*3.2. Statistical Analysis*

The results of the ANOVA on the yield of the reaction response (see Table S1a) indicated that the factors (A, B, and C) individually did not meet the criteria that represented a

linear model of the data for the response of interest. Meanwhile, the value of the *p* statistic (significance value of 0.05) was found to be 0.037, which was less than 0.05. This result confirmed the prior result obtained through the F statistic. Therefore, the performance of the polymerization reaction for hydrogel synthesis was independent of factors A, B, and C. Furthermore, when analyzing factor B (CNF w w−<sup>1</sup> %), it was observed that the value of the F statistic was greater than that of Fcritical and the *p* statistic (0.008 < 0.05), considering the results obtained from the double and triple interactions of the factors. The values of the F statistic were less than the Fcritical values, and the values of the *p* statistic were less than 0.05.

The ANOVA results for the response of the water absorption capacity (see Table S1b), including all the factors (individually as well as in their binary and ternary combinations), generated an F statistic value that was less than the Fcritical statistic value, and the *p* values were greater than the significance value of 0.05. All parameters significantly influenced the water absorption capacity.

The ANOVA results for the response of the effective cross-link density of a cross-linked structure can be observed in Table S1c, where the F statistic value was greater than the Fcritical value; similarly, the value of the *p*-statistic (0.027) was less than the significance value (0.05). Considering the results of the individual factors and their double combinations when comparing the values of the F statistic with their Fcritical values, F < Fcritical. Furthermore, the values of the *p* statistic for these factors and their double combinations generated values less than 0.05; therefore, they were significant and influenced the response of the effective cross-link density of a cross-linked structure.

To establish the relationship between the investigated factors and yield of the reaction, water absorption capacity, and the effective cross-link density of the cross-linked structure, the data were graphically displayed using standardized Pareto charts. The changes in the level of a factor that influenced the different system responses were represented as the effects of the factor. The standardized Pareto chart compares the absolute values and the significance of the effects.

Figure 2 shows standardized Pareto charts for the response variables, highlighting that the critical value obtained from the Student's *t*-test statistic is *t*0.25,8 = 2.776.

From the ANOVA analysis for the three factors, it was found that, for the treatment of the data obtained for the reaction yield, only the CNF factor w w−<sup>1</sup> % generated a value of the *p* statistic at 0.008 << 0.05; therefore, this factor significantly influences the performance results. Furthermore, the error had eight degrees of freedom, and we worked with α = 0.05; therefore, the critical value obtained from the Student's *t*-test was *t*0.25,8 = 2.776. Figure 2a shows that the bar of the CNF factor w w<sup>−</sup>1% exceeds the critical value of 2.776; therefore, it was concluded that the reaction performance was affected significantly. Other factors did not significantly affect the yield of the reaction. The ANOVA analysis of the results obtained for the effective cross-link density of a cross-linked structure and the water absorption capacity also presented eight degrees of freedom in the error value; therefore, the critical value of the Student's *t*-test statistic was *t*0.25,8 = 2.776. It can be observed from Figure 2b that none of the factors or the combination of these factors had a significant effect on the water absorption capacity, as none of the bars exceeded the critical value of *t*0.25,8 = 2.776.

The effective cross-link density of the cross-linked structure was not strongly affected by the interactions of the three factors (MBA mol% × APS mol% × CNF w w−<sup>1</sup> %) because the value shown in Figure 2c does not exceed the critical value of *t*0.25,8 = 2.776.

**Figure 2.** Result of the standardized Pareto analysis for the response variables (parameters A: MBA mol%; B: APS mol%; and C: CNF w w<sup>−</sup>1%): (**a**) yield of the reaction, (**b**) water absorption capacity, and (**c**) the effective cross-link density of the cross-linked structure.

The analysis of the effect of APS mol% in these tests revealed a random behavior. APS affects the length of the polymer chains, making them longer or shorter depending on the concentration used. Increasing the initiator percentage produces more free radicals that accelerate the chain termination reaction, resulting in shorter chains and thus increasing the possibilities for water entry [39]. The incorporation of CNF helps ensure that the hydration of the polymer does not lead to swelling drastically, since the fibrils become a part of the interpolymeric network when physically mixed and by electrostatic interactions, and continue to absorb significant percentages of water. An increase in hydrogel swelling is observed when the concentration of CNF is increased, which may be explained by the increased number of carboxyl groups in the hydrogels [40].
