Major topics covered in this chapter
Uncertainty 99 to each, and then combines these components using the rules summarised in
5.6 Calculation of a concentration and its random error
Once the slope and intercept of the regression line have been determined, it is very simple to calculate the concentration (x-value) corresponding to any measured in-strument signal (y-value). But it will also be necessary to find the error associated
sy/x
b = y>x (x, y)
yN
122 5: Calibration methods in instrumental analysis: regression and correlation
with this concentration estimate. Calculation of the x-value from the given y-value using Eq. (5.2.1) involves the use of both the slope (b) and the intercept (a) and, as we saw in the previous section, both these values are subject to error. Moreover, the instrument signal derived from any test sample is also subject to random errors. As a result, the determination of the overall error in the corresponding concentration is extremely complex, and most workers use the following approximate formula:
(5.6.1)
In this equation, y0 is the experimental value of y from which the concentration value x0is to be determined, is the estimated standard deviation of x0, and the other symbols have their usual meanings. In some cases an analyst may make several readings to obtain the value of y0: if there are m such readings, then the equation for
becomes:
(5.6.2)
As expected, Eq. (5.6.2) becomes the same as Eq. (5.6.1) if m 1. Confidence limits can be calculated as , with (n 2) degrees of freedom. Again, a simple computer program will perform all these calculations, but most calculators will not be adequate.
x0 ; t(n - 2)sx0 sx0 =
sy>x b Q
1 m +
1 n +
(y0- y)2 b2a
i
(xi - x)2 sx0
sx0 sx0 =
sy>x b Q
1 + 1 n +
(y0 - y)2 b2a
i
(xi - x)2
Example 5.6.1
Using the data from Example 5.3.1, determine x0- and -values and x0
confidence limits for solutions with fluorescence intensities of 2.9, 13.5 and 23.0 units.
The x0-values are easily calculated by using the regression equation determined in Section 5.4, y 1.93x 1.52. Substituting the y0-values 2.9, 13.5 and 23.0, we obtain x0-values of 0.72, 6.21 and 11.13 pg ml-1respectively.
To obtain the -values corresponding to these x0-values we use Eq. (5.6.1), recalling from the preceding sections that n 7, b 1.93, sy/x 0.4329,
13.1 and . The y0-values 2.9, 13.5 and 23.0 then yield -values of 0.26, 0.24 and 0.26 respectively. The corresponding 95% confi-dence limits (t5 2.57) are 0.72 ; 0.68, 6.21 ; 0.62 and 11.13 ; 0.68 pg ml-1 respectively.
sx0
ai
(xi - x)2 = 112 y
sx0
sx0
This example illustrates an important point. Although the confidence limits for the three concentrations are similar (we expect this, as we have used an unweighted regression calculation), the limits are rather smaller (i.e. better) for the
Calculation of a concentration and its random error 123
(x, y )– –
Signal
Concentration
Figure 5.6 General form of the confidence limits for a concentration determined by using an unweighted regression line.
result y0 13.5 than for the other two y0-values. Inspection of Eq. (5.6.1) confirms that as y0approaches , the third term inside the bracket approaches zero, and thus approaches a minimum value. The general form of the confidence limits for a calculated concentration is shown in Fig. 5.6. Thus in a practice a calibration exper-iment of this type will give the most precise results when the measured instrument signal corresponds to a point close to the centroid of the regression line.
If we wish to improve (i.e. narrow) the confidence limits in this calibration experi-ment, Eqs (5.6.1) and (5.6.2) show that at least two approaches should be consid-ered. We could increase n, the number of calibration points on the regression line, and/or we could make more than one measurement of y0, using the mean value of m such measurements in the calculation of x0. The results of these approaches can be assessed by considering the three terms inside the brackets in the two equations. In the example above, the dominant term in all three calculations is the first one – unity. It follows that in this case (and many others) an improvement in precision might be made by measuring y0several times and using Eq. (5.6.2) rather than Eq.
(5.6.1). If, for example, the y0-value of 13.5 had been calculated as the mean of four determinations, then the -value and the confidence limits would have been 0.14 and 6.21 ; 0.36 respectively, both results showing much improved precision. Of course, making too many replicate measurements (assuming that sufficient sample is available) generates much more work for only a small additional benefit: the reader should verify that eight measurements of y0would produce an -value of 0.12 and confidence limits of 6.21 ; 0.30.
The effect of n, the number of calibration points, on the confidence limits of the concentration determination is more complex. This is because we also have to take into account accompanying changes in the value of t. Use of a large number of calibration samples involves the task of preparing many accurate standards for only marginally increased precision (cf. the effects of increasing m described in the previous paragraph). On the other hand, small values of n are not permissible. In such cases 1/n will be larger and the number of degrees of freedom, (n 2), will become very small, necessitating the use of very large t-values in the calculation of the confidence
sx0 sx0
sx0 y
124 5: Calibration methods in instrumental analysis: regression and correlation
limits. As in the example above, six or so calibration points will be adequate in many experiments, the analyst gaining extra precision if necessary by repeated measure-ments of y0. If considerations of cost, time or availability of standards or samples limit the total number of experiments that can be performed, i.e. if m n is fixed, then it is worth recalling that the last term in Eq. (5.6.2) is often very small, so it is crucial to minimise (1/m 1/n). This is achieved by making m n.
An entirely distinct approach to estimating uses control chart principles (see Chapter 4). We have seen that these charts can be used to monitor the quality of lab-oratory methods used repeatedly over a period of time, and this chapter has shown that a single calibration line can in principle be used for many individual analyses.
It thus seems natural to combine these two ideas, and to use control charts to mon-itor the performance of a calibration experiment, while at the same time obtaining estimates of . The procedure recommended by the ISO involves the use of q (2 or 3) standards or reference materials, which need not be (and perhaps ought not to be) from among those used to set up the calibration graph. These standards are mea-sured at regular time intervals and the calibration graph is used to estimate their analyte content in the normal way. The differences, d, between these estimated concentra-tions and the known concentraconcentra-tions of the standards are plotted on a Shewhart type control chart, the upper and lower control limits of which are given by 0 ; (tsy/x/b).
Here sy/xand b have their usual meanings as characteristics of the calibration line, while t has (n 2) degrees of freedom, or (nk 2) degrees of freedom if each of the original calibration standards was measured k times to set up the calibration graph.
For a confidence level of a (commonly a 0.05), the two-tailed value of t at the (1a/2q) level is used. If any point derived from the monitoring standard materials falls outside the control limits, the analytical process is probably out of control, and may need further examination before it can be used again. Moreover, if the values of d for the lowest concentration monitoring standard, measured J times over a period, are called dl1, dl2, . . ., dlJ, and the corresponding values for the highest monitoring stan-dards are called dq1, dq2, . . ., dqJ, then is given by:
(5.6.3)
Strictly speaking this equation estimates for the concentrations of the highest and lowest monitoring reference materials, so the estimate is a little pessimistic for concentrations between those extremes (see Fig. 5.6). As usual the -value can be converted to a confidence interval by multiplying by t, which in this case has 2J degrees of freedom.