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Tutorial 13.1 - General Weighting Scheme

When you preform a regression analysis you should be aware of the relative uncertainty or error in each of the data points used in the analysis. If we plot data on linear graph paper and put a line through the data we tend to assume that the uncertainty in each data point is the same. Typically we give each data point equal weight in where we draw the best-fit line. What we are trying to do is make the total residual between the line and each data point as small as possible. Again, on linear paper we tend to think of each point as having the same uncertainty or error.

Best fit line on linear graph

Figure 13.1(t) Best-fit line on Linear Graph paper

Notice the residuals (only 1 and 2 are marked) in Figure 13.1 represent distances that are about the same magnitude. That is, about one unit on the y scale. The line is drawn to minimize the total residual.

The situation changes somewhat if we use semi-log graph paper. Remember on semi-log graph paper each decade takes up the same space on the graph paper. That is, data values between 100 and 10 are spread out (in a vertical direction) and take up as much space as data that might fall between 1 and 0.1. Therefore when we put a line through data drawn on a semi-log plot we tend to emphasize to points with the lower values. Residuals at higher concentrations are much higher than the residuals at lower concentrations even though they look to be the same on semi-log graph paper.

Best-fit line on semi-log graph paper

Figure 13.2(t) Best-fit line on Semi-log Graph paper

In Figure 13.2, residual 1 is the difference between 27 and 54 or 27 units whereas for residual 2 the difference is between 1 and 0.5 or 0.5 units. However, on semi-log graph paper these difference appear to be the same. The best-fit line is drawn to minimize 'weighted' residuals. In the case of the semi-log plot the lower values have more emphasis.

Data from Figure 13.2 on linear plot

Figure 13.3(t) Data in Figure 13.2 Plotted on Linear Graph Paper

In Figure 13.3 the data from Figure 13.2 are plotted on linear graph paper. The green line represents the 'linear' best-fit line. The blue line is the best-line from the semi-log plot.

When we use a computer program to 'draw' the best-fit line through the data we need to be able to tell the program how much emphasis or 'weight' we want to give to each data point. If we know or can estimate the variance (= standard deviation squared) of each data point we can weight the data by the reciprocal of the variance.

Weight as 1/Variance

Equation 13.1(t) Weight as a function of Variance

For some data sets the variance is essentially the same for each value for others the variance varies with value measured. Wagner (Wagner, J.G. 1975. Fundamentals of Clinical Pharmacokinetics, Drug Intelligence Publications, Hamilton, IL p289) describes a general equation for variance.

Weight as a function of Data Value

Equation 13.2(t) A General Equation for Weight as a Function of the Data Value

Variance as a function of Data Value

Equation 13.3(t) A General Equation for Variance as a Function of the Data Value

Taking the log of both sides of Figure 13.3 gives Figure 13.4


Equation 13.4(t) log(Variance) as a Function of the log(Data Value)

As described by Wagner (1975) a plot of log(Variance) versus log(Data Value) can provide a straight line with a slope of 'b' and an intercept of log(a).

Determining the 'Correct' Weighting Scheme for the PHAR 7633 Project

The data provided for the PHAR 7633 project included more one or more data sets consisting of time, value, and standard deviation. The weight equation (similar to Equation 13.2) must be determined for each data set. The values of 'a' and 'b' can be determined by plotting the log of the square of the standard deviation versus the log of the (y) value. The slope is 'b' and the intercept is log(a). A spreadsheet program such as Excel may be useful in this analysis.

Data Entered on an Excel Spreadsheet

Figure 13.4(t) Data Entered on an Excel Spreadsheet

Notice cell D2 is highlighted with the equation for this value shown as =C2*C2. The equations for E2 and F2 are =log(B2) and =log(D2), respectively. Click on Cell D2 and move the cursor until you see the part-square shape then drag to cell D12 (or the last row of your data). When you release the button Excel will fill down values in column D. Repeat this process for E2 and F2 for columns E and F. The result is shown in Figure 13.5. [If you are in a hurry you can do this fill-down on all three columns at once].

Figure after calculate down

Figure 13.5(t) After fill-down in Columns D, E, and F

The next step is to select cells E2 through F12 and use the Chart Wizard Chart Wizard to generate a XY (Scatter) plot of the data. Don't forget to select XY (Scatter) plot as the type. Selecting all the defaults should result in a graph similar that in Figure 13.6.

XY Scatter plot

Figure 13.6(t) XY (Scatter) Plot of log(Variance) versus log(Cp)

With the Chart (graph) selected choose 'Add Trendline...' from the Chart menu. Select the linear trend and click on the Option tab. Check 'Display equation on chart" in the options and click 'OK'. This produced the equation:

y = 2.0211x - 2.6238

Thus the slope 'b' is 2.0211 and the intercept log(a) is -2.6238 leading to a value for 'a' of 0.002378. In Boomer choose the fourth weighting scheme ('3') and enter the a and b when asked. In SAAM II use the GEN weighting function and include (GEN 0 a b) at the top of the data table in the Data window, replacing the a and b values with your numbers.

Does this help? Let me know if you find any mistakes or unclear sections. David Bourne

P.S. You can download my Excel Spreadsheet here.