## Training and consultancy for testing laboratories. ### Linear regression for calibration – Part 2

The equations (7) and (8) stated in the previous Part 1 article revealed an important message.  That is when the measured value of y is closer to the mean value of y, the confidence limit approaches to a minimum value. Hence, in practice, a calibration experiment 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.

This implies that we should always aim to prepare a calibration curve with a range that can effectively measure the analyte of sample solution nearer to its regression line central for better precision.   This can be illustrated in the following worked example.

A series of standard aqueous solutions of fluorescein was used to calibrate a fluorescence spectrometer. The following fluorescence intensities (in arbitrary units) were obtained:

The least-squares regression equation was found to be y = 1.518 + 1.930x and its calibration graph plotted in Figure 1 is shown below:

The important parameter for estimating the confidence intervals, i.e. standard error of y in x, sy/xwas found to be 0.433, with the number of points, n as 7.  The concentrations in terms of x-values are easily calculated using the regression equation by substituting various intensities y-values, and also their corresponding confidence intervals using equation (7) as follows:

The changes of magnitude in confidence intervals against the calculated concentrations can be visualized by the following Figures 2 and 3:

It is thus obvious from both graphs that the confidence intervals become narrowest (+/- 0.62 pg/ml) at the centroid of the calibration curve, indicating a better measurement precision at this point.

An approach to improve the confidence limits in this calibration experiment is by increasing n, the number of calibration points on the regression line. Another way is by making more than one measurement of y-values, as using equation (8) with R greater than 1. Of course, we need to weigh the underlying benefits if we are to make too many replicate measurements (assuming that sufficient sample is available) or increasing the number of calibration points as compared with the additional costs, and time involved.

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