Analysis of Magnetic Force (FM)
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Key Findings and Interpretation
The regression model for Magnetic Force (FM) is statistically significant (P-Value = 0.000) and demonstrates an exceptionally strong fit to the experimental data, explaining 95.14% of the response variability (R-sq). This high value indicates that the formulation components have a very predictable and well-defined impact on the material's magnetic force.
The Pareto chart clearly shows that the linear term for NPs is the most dominant factor, with a large positive coefficient, indicating that a higher concentration of nanoparticles is the primary driver for increasing magnetic force. The model also reveals complex interactions: the NPs*Calcium interaction (P=0.000) is highly significant. Furthermore, the model includes a borderline significant two-way interaction, Alginate*NPs (P=0.050), and a three-way interaction, Alginate*NPs*Calcium (P=0.062), highlighting a sophisticated relationship where the effect of nanoparticles is modulated by both the alginate and calcium levels. Crucially, the Lack-of-Fit is not significant (P=0.165), which validates the model's form and confirms that no major terms are missing.
Regression Equation
The relationship between the components and the Magnetic Force response is described by the following equation, where the process variable Calcium is coded (-1 for 1%, +1 for 5%):
Model Goodness-of-Fit
The statistical model provides an excellent and reliable fit to the experimental data. The key metrics from the "Model Summary" table are outstanding:
- R-sq = 95.14%: The model explains over 95% of the variation in the magnetic force data, indicating an extremely strong correlation.
- R-sq(adj) = 94.20%: The adjusted R-squared is very high and close to the R-squared value, confirming the model's efficiency.
- R-sq(pred) = 93.34%: The predicted R-squared is exceptionally high, demonstrating robust predictive power for new observations.
- S = 0.0704066: The standard error of the regression is low, signifying high precision of the model.
The overall regression model is highly significant (P-Value = 0.000), and the non-significant Lack-of-Fit (P=0.165) provides strong evidence that the model is well-specified and accurately captures the underlying relationships.
Model Summary: Stepwise Selection
The following table shows the stepwise selection process for the final model. The last row, highlighted, represents the chosen model with the best combination of explanatory and predictive power.
| Step | S | R-sq (%) | R-sq(adj) (%) | R-sq(pred) (%) |
|---|---|---|---|---|
| 1 | 0.139858 | 78.99 | 77.13 | 72.54 |
| 2 | 0.0764592 | 93.90 | 93.17 | 91.96 |
| 3 | 0.0733593 | 94.56 | 93.71 | 92.46 |
| 4 | 0.0704066 | 95.14 | 94.20 | 93.34 |
Model Diagnostic Plots
To ensure the validity of the statistical model, a series of diagnostic plots were generated. These plots help confirm that the assumptions of the regression analysis are met. Below is a guide to interpreting each plot:
- Normal Probability Plot: This plot checks if the residuals are normally distributed. The goal is to see our experimental points fall closely to the theoretical straight line. Significant deviations may indicate that the assumption of normality is not met.
- Residuals vs Fits: This plot is used to detect non-constant variance, missing terms, or outliers. The points should be randomly scattered around the horizontal line at zero. Any clear pattern would suggest a problem with the model.
- Histogram of Residuals: This provides another visual check for the normality of residuals. The distribution should be roughly symmetric and bell-shaped, centered around zero.
- Residuals vs Order: This plot helps to verify that the residuals are independent of one another. The data points should show no discernible trend or pattern. Any systematic pattern could suggest that the order of the experiments influenced the results.
Pareto Chart of Effects
The Pareto chart visually ranks the importance of each factor and interaction on the Magnetic Force response. The red line indicates the threshold for statistical significance (α=0.05). Effects that cross this line are considered the most influential drivers of the process.
2D Contour Plots
The following interactive 2D contour plots show how pairs of variables influence Magnetic Force while holding the other factors at constant levels. These maps are essential for identifying optimal regions in the formulation space.
3D Surface Plots
These interactive 3D plots provide an intuitive view of the response surface. Each colored surface represents the predicted Magnetic Force response based on the model for a specific combination of held factors.
Overlaid on the surfaces are the data points from the actual experiments. The solid dots (●) represent the actual, measured Magnetic Force values, while the crosses (+) show the values predicted by the model for those same experimental conditions. The vertical distance between a dot and its corresponding cross represents the residual error for that point. A good model will have these points lying close to the surface, indicating small errors.