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Weighting Schemes

Much of the material on this page is Boomer specific but most of the weighting schemes are available with other non linear regression programs.

Equal Weight

In its simplest form equal weighting mean that each data point is assigned a weight of 1.

Weight Proportional to the Inverse of the Value

In this case the variance in the data point is proportional to the value of the data point. This approach or method is quite useful when the data has been determined by measuring radioactivity counts.

Equation 13.5.1 Variance Proportional to Observed Data

Weight Proportional to the Inverse of the Square of the Value

More commonly data are measured by methods, such as HPLC, that provide standard deviation values that are proportional to the value. That is, the coefficient of variation (CV) of these data are similar. This can be a reasonable approach as long as the limits of assay sensitivity are not approached too closely.

Variance proportional to observed value squared

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

None of the fitted lines in Figure 13.5.1 are particularly good, maybe another model is required, but the effect of weighting scheme can be readily determined. The equal weight scheme fits the high data points well. In contrast the weight scheme proportional to the value squared fit the low data points. A better model with a suitable weighting scheme should provide a much better fit.

Variance Equal to a • Observed Value^b

A more general approach to defining the weighting scheme is to use the scheme shown as Equation 13.5.3. This approach was described by Wagner (Wagner, J.G. 1975 Fundamentals of Clinical Pharmacokinetics, Drug Intelligence Publications, Hamilton, IL, page 289).

Variance is a times value to the power b

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.

Spreadsheet illustrating the Determination of Values for a and b

Figure 13.5.2 Spreadsheet Illustrating the Determination of a and b

Click on the figure to download the Excel^tm Spreadsheet or here for a Numbers^tm Spreadsheet

This approach can be explored with the spreadsheet above in Figure 13.5.2

Weighting Scheme Tutorial

Variance Equal to c^b + a • Observed Value^b

Although the weighting scheme where variance is a • Observed Data^b can be very useful in describing the constant CV part of assay standard curve with smaller data values the CV can rise dramatically. The inclusion of an additional term representing the assay sensitivity (c) can compensate for this increase in CV.

Variance with an additional assay sensitiivty term

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.

Variance Estimate taking the 'Age' of the Data into Account

When data are collected over a period of time, such clinical samples collected over a few days or weeks it may be useful to discount older samples. This can be achieved by including time in the variance equation or weighting scheme.

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, t_last, 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%.

References

General Weighting Scheme Tutorial

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