Analysis of Coercive Field (Hc)
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Key Findings and Interpretation
The regression model for the Coercive Field (Hc) is statistically significant (P-Value = 0.000) and provides a very strong fit for the experimental data, explaining 85.11% of the variability (R-sq). This demonstrates a clear and predictable relationship between the formulation components and the material's coercivity.
The Pareto chart reveals that the interaction between Water*Spiruline (P=0.000) is by far the most dominant factor, with a large positive effect on the coercive field. The linear term for NPs is the second most influential factor. However, a critical finding from this analysis is the highly significant Lack-of-Fit (P=0.000). This indicates that while the model has strong explanatory power, it does not fully capture the underlying relationships. The "Residuals vs Fits" plot confirms this, showing a distinct curve instead of a random scatter, which suggests that higher-order terms (like quadratic effects) may be needed to perfectly model the response.
Regression Equation
The relationship between the components and the Coercive Field response is described by the following equation:
Model Goodness-of-Fit
The statistical model provides a strong fit to the experimental data, but the significant Lack-of-Fit warrants careful interpretation. The key metrics are:
- R-sq = 85.11%: The model explains a high percentage of the variation in the Coercive Field data.
- R-sq(adj) = 82.79%: The adjusted R-squared is also very high, confirming the model's efficiency with the selected terms.
- R-sq(pred) = 80.51%: The predicted R-squared is strong, indicating the model is reliable for predicting new observations within the design space.
- S = 0.161171: The standard error of the regression is at an acceptable level.
Despite the excellent R-squared values, the significant Lack-of-Fit (P=0.000) is a crucial finding. It suggests that while the current model is very useful, a more complex model (e.g., one including quadratic terms) could potentially improve the fit and resolve the patterns seen in the residual plots.
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 | 0.287366 | 49.71 | 45.28 | 35.01 |
| 2 | 0.165868 | 83.74 | 81.77 | 79.23 |
| 3 | 0.161171 | 85.11 | 82.79 | 80.51 |
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 Coercive Field 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 the Coercive Field 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 Coercive Field 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 Coercive Field 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.