Once we know the equation of the regression line, we can easily calculate the concentration x0 from a given signal y0. However, because we are now going from a y-value to an x-value (instead of the other way around), we need to find the standard deviation in x, sx0.
The error in x will depend on various factors, two of which we will consider here. The first is whether there are replicates and the second is whether the y-value is interpolated or extrapolated. In this section, we cover three situations:
- No Replication, Interpolated Value
- With Replication, Interpolated Value
- No Replication, Extrapolated Value
No Replication, Interpolated Value
With no replications and a measured signal within the calibrated range of the instrument (interpolation), sx0 is given by
Here, y0 is the experimental signal from the instrument for which x0 is to be determined, and n is the number of samples. This formula only applies if there is no replication of each measurement.
To calculate the concentration of an uknown sample where the measured fluorescence intensity is 16.9,
- Use the calibration equation determined previously, y=1.88x+1.8, with y0=16.9, giving x0=8.06 pg·ml-1.
- Calculate the standard deviation sx0 using the equation above. For n=9, sy/x=0.5415, and b=1.88, we obtain sx0=0.3.
- Obtain a 95% confidence interval in the interpolated concentration by determining the two-tailed t-statistic for n-2 degrees of freedom. It is important to note that a two-tailed test is required for the interpolated results (d.o.f.: ν = n-2), compared to the one-tailed test for the mean. From table of t-values, for ν=n-2=7, t7=2.58. The interpolated concentration with 95% confidence interval is then reported as C=x0±tνsx0=8.0±0.8 pg·ml-1.
Note that with such an error, lower measured signals, and hence lower concentrations, may lead to error values on the order of the estimated concentration. This concept will be explored further when we consider limits of detection.
With Replication, Interpolated Value
When you perform a sample measurement, you would normally perform more than one measurement of each sample, which is called replication. Replication is important in the statistical determination of your answer, in order to reduce the uncertainty and improve the accuracy of your measurement. Random fluctuations, which occur in any system, can lead to small errors in each measurement. By performing replications at each measurement, some or most of the error due to random fluctuations can be averaged out.
If replications are performed, the computation of the error in x must be modified to account for the extra degrees of freedom, as a result of the extra measurements. The formula for the standard deviation in x0 with m replications is
where the variable are the same as before. When working with a calibration curve with n measurements and a sample measurement y0, the concentration with error as read from the calibration curve is x0 ±tνsx0,R.
No Replication, Extrapolated Value
In some cases, the measurement value for the sample will be outside the measured range of you calibration curve. While this situation is not desirable, due to the possibility of nonlinear effects outside the measurement range, it is sometimes unavoidable, and the results can still be used! All this requires is knowledge of a different way to calculate the standard deviation for extrapolation,
where n is the number of calibration values. The differences between this equation and the previous ones is that replications are not taken into account, and y0 = 0, which is shown as part of the numerator in the square root. y0 is shifted to the x-axis, and all calibration values are calculated from there. The reported sample concentration is then xE ±tνsx,E.
Following the previous example, assume the measured fluoresence intesnity of an uknown sample is yE=34.9. From the calibration curve, the concentration is 17.6 pg·ml-1, which is outside the calibration range. Using the above equation, the standard deviation would be sxE=0.457 and the concentration with error would be 18±1 pg·ml-1.
We have now seen how to determine the calibration curve, and use it to convert a measured signal to a concentration value. In the next section, we will see how to use statistics to determine the limit of detection for an instrument.