XLSTAT: Multiple nonlinear regression

 

Dataset for Example

Nonlinear regression is used to model complex phenomena which cannot be handled by linear models. Our purpose is to study the effect of the concentration of two components, C1 and C2, on the viscosity of a yogurt. The model that we want to fit writes:

 

Where β1, ..., β5 are the parameters of the model. This logistic-like model allows taking into account both the concentrations of the components and their interaction.

 

Setting Up

• Open XLSTAT.

• Click on the Modeling data menu and select the Nonlinear regression feature.

• The nonlinear regression dialog box pops up.

In the General tab:

• Select the Viscosity variable as the Dependent variable  (or response variable).

• Select the concentration of the two components C1and C2 as the quantitative  explanatory variable.

• Activate the option Variable labels as we selected the column headers.

In the Options tab:

• Select the initial values of our five parameters.

In the Functions tab: 

• Check the User defined functions option.

• Click Add and enter the equation f(C1, C2) of our model (See the background section). Note: you have a second option to choose one of the build-in functions.

• Check the Derivatives option and select their values from the Excel sheet.

• Click Save to add the model to the library of user-defined functions.

•  In the Outputs tab, select the Residuals option as well in order to analyze the predictions and the residuals.

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results

The first table gives the basic statistics for the selected variables. The second table (see below) displays the goodness of fit coefficients, including the SSE (sum of square of errors), the latter being the criterion used for the model optimization.

The next table shows the results for the model parameters. As we can see, the ratios (parameter)/(std deviation) are larger for pr5 and pr4. As the same ratio is the largest for pr5 we deduce that the interaction between the two components has a greater effect on the viscosity than the concentrations themselves.

The following chart visualizes the quality of the fit by comparing the predicted values to the observed values.