Tips for understanding models used in nonlinear regression.
The first step in nonlinear regression is to choose a model (equation) to fit. These tips will help you understand what an equation means.
As an example, let’'s use the Michaelis-–Menten equation that describes enzyme activity (Y) as a function of substrate concentration (X):
Y=Vmax*X/(Km + X)
Tip: Make sure you know the meaning and units of X and Y
Here Y is enzyme activity, which can be expressed in various units, depending on the enzyme. X is the substrate concentration in some unit of concentration.
Tip: Figure out the units of the parameters
In the example equation, the parameter Km is added to X. It only makes sense to add things that are expressed in the same units, so Km must be expressed in the same concentration units as X. This means that the units cancel in the term X/(Km +X), so Vmax is expressed in the same units of enzyme activity as Y.
Tip: Figure out the value of Y at extreme values of X
Since Because X is concentration, it cannot be negative. But it can be zero. Substitute X=0 into the equation, and you will see that Y is also zero.
What happens when X is very large?. As X gets large compared to with Km, the denominator (X+Km) has a value very similar to X. So the ratio X/(X+Km) approaches 1.0, and Y approaches Vmax. So the graph of the model must level off at Y=Vmax as X gets very large. Vmax is the maximum enzyme velocity.
Tip: Figure out the value of Y at special values of X
Since Because Km is expressed in the same units as X, what is Y when X equals Km? The ratio X/(Km + X) equals 0.5, so Y equals half of Vmax. This means the Km is the concentration of substrate that leads to a velocity equal to half the maximum velocity, Vmax.
Tip: If you are entering your own equation, use plenty of parentheses
It is not wise to make assumptions about how Prism (or any program) will interpret ambiguous equations. Your understanding of the order of precedence may not be correct. Or different programs may do things differently. Use plenty of parentheses so you are sure the calculations being done are those you intend. The example equation would be very different if you left out the parentheses:
Y=Vmax*X/ Km + X
You may think that is clear. You may want the program to first multiply Vmax times X and then divide that product by the sum of Km and X. But that isn't what Prism (or any program will do). It will divide the product of Vmax and X by Km, and then add X to that quotient. The result will not be what you want. This example is pretty simple, but it easy to get confused by more complicated equations. When it doubt, add more parentheses.
Tip: Use asterisks to show multiplication
In the context of a math textbook, AB may be the product of A times B. But Prism will see AB as a parameter name. If you want Prism to multiply, be explicit and use an asterisk. Enter "A*B".
Tip: Use a caret to show power
In a typeset book, AB means A to be power of B. Copy and paste that into an equation editor (in Prism or elsewhere), and the superscripting will be gone. What's left is "AB", which Prism will assume is the name of a parameter. It will not take A to the power of B unless you enter "A^B".
Tip: Once you are sure the equation is correct, think about initial values
The first half of this page was adapted from Chapter 36 of Intuitive Biostatistics (reference below).