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Weighting Example

The selection of appropriate weighting schemes becomes even more important when more than one data set is fit simultaneously. As example consider fitting drug concentrations in plasma and drug amount in urine. The magnitude of the data may be quite different. Also, the variance function may be different. Drug concentration data points are essentially independent. That is the uncertainty in one sample is independent of the uncertainty or error in the other samples. The actual values may be dependent on the underlying processes controlling the absorption, distribution, metabolism and excretion but that is the subject of pharmacokinetics. Cumulative amounts excreted into urine values, however are dependent on all the preceding values. A variance or weighting scheme proportional to the square of the value may be quite suitable for plasma concentration data measured by HPLC. For cumulative amount of drug excreted into urine a constant variance or standard deviation scheme may be more useful.

Plasma concentration weighting scheme - constant CV

Figure 13.8.1 A Suitable Weighting Scheme for these Plasma Data may be Constant CV

As shown in Figure 13.8.1 a constant coefficient of variation (CV) for a series of data points translates into a weighting scheme described as the reciprocal of the observed value squared. Since there are two data sets fitted simultaneously it is important to scale the weight for each line appropriately. Thus, the b value of power is 2 and the a value is the square of the CV.

Urine data weighting scheme - constant SD

Figure 13.8.2 A Suitable Weighting Scheme for these Urine Data may be Constant SD

For cumulative amount of drug excreted into urine a constant standard deviation (SD) weighting scheme may be useful. Thus, variance also is constant for each of these data values. Thus, a standard deviation of 5 mg translates into a variance of 25. The a value is 25 and the b values is 0 since the weight is independent of the observed data value.

If these estimates of CV (for the plasma data) and SD (for the urine data) are reasonable then a good optimization should be possible with reasonable scatter in the weighted residual plots, Chapter 9.

Boomer Output

One example using Boomer produced:

 ** FINAL OUTPUT FROM Boomer (v3.0.8) **      09-Feb-2003 --- 11:38:29 am

 Title:  Fit to two lines simultaneously                             
 Input: From Ch1308.BAT                                                      
 Output:  To Ch1308.OUT                                                      
 Data for [Drug] came from Ch1308p.DAT                                                     
 Data for Drug in Urine came from Ch1308u.DAT                                                     
 Fitting algorithm: DAMPING-GAUSS/SIMPLEX      
 Weighting for [Drug]          by 1/a*Cp(Obs )^b                                    
      With a = 0.2500E-02 and b =  2.000    
 Weighting for Drug in Urine   by 1/a*Cp(Obs )^b                                    
      With a =  25.00     and b = 0.0000    
 Numerical integration method: 2) Fehlberg RKF45                          
          with  2 de(s)
 With relative error   0.1000E-03
 With absolute error   0.1000E-03
 DT =   0.1000E-02     PC =   0.1000E-04 Loops =     2
 Damping =     1

                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.14030      0.465E-03  0.33      0.00       10.    
  2) km                    0.68668E-01  0.627E-03  0.91      0.00       10.    
  3) V                      25.045      0.159      0.63      0.10      0.10E+03

 AIC =   -1.28240      Log likelihood =  -12.3     Schwartz factor =   -16.2293    
 Final WSS =   0.594419     R-squared =    1.000     Correlation Coeff =    1.000

Try it out yourself with the Boomer .BAT and .DAT files (Macintosh (OS 9), Macintosh (OS X), and DOS versions as zip Archives

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