RSM data modelling and interpretation

Upon completion of the experiments according to a selected experimental design, a model equation is established and regression coefficients are predicted (Bezerra et al., 2008). In RSM, a second order polynomial model is typically used for fitting data:

𝑦 = 𝛽0+ Σ𝑗=1𝑘 𝛽𝑗𝑋𝐽+ Σ𝑗=1𝑘 𝛽𝑗𝑗 𝑋𝑗2+ ΣΣ𝑖<𝑗𝛽𝑖𝑗𝑋𝑖 𝑋𝑗+ ɛ

where 𝛽0, 𝛽𝑖, 𝛽𝑖𝑖, and 𝛽𝑖𝑗 represent regression coefficients for intercept, linear, quadratic and

interaction terms, respectively . The response variable (dependent variable) is represented by 𝑦, and 𝑋𝑖 and 𝑋𝑗 represent the level of the independent variables (factors). The term k represents the number

of factors to be investigated, while ɛ represents the residual error associated with the experiment (Baş & Boyacı, 2007; Bezerra et al., 2008; Dejaegher & Vander Heyden, 2011). Thereafter, analysis of variance (ANOVA) is applied to determine how well the generated model fits the data. ANOVA

58 compares the variation due to different treatments with variation due to random errors inherent to the measurements of the generated responses, which is represented by the coefficient of variation (R2_{) (Bezerra et al., 2008). Additionally, the statistical significance of the factors and their interactions} on the measured responses are estimated by ANOVA. The experimental procedure is repeated to validate the proposed model and to compare experimental results to predicted values.

The regression (predicted model) equation is graphically depicted by two dimensional contour plots or three dimensional response surface plots. They demonstrate the type of interaction (significant vs. negligible) between the factors that are tested and their relationship to measured responses (Baş & Boyacı, 2007; Bezerra et al., 2008). In addition, the response surface plot demonstrates the scale of a response value as a result of the combined effect of two factors at a particular time or condition (Yang et al., 2010). If three or more variables are present, the plot visualisation is only possible if one or more variables are kept at a constant level. If a slope on the response surface plot is steep, showing a maximum response, minimum response or a saddle point, the response is significantly affected by a change in factor levels, whereas a flat surface indicates a non- significant effect. The optimum value in a range of tested parameters is also referred to as the critical/stationary point and it can be determined by using the second order polynomial equation. A single optimum value in RSM cannot always be identified. Therefore, in such cases an optimum region of values can be displayed on the response surfae plot instead (Bezerra et al., 2008, Granato & De Araújo Calado, 2014).

Figure 2.9a and Figure 2.9b show response surfaces where the maximum response is found within the space of the experimental design. However, Figure 2.9b differs as it demonstrates a plateau in relation to variable X2, which means that the change in its levels has no effect on the level of the response (y). In Figure 2.9c, the maximum response does not lie completely within the experimental region. In this case, the experimental design would have to be altered to get an optimum response, i.e. the experimental design would have to include extended ranges of the independent variable. A minimum point found within the experimental region is shown in Figure 2.9d, and a saddle point, which represents an inflexion point between a relative maximum and minimum, is shown in Figure 2.9e. Saddle point coordinates, however, are not considered as valid optimal values when the aim is to obtain a minimum or a maximum response in a system (Bezerra et al., 2008).

59 Figure 2.9 Examples of response surface plots obtained in the optimisation of two variables, X1 and X2, showing (a) maximum, (b) plateau, (c) maximum outside the experimental region, (d) minimum, and (e) saddle surfaces (Bezerra et al., 2008).

60 Zhang et al. (2012) successfully used RSM for optimising the extraction of tea polyphenols, (-)- epigallocatechin gallate and theanine from summer green tea. The combined effect of extraction temperature (axis A) and ratio (axis B), amongst others, on the yield of polyphenols (vertical axis/y axis) was visualised on a three dimensional response surface plot which demonstrated that a maximum response was attained within the experimental range (Figure 2.10).

Figure 2.10 Response surface plot demonstrating the effect of two process variables on the yield of summer green tea polyphenols (Zhang et al., 2012).

Two-dimensional contour plots can also be used to illustrate response surfaces, where plotlines that are close together (i.e. darker areas) indicate that minor alterations to the input factors are associated with significant changes in the response value. Significant interactions are represented by elliptical contour plots, while non-significant interactions are represented by circular plots. (Steinberg & Bursztyn, 2010). Zhu et al. (2014) made use of RSM for the optimisation of enzyme- assisted extraction and characterisation of polysaccharides from Hericium erinaceus, using two- dimensional contour plots for the illustration of the effect of three process variables on the extraction efficiency. Figure 2.11 shows the effect of pH (X1) and extraction time (X3) on the polysaccharide yield at a fixed temperature. The elliptical shape of the contour plot indicates a significant quadratic effect where the pH was 5.7 and the extraction time was 33.79 min.

A: Extraction temperature (⁰C) B: Ratio (mL 𝐠−𝟏) Yi e ld o f tea po ly p h e n o ls ( mg .g -1 )

61 Response surfaces are also typically visually illustrated as three-dimensional (3D) surface plots in combination with their corresponding two-dimensional (2D) contour plots (Figure 2.12). Chen et al. (2012) optimised the ultrasound assisted extraction of water-soluble polysaccharides from Boletus edulis mycelia and utilised combined plots to demonstrate the effect of process variables X1 (ratio of dried mycelia to water) and X2 (extraction time) on extraction yield. The elliptical shape of the contour plots shows that there was a significant interaction between these two variables, and the optimal ranges were identified as 1:55 (ratio of dried mycelia) and 8.4 min (extraction time).

Figure 2.11 Elliptical contour plot demonstrating the combined effect of pH (X1) and extraction time (X3) on the extraction yield of polysaccharides from Hericium erinaceus (Zhu et al., 2014).

62 Figure 2.12 Combined response surface plot (A) and corresponding contour plot (B) demonstrating effects of two variables (X1 = ratio of dried mycelia to water; X2 = extraction time) on extraction yield (Chen et al., 2012).

A standardised Pareto chart ranks the response, identifying the most critical parameters. It is also used to graphically demonstrate the significance of linear, quadratic and interaction effects. Factors and their interactions are represented by horizontal bars, and those bars which intersect the vertical line represent significant effects at 5% level of significance (P=0.05). The magnitude of its effect is proportional to the length of a horizontal bar, where a negative effect on the measured response is indicated by a negative value bar length (Das et al., 2014).

Silva et al. (2011) studied the effects of temperature, catalyst concentration, reaction time and molar ratio of alcohol in relation to oil on the transesterification of soybean oil with ethanol, and presented their ANOVA data in a standardised Pareto chart (Figure 2.13). The magnitude and significance (P<0.05) of the linear, quadratic and interaction effects of the four tested factors on the ethyl esters produced was depicted graphically in the form of a Pareto chart. Six bars with positive effect estimate values out of the fourteen horizontal bars representing these terms crossed the vertical black line which denotes the 5% significance level. The linear effects of molar ratio (1) and catalyst concentration (2) had the most significant positive effect on the response, as portrayed in the relative size of their bars and the standardised effect values (12.701 – 16.463). The interaction effects of molar ratio and time, molar ratio and temperature and the quadratic effects of temperature and time also had a significant, positive effect on the response, but these factors were considerably smaller than those of the two linear terms. However, the interaction effect of molar ratio and catalyst concentration, as well as the linear effect of temperature had a significant, negative effect on the

63 response. The six terms represented by the remaining six bars on the Pareto chart did not have significant effects on the response.

Figure 2.13 Standardised Pareto chart for effects on ethyl esters production. (L) is the linear and (Q) is the quadratic interaction of variables (Silva et al., 2011).

If several responses need to be optimised simultaneously, then multi-criteria methodology such as desirability profiling can be utilised. The levels of factors that result in maximum overall desirability for the process in terms of output are determined by this method (Bezerra et al., 2008). Therefore, an optimal compromise needs to be made since factors can have opposite effects on the measured responses. The individual desirability function for each response is determined by assigning a dimensionless number to the predicted scores, ranging from 0 (very undesirable) to 1 (very desirable), from which an overall desirability function can be obtained. Therefore, this method indicates the level of factors that display optimal overall desirability (Bezerra et al., 2008). Multi- response desirability profiling was utilised by Miller et al. (2017) for the optimisation of aspalathin and soluble solids yield in a hot water extraction process for green rooibos (Aspalathus linearis). Prediction profiles were generated which demonstrate the effect of the three independent variables under investigation (extraction time, extraction temperature and water-to-plant-material ratio) on the desirability of predicted aspalathin and soluble solids yield (Figure 2.14). The assessment of prediction reliability is aided by the blue horizontal lines on the prediction profiles which indicate 95% confidence intervals. The levels of the independent variables which would result in the most desirable (i.e. optimal) extract (soluble solids) and aspalathin yields are indicated by the vertical red lines that intersect the x- axes and apices of the desirability curves (green). In this study, the optimal levels were an extraction

64 time of 37 min, an extraction temperature of 93 °C and a water-to-plant material ratio of 23.4:1 (v.m- 1_{). After consideration of operation costs and energy input, the extraction time and water-to-plant} material ratio could be reduced to 30 min. and 10:1 v.m-1_{, respectively, without affecting the desired} responses. Therefore a range of values (29-31 min, 90-95 °C and 9-11:1 v.m-1_{) is more feasible from a} process control perspective as it allows for minor deviations in operations.

Figure 2.14 Multi-response desirability profiles for maximum extract yield (soluble solids) and aspalathin extraction efficiency in green rooibos (Aspalathus linearis) hot water extraction process (Miller et al., 2017).