*3.3. Statistical Analysis and Optimization*

A single block D-optimal mixtures design was launched using the design constraints summarized in Table 2 on Design expert software version 12.0 software (Stat-Ease Inc., Minneapolis, MN, USA) to estimate the effects of the proportion of each of the surfactant-mixture components. To facilitate an understanding of the relationship(s) between one or more measured responses to input factors, a D-optimal design uses a sequential strategy that results in either first or second order polynomial mathematical relationships that can be best fit to statistical models such as quadratic, linear and cubic models for point prediction or optimization. The design produced was a simplex-lattice design with A + B + C = 1 or surfactant mixtures of 100% [46] which were then used for the manufacture of 10% *m*/*m* flaxseed oil containing nanoemulsions. A total of 16 runs with the compositions are summarized in Table 4 and the resulting droplet sizes, PDI and zeta potential. The most common empirical models fitted to experimental data include linear, quadratic or cubic models which increase in complexity of the polynomial from a 1st, 2nd, to 3rd degree, respectively. A 4th degree polynomial

for systems involving composition, with the sum of the proportions by volume and weight has also been applied and reported [47]. Special quartic models are useful for modelling data generated from multicomponent mixtures and can be used to estimate multiple effects and the curvature of a response surface in the interior of a triangle to produce contour like effects [48,49]. All models were automatically fit by the design software for these data including linear, quadratic, cubic, special cubic, quartic and special quartic models were applied to the data then analyzed for the response variables monitored. The ANOVA analysis results are summarized in Table 5 for the suggested models and Table S2 in the Supplementary Material for all the models tested. The predicted residual sum of squares (PRESS) was used to establish the suitability of each model in respect of data fitting and the model with the lowest value for PRESS was identified as suitable for that response. The PRESS value is said to analyze the prediction ability of models and the model with the minimum PRESS is usually considered the best predictive model for a set of data [50]. ANOVA analysis following fitting of responses to each models are summarized in Table 5 to which droplet size and PDI were best fit to special quartic models while zeta potential was best fit to a linear model. Prior to predicting the optimized nanoemulsion formulation, composition residual analysis was undertaken to confirm that the assumptions for ANOVA analysis had been met. For this purpose, diagnostic plots viz., box Cox plots for residuals were plotted for all three responses and confirmed that data transformation was not required for droplet size and zeta potential and PDI. Diagnostic box Cox plots are depicted in Figures S3, S4 and S5 respectively in the Supplementary Material. When the statistical data are analyzed, input variables Span® 20 (A) Tween® 80 (B) and ethanol (C) are used to produce effects terms; A, B, C, AB, AC, BC, A2BC, AB2C, ABC<sup>2</sup> which are tested to asses which ones are significantly different from 0, i.e., >0 therefore estimated to give coefficient values of correlation for prediction of a specific response. The largest positive coefficient of model terms represents the model term with the largest effect on a specific response.


**Table 4.** Surfactant-mixture compositions generated by the D-optimal design and the experimental responses of 10% *m*/*m* flaxseed oil nanoemulsion in each run observed.


**Table 5.** ANOVA data for D-optimal responses and best-fit model.
