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Least Squares Criteria

Least squares criteria refers to the formula used as a measure of how well the computer generated line fits the data. Thus it is a measure of the total of the differences between the observed data and the calculated data point. Most commonly with pharmacokinetic modeling these differences are measured in the vertical direction. That is, in the y axis values. Usually time is the x or independent variable and it should be possible to measure time accurately. The y axis or dependent variable, usually concentration, often involves an assay method which means there may be error (or variation) in each result.

Plot of Cp versus time ilustrating error in y axis value

Figure 6.7.1 Linear plot of Cp versus time illustrating error between observed data and calculated line

Again, usually the residual or error is assumed to be in the vertical direction although there are programs available that are capable of looking at oblique error in both the x and y direction. For the rest of our modeling discussion we will assume that the error is in the y axis variable only.

Looking at an individual data point and the calculated value with the same x value the residual can be expressed as a simple subtraction.

Residual = Yobserved - Ycalculated

Equation 6.7.1 Residual in the y direction

The problem is that over all the data points there might be high positive and high negative residuals that might cancel out. An absolute difference would solve this problem but squaring the residual is better statistically and achieves the same result.

Residual2 = (Yobserved - Ycalculated)2

Equation 6.7.2 Residual in the y direction squared

This gives us an equation of the residual for one data points. To complete the calculation we need to include the residuals for all the data points. This is called the sum of the squared residuals (SS).

SS = Sum of the squared residuals

Equation 6.7.3 Sum of the squared residuals

Finally we need to take the error in each data point as a separate value. That is the error may be different for each measured, observed data point. We can compensate for this by applying a weight to each residual thus the usual criteria for a best fit is a minimum sum of the weighted, squared residuals (WSS).

WSS = Weighted residual

Equation 6.7.4 Weighted sum of squared residuals

The job of the computer program is produce a minimum value for WSS which represents the best fit according to the least squares criteria. Inspection of Equation 6.7.4 leads to the conclusion that this can be achieved by changing the calculated values (Ycalculated,i) by changing the parameter values.

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