Analysis of Saturation Magnetization (Ms)
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Key Findings and Interpretation
The regression model for Saturation Magnetization (Ms) is statistically significant (P-Value = 0.000) and provides an exceptionally strong fit for the experimental data, explaining an impressive 96.89% of the variability (R-sq). This indicates that the magnetic properties of the material are very well-defined by the formulation.
The most dominant factor by far is the interaction between Alginate*NPs (P=0.000), which has a very large negative effect, suggesting it is the primary driver for reducing magnetization. The interaction between Water*NPs (P=0.002) is also highly significant with a negative coefficient. The Alginate*Spiruline (P=0.008) interaction also shows a significant negative effect. Notably, the linear term for NPs is the most significant positive factor, indicating that increasing nanoparticle concentration, in the absence of interactions, increases magnetization. However, it's important to note the significant Lack-of-Fit (P=0.024), which suggests that while the model has high explanatory power, there may be underlying complexities it does not fully capture.
Regression Equation
The relationship between the components and the Saturation Magnetization response is described by the following equation:
Model Goodness-of-Fit
The statistical model provides an excellent fit to the experimental data, though the Lack-of-Fit requires consideration. The key metrics from the "Model Summary" table are:
- R-sq = 96.89%: The model explains nearly 97% of the variation in the Saturation Magnetization data, indicating an extremely strong relationship.
- R-sq(adj) = 96.29%: The adjusted R-squared is very high and close to the R-squared value, confirming the model's efficiency.
- R-sq(pred) = 94.84%: The predicted R-squared is also exceptionally high, demonstrating outstanding predictive power for new observations.
- S = 0.0464827: The standard error of the regression is very low.
While the overall regression is highly significant (P-Value = 0.000), the significant Lack-of-Fit (P=0.024) suggests that there may be other terms or higher-order interactions that could further improve the model, even with its already high performance.
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 | 0.0990768 | 84.52 | 83.16 | 78.04 |
| 2 | 0.0573737 | 94.96 | 94.35 | 92.61 |
| 3 | 0.0514332 | 96.07 | 95.46 | 93.86 |
| 4 | 0.0464827 | 96.89 | 96.29 | 94.84 |
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 Saturation Magnetization 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 Saturation Magnetization 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 Saturation Magnetization 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 Saturation Magnetization 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.