times termed as linear rangeor dynamic range, should be as large as possible; usually it is defined in terms of orders of magnitude. An analytical range of three orders of magnitude implies the coverage of 100and 103units (or 101and 104,…) for analyte quantity or concentration in the same calibration plot. A large range has a significant practical value, especially when autosamplers are used. The samples having analyte concentrations lower than LOQ should be analysed by using another analytical approach or preconcentration techniques must be employed to bring the analyte con- centration up into the analytical range. On the other hand, a large variety of concen- trations can be handled in a large range. In cases where the analytical calibration has a rather small range, the samples with analyte concentrations higher than the upper limit will need to be rehandled and diluted; this will significantly reduce the effectiveness of using an autosampler. Recently, some commercial autosamplers used in atomic spec- trometry are capable of properly diluting the sample until the analyte concentration falls in the range. The factor for this dilution is then used in computations.
Regarding the different concentrations each analyte will exhibit, the time saved by a large range will be more significant if a multi-element system is being used.
3.2.5.3 Sensitivity
Sensitivityis the degree of ability of a method to differentiate between two concentra- tions or amounts of analyte. When applied to comparative analytical methods, theslope of the calibration curve is used as the best measure of sensitivity, and is termed as cal- ibration sensitivity. One should be reminded that for a linear calibration, the slope and thus the sensitivity is constant, where for a curved calibration, calibration sensitivity will be a function of analyte concentration. In other words, the precision is also very effec- tive on the ability of differentiating between two close concentration values. Two cali- bration lines may have the same slope value, but if the one of them has better precision for both standard and sample measurements, the sensitivity will be superior as com- pared to the other line, because having a small standard deviation, closer analyte con- centrations will be statistically different than each other when a test such as Student’s t is applied (see Chapter 2). Some of the calibration slope values may depend on elec- tronic amplification or attenuation of signal; this further complicates the definition of sensitivity in a universally comprehensible manner. The difficulties and confusions caused by both the electronic alterations of signal and precision problems could be solved by using another definition,analytical sensitivity, which is obtained by dividing the calibration slope value at any point by the standard deviation of analytical signal at that point. During any simple electronic alteration of signal such as amplification or attenuation, both signal and noise are subjected to the same change and therefore sig- nal/noise (S/N) ratio remains constant. Signal refers to the instrumental response to ana- lyte concentration where noise is the standard deviation of blank value (see Section 3.2.5.4). There are electronic hardware and software approaches to improve S/Nratio, but these subjects are beyond the scope of this chapter and can be found elsewhere.
Calibration sensitivity⫽m (3.6)
where mis the slope of the calibration line
Analytical sensitivity,γ⫽m/s (3.7) where sis the standard deviation on the calibration line where the slope is measured.
By using analytical sensitivity values, laboratories and analysts at distant locations can have a more universal definition for their results.
Sometimes other definitions for the term sensitivity are employed. For example, in atomic absorption spectrometry, it has been customary to use “the concentration or amount which will cause a signal of 1% absorption (0.0044 absorbance)” as sen- sitivity; this is sometimes called as reciprocal sensitivitysince it is inversely propor- tional to the slope of calibration line.
3.2.5.4 Signal to Noise Ratio
Signal, S, is the net response by an instrument induced ideally by the presence of analyte. Noise, N, is the standard deviation of many randomly distributed signals induced in the absence or presence of analyte. Sometimes, peak-to-peak noise is defined as shown in Figure 3.5. A ratio of these two quantities in a measurement is known as signal to noise (S/N) ratio. A high S/Nis always desired. In this case, the uncertainty in Swill be smaller and detection of smaller quantities can be detected.
3.2.5.5 Relations between Precision, Sensitivity, DL and S/N
Precision is most commonly expressed as RSD. Therefore, precision, analytical sensi- tivity, DL and S/Nare all functions of s, standard deviation, for a defined set of meas- urements. If for Nmeasurements, the standard deviation is sand the mean is x¯, then
RSD⫽s/x¯ (3.8)
Since for a set of measurements S⫽x¯, and N⫽s, then
S/N⫽x¯/s⫽1/RSD (3.9)
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Figure 3.5 Practical definition of signal (S), noise (N⫽s) and S/N
On the other hand, analytical sensitivity is
γ⫽m/s, i.e. Equation(3.7) If m⫽∆signal/∆concentration, for unit concentration,
m⫽Sand γ⫽m/sor γ⫽S/N (3.10) It must also be remembered that DL3s⫽3s(1/m) or proportional to s/mor 1/γ.
Therefore, we can write the following relation:
S/N⫽1/RSD ∝ γ ∝1/DL (3.11)
The relation above emphasises the fact that the precision of a signal is very impor- tant. The requirements of high S/N, low RSD, high γ(sensitivity) and low DL all depend on a small standard deviation, the target of a high-quality signal and thus the result.