Preparing to Fit an Asymptotic Regression Model

The asymptotic regression model has the form:

Figure 1. Asymptotic regression model
Asymptotic regression model

When b1>0, b2<0, and b3<0, it gives Mistcherlich's model of the "law of diminishing returns". This model initially increases quickly with increasing values of x, but then the gains slow and finally taper off just below the value b1.

Choosing starting values

The Nonlinear Regression procedure requires that you supply starting values for the parameters in the model. This seems a daunting task at first, but becomes easier with some familiarity with the model.

  • b1 represents the upper asymptote for sales. Looking at the chart, even the largest sales values fall justs short of 13, so that's a reasonable starting value.
  • b2 is the difference between the value of y when x=0 and the upper asymptote. A reasonable starting value is the minimum value of y minus b1. Looking at the chart, say that's about 7-13=-6. It is possible to reparametrize the model by replacing b2 with (b4-b1), where b4 represents the sales when no advertising money is spent. b4 might be easier to interpret than b2, but doesn't improve estimation.
  • b3 can be roughly initially estimated by the negative of the slope between two "well separated" points on the plot. Looking at the chart there are a few points about x=2, y=8, and about x=5, y=12. The slope between these points is (12-8)/(5-2)=1.33, thus a rough initial estimate for b3 is -1.33.