2.6 RESULTS AND DISCUSSION
2.6.2 Central Composite Design
2.6.1.1 Evaluation of Model Adequacy for Retention Time of NVP
The values of the factors that were assessed to determine the adequacy of the model are summarised in Table 2.6. The most important parameters in the evaluation of model adequacy are the model F-value, coefficient of variation, adequate precision, PRESS and R2 values. The importance of these parameters in Table 2.6 and their respective values are discussed in detail.
Table 2.6. Summary of model adequacy parameters and associated values
Parameter Value Parameter Value
Std. Deviation 0.15 R-Squared 0.9917
Mean 3.87 Adjusted R-Squared 0.9842
% Coefficient of variation 3.99 Predicted R-Squared 0.9369
PRESS 1.81 Adequate precision 38.864
F-Value 132.67
2.6.1.1.1 Model F-Value
The model F-value is used to ascertain the utility of a model that the data has been fitted to and to determine whether a model best fits the data set. The F-value produces a ratio of explained and unexplained variability and the larger the value for F, the more useful the model [117, 160]. A model F-value of 132.67 was obtained and represents a 0.01% chance that a model F-value this large could occur due to noise. This implies that the model would be able to describe the fitted data set accurately and there is only 0.01% chance of the model being inaccurate.
2.6.1.1.2 Coefficient of Variation
The coefficient of variation is the ratio of the standard deviation and mean and is indicative of the normalised measure of dispersion of a probability distribution [161, 162]. The % C.V is a measure of the reproducibility of a model and a value of < 10% is desirable [163]. The value of 3.99 was within this limit and implies that the model would be able to produce reproducible results over time, which is important in any laboratory analytical procedure.
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2.6.1.1.3 Adequate Precision
Adequate precision compares the range of predicted values at the design points of the CCD to the average prediction error and a ratio of > 4 indicates that there is adequate model discrimination [119, 160]. The value of 38.864 was well above the limit implying that the model can be used to predict possible experimental outcomes with acceptable accuracy.
2.6.1.1.4 R2, Predicted R2 and Adjusted R2Values
It is important to determine whether a model is able to describe the experimental data under consideration adequately [119, 160]. The process of evaluation includes the determination of different coefficients of correlation. The R2 coefficients have values between 0 and 1 and the closer the value to 1, the more reliable the model. All values that were obtained were > 0.9 signifying that the model is reliable and when used to predict experimental outcomes, the actual values will be reasonably close to the predicted values.
2.6.1.1.5 Residual Analysis
The adequacy of a model is also investigated by examination of residuals. Residuals are the difference between the respective observed and predicted responses [119]. Residuals are examined using normal probability plots of residuals and the plot of residuals versus predicted responses. If a model is adequate, the points of the normal probability plot of residuals should fall on a straight line. On the other hand, the location of points on the plots of residuals versus the predicted responses should be structureless, i.e. should not be seen to follow a specific pattern [117, 119, 160]. The data depicted in Figures 2.1 and 2.2 clearly indicate that these criteria have been met and the model was therefore deemed adequate.
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Figure 2.1. Normal probability plots of residuals for the retention time of NVP.
Figure 2.2. Plot of residuals versus predicted responses for the retention time of NVP.
Design-Expert® Software Retention time Color points by value of Retention time:
6.78 2.37
Internally Studentized Residuals
Normal % Probability
Normal Plot of Residuals
-2.53 -1.28 -0.04 1.21 2.45
1 5 10 20 30 50 70 80 90 95 99
Design-Expert® Software Retention time
Color points by value of Retention time:
6.78 2.37
6
Predicted
Internally Studentized Residuals
Residuals vs. Predicted
-3.00 -1.50 0.00 1.50 3.00
2.31 3.37 4.43 5.49 6.55
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2.6.1.1.6 PRESS
The PRESS value is the sum of squares of residuals and is a measure of the discrepancy between experimental data and those estimated by the model. A small PRESS value indicates a good fit of the data under investigation to the model selected [160]. The value of 1.81 suggests that the data was well fitted to the model, indicating that the model selected was accurate and can be used to describe the data set adequately.
2.6.1.1.7 Box-Cox Plot for Power Transformations
Most statistical tests and intervals are based on the assumption of normality of distribution of data as this leads to tests that are simple, mathematically tractable and powerful compared to tests that do not make any assumption of normality [117, 160]. However, many data sets are neither in fact, nor approximately normal and transformation of results may be necessary to yield data that follow a normal distribution, albeit approximately in some cases. Box-Cox plots of normality are used when transformation is required to increase the applicability and usefulness of an applied statistical test [164]. In this case, inspection of the Box-Cox plot suggests that transformation of the power λ = -1.76 should be made to the data set and this is depicted in Figure 2.3.
Figure 2.3. Box-Cox plot for power transformation of retention time data for NVP.
Design-Expert® Software Retention time Lambda Current = 1 Best = -1.76 Low C.I. = -2.03 High C.I. = -1.47 Recommend transform:
Power
(Lambda = -1.76)
Lambda
Ln(ResidualSS)
Box-Cox Plot for Power Transforms
-5.02 -3.55 -2.08 -0.60 0.87
-3 -2 -1 0 1 2 3
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The resultant response range was between 2.39 and 6.78 and the ratio of maximum to minimum response was 2.8368. A ratio > 10 indicates that transformation is required.
However ratios of < 3 indicate power transformations have little effect on the experimental responses, therefore transformation was not performed for these data.