Most often nonlinear regression is done without weighting. The program minimizes the sum-of-squares of the vertical distances of the data from the curve. This method gives equal weight to all points, as is appropriate when you expect experimental scatter to be the same in all parts of the curve.If you expect experimental scatter to vary along the curve, you can weight points differentially.

Relative weighting (weighting by 1/Y2)

The weighting method used most often is called weighting by 1/Y2. It is easier to think of this method as minimizing the sum-of-squares of the relative distances of the data from the curve. This method is appropriate when you expect the average distance of the points from the curve to be higher when Y is higher, but the relative distance (distance divided by Y) to be a constant. In this common situation, minimizing the sum-of-squares is inappropriate because points with high Y values will have a large influence on the sum-of-squares value while points with smaller Y values will have little influence. Minimizing the sum of the square of the relative distances restores equal weighting to all points.

There are two ways to express the equation describing the quantity that nonlinear regression minimizes, shown below. The form on the left is, I think, easier to understand. You divide the distance of the data from the curve by the Y values of the data to obtain the relative distance, and then square that result. Most books on nonlinear regression use the equivalent form shown on the right - you first square the distance of the data from the curve, and then multiply that value times a weighting constant equal to 1/Y2. That explains why relative weighting is often called weighting by 1/Y2.

Weighting by 1/Y

Weighting by 1/Y is a compromise between minimizing the actual distance squared and minimizing the relative distance squared. One situation where 1/Y weighting is appropriate is when the Y values follow a Poisson distribution. This would be the case when Y values are radioactive counts and most of the scatter is due to counting error. With the Poisson distribution, the standard error of a value equals the square root of that value. Therefore you divide the distance between the data and the curve by the square root of the value, and then square that result. The equation below shows the quantity that Prism minimizes, and shows why it is called weighing by 1/Y

Weighting by 1/X or 1/X2

The choices to weight by 1/X or 1/X2 are used rarely. These choices are useful when you want to weight the points at the left part of the graph more than points to the right.

Other weighting schemes

If you understand how the scatter (or errors) arise in your experimental system, you may be able to calculate appropriate weighting factors based on theory. You want to choose a weighting scheme to account for systematic differences in the predicted amount of variability if you were to repeat the experiment many times. You should not choose weighting based on variability you happened to observe in one small experiment. Random scatter can cause some SD values to be high and some low, and these differences may not reflect consistent differences in variability. Don't weight by the SD of replicate values, unless you have collected data from many replicates.

To weight by factors you calculate using GraphPad Prism, choose the option to weight by the reciprocal of the standard deviation squared. Format a data table for entry of mean and SD, and enter (or paste) the weighting factors into the SD column. Prism minimizes this quantity shown below. Note that the weigting factor (the SD) is in the denominator. This means that rows where you enter a small weighting factor (small standard deviation) get more weight than rows where you enter a high weighting factor.

General comments on weighting with Prism

If the weighting scheme you chose would result in a division by zero for any value, Prism will not fit the data set and reports "Weighting impossible" at the top of the results page.

Prism also considers sample size when weighting. If you entered individual replicates, and chose to treat the replicates separately, then no special calculations are needed. If you entered replicate values, but chose to fit to the mean of the replicates, then Prism always multiplies the weighting factor by N. The means computed from a large number of replicates get more weight than means computed from a few replicates. Similarly, if you enter mean, SD (or SEM) and N, Prism multiplies the weighting factor by N.