Comment effectuer une régression non linéaire multiple avec XLSTAT ?
Une feuille Excel contenant les données et les résultats de cet exemple peut être téléchargée en cliquant ici. Le but de l'étude est d'étudier l'effet de la concentration de deux composants sur la viscosité d'un yaourt. Le modèle que nous voulons ajustés est défini par :
F(C1, C2) = pr5 / (1+Exp(-pr1-pr2*C1-pr3*C2-pr4*C1*C2))
pr1, ..., pr5 sont les paramètres du modèle. Ce modèle dont la forme est sigmoïde (comme la fonction logistique) permet de prendre en compte à la fois la concentration des composants et leur interaction.
Une fois XLSTAT lancé, choisissez la commande XLSTAT/Modélisation/Régression non linéaire ou cliquez sur le bouton "Régression non linéaire" de la barre d'outils "Modélisation".
Once you've clicked on the button, the nonlinear regression dialog box appears. Select the data on the Excel sheet. The "Dependent variable" (or response variable) is in our case the "Viscosity". The quantitative explanatory variables are the concentration of the two components C1 and C2. As we selected the column headers, we left the option "Variable labels" option activated. We left the "Residuals" option activated as well, because we want to analyze the predictions and the residuals.
In the "Options" tab we selected the values of the initial values of the five parameters.
In the "Functions" tab, the various functions are displayed. As the function we want to use is not listed in the "Preprogrammed functions" (you can notice the univariate version of the function in the list), we needed to enter the model: we first clicked on "Add", then entered the function, then checked "Derivatives", then selected them on the Excel sheet. In order to add this function to the user functions library, we clicked on "Save". The function is then automatically added and selected.
The computations begin once you have clicked on the "OK" button. The results will then be displayed. The first table gives the basic statistics of the selected variables.
The second table (see below) displays the goodness of fit coefficients, including the R² (coefficient of determination), and the SSE (sum of square of errors), the later being the criterion used for the model optimization. The R² corresponds to the % of the variability of the dependant variable (the dry weight) that is explained by the explanatory variable (the time). The closer to 1 the R² is, the better the fit.
In our case, 99% of the variability is explained by the two variables and their interaction, which is an excellent result that confirms that the selected model is appropriate.
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 allows to visualize the quality of the fit by comparing the predicted values to the observed values.
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