How to specify the expectation function in nonlinear regression

You must specify the expectation function that Minitab uses to perform nonlinear regression. Your choice for the function often depends on previous knowledge about the response curve's shape or the behavior of physical and chemical properties in the system. Potential nonlinear shapes include concave, convex, exponential growth or decay, sigmoidal (S), and asymptotic curves. You must specify the function that satisfies both the requirements of your previous knowledge and the nonlinear regression assumptions.

If you specify a new function, it must contain at least one of each of the following three basic components:

Parameters

Minitab estimates parameters by fitting the expectation function to the data using an iterative algorithm that minimizes the sum of squares of the residual error (SSE). In the function, enter text to identify a parameter. For example, you can enter b1, b2, Theta1, Theta2, and so on. However, you cannot enter text that matches a constant name, a column name, or a mathematical operation.

Unacceptable parameters	Example	Result
Names of constants, such as K1, K2, K3…	1/(1 + K1 * X)	K1 acts as a number instead of a parameter, so the value of K1 is fixed.
Names of columns, such as C1, C2, C3…	1/(1 + C1 * X)	C1 acts as a variable instead of a parameter.
Mathematical parameters, such as +, /, and *.	1/(1 + B+ * X)	The symbol creates an incorrect function.

Predictors

Variables that you enter in worksheet columns. Enter the column name in the function. If the name contains more than one word, use single quotation marks (for example, 'Density Ln').

Mathematical operations and functions

Specify the mathematical relationship between the parameters and predictors that produces the expected value of the response variable. You can use the Nonlinear Regression Calculator to easily enter the operations and functions (for example, *, +, COS, EXP, and so on). Or, you can type them directly into the Edit directly field.

The following examples from the expectation function catalog are acceptable functions. Thetas represent parameters and X's represent predictors. You replace the X's with the variable names. Each time you perform nonlinear regression using a new function, Minitab automatically adds the function to the catalog.

Expectation function	Model name	Model contains
1 / (1 + Theta *X )	Convex 1	One parameter and one predictor
Theta1* X / ( Theta2 + X )	Michaelis-Menten	Two parameters and one predictor
Theta1 * cos ( X + Theta4 ) + Theta2 * cos ( 2 * X + Theta4 ) + Theta3	Fourier 1	Four parameters and one predictor
Theta1 - Theta2 * ( ln ( X1 + Theta3 ) - ln ( X2 ) )	Nernst equation	Three parameters and 2 predictors
X1 * X2 / ( Theta1 + Theta2 * X1 + Theta3 * X1 * X2 + Theta4 * X1 * X3 )	Enzyme reaction	Four Parameters and 3 predictors