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Equation 13.5.1 Variance Proportional to Observed Data
Equation 13.5.2 Variance Proportional to the Square of Observed Value
The effect of different weighting schemes on the result of non linear regression are illustrated in Figure 13.5.1.
Figure 13.5.1 Effect of Weighting Scheme on the Fit to Data
Equation 13.5.3 Variance as a Function of Observed Value
Values of the parameter a and b may be determined from the data or knowledge of the assay method. A plot of data variance versus observed value on log-log graph paper should reveal a straight line. The slope and intercept of this line can be used to determine suitable values for a and b.
Alternately, if this information is not available, that is the variance of each data point it might be possible estimate a value. (Of course if this information was available either as a variance value or standard deviation an appropriate weight could be calculated directly). One approach to estimating a suitable weight might be to fit the data with an arbitrary polynomial or other suitable empirical function. The objective would be to put a smooth line through the data without regard to the 'theoretical' model. Deviations from this arbitrary line might be used to estimate the standard deviation and thus the variance of the data with respect to the data value.
Figure 13.5.2 Spreadsheet Illustrating the Determination of a and b
Click on the figure to download the Exceltm Spreadsheet or here for a Numberstm SpreadsheetThis approach can be explored with the spreadsheet above in Figure 13.5.2
Weighting Scheme Tutorial
Equation 13.5.4 Variance as a Function of Observed Value with an Assay Sensitivity Term
This weighting scheme provides information about the error in the data at both low and high concentrations. Further extension of this type of scheme could include other polynomial terms and further refinement.
Equation 13.5.5 Variance as a Function of Observed Value with Time Information
As the difference between a sample time and the last sample time, tlast, becomes larger the weight for that data point becomes smaller. Therefore older data points, collected when patient status may be different, are given less emphasis in the current analysis. A value 1.002 for c (with t in hours) can serve as a starting point and discounts samples older than 24hours by 5%.
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