## 3 Results
### Determination of Muffin Texture
Hardness (N) = 8.81444 - 0.399673 \(\times\) SS + 0.0348701 \(\times\) SO + 0.288479 \(\times\) T + 0.218658 \(\times\) SS \(\times\) SO - 0.186782 \(\times\) SS \(\times\) T - 0.120904 \(\times\) SS \(\times\) T - 0.68025 \(\times\) SS \(-\) 0.52308 \(\times\) SO \({}^{2}\) - 0.0662622 \(\times\) T\({}^{2}\)  
\[\begin{array}{c}\text{Chewiness (N) = 5.93033 + 0.0587074 \(\times\) SS - 0.045006 \(\times\) SO - 0.100777 \(\times\) T - 0.0337358 \(\times\) SS \(\times\) SO - 0.207557 \(\times\) SS \(\times\) T - 0.550314 \(\times\) SS \(\times\) T + 0.0881037 \(\times\) SS \({}^{2}\) - 0.472288 \(\times\) SO \({}^{2}\) + 0.0571753 \(\times\) T\({}^{2}\)  
\[\begin{array}{c}\text{Cohesiveness (/) = 0.643195 - 0.00534385 \(\times\) SS + 0.0214696 \(\times\) SO + 0.0177404 \(\times\) SS - 0.0362734 \(\times\) SS \(\times\) SO + 0.0426371 \(\times\) SS \(\times\) T - 0.0254079 \(\times\) SO \(\times\) T + 0.0521497 \(\times\) SS \({}^{2}\) - 0.00429425 \(\times\) SO \({}^{2}\) + 0.00680574 \(\times\) T\({}^{2}\)  
The linear term T has the greatest positive effect on hardness (Equation (3)), followed by the linear terms SS (substituted sugar) \(>\) the interaction of the linear terms SS \(\times\) SO (substituted sugar \(\times\) substituted oil) \(>\) SO (substituted oil). On the other hand, responsible for the greatest negative effect on hardness is followed by the quadratic terms SO\({}^{2}\) (substituted oil) \(>\) linear term SS \(>\) SS\({}^{2}\) (substituted sugar) and, respectively, the interaction of the linear terms SS \(\times\) T, SO \(\times\) T and the quadratic term T\({}^{2}\). Regarding chewability (Equation (4)), the quadratic term SS\({}^{2}\) has the greatest positive effect, followed by the linear term SS (substituted sugar) \(>\) T (temperature), and the interaction of the terms has the greatest negative effect linear SS \(>\) T\({}^{2}\) (substituted  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Run**} & \multicolumn{2}{c}{**Independent Variables**} & \multicolumn{2}{c}{**Measured Response**} & \multicolumn{2}{c}{**Predicted Response**} \\ \cline{2-10}  & **SS, \%** & **SO, \%** & \( \begin{array}{c}\text{T}_{c}\\ \text{}^{\circ}\text{C}\end{array} \)** & \( \begin{array}{c}\text{Hardness,}\\ \text{N}\end{array} \)** & \( \begin{array}{c}\text{C}\text{C}\text{hewiness,}\\ \text{N}\end{array} \)** & \( \begin{array}{c}\text{Cohesiveness,}\\ \text{-}\end{array} \)** & \( \begin{array}{c}\text{Hardness,}\\ \text{N}\end{array} \)** & \( \begin{array}{c}\text{C}\text{hewiness,}\\ \text{-}\end{array} \)** \\ \hline
1 & 75 & 25 & 200 & 7.86 & 5.51 & 0.63 & _7.97_ & 5.54 & 0.63 \\ \hline
2 & 25 & 75 & 200 & 8.3 & 6.01 & 0.71 & 8.41 & 5.90  
sugar \(\times\) temperature). In the case of cohesiveness (Equation (5)), the positive effect with the greatest impact is represented by quadratic term ZS\({}^{2}\), followed by the linear terms US > ZS > ZS \(\times\) T.  
Hardness, the force required to deform or crush a muffin, can be influenced by the ingredients or the proportion of the ingredients and it is an important parameter that can affect the acceptability and quality of the final product. Regarding the substitution of sugar and oil with apple puree from muffins, the hardness did not show significant differences (\(p\) < 0.01), and a similar result was obtained by [PERSON] and [PERSON] (2013) following the substitution of fat from cakes with inulin and pectin [51]. The hardness showed an increase in the case of samples where the percentage of substitution was around 50%, and a similar variation was obtained by [PERSON] et al. (2019), in the case of cakes where fat was substituted with psyllium fiber, and [PERSON] et al. (2018), who substituted sugar and fat in cakes with rebaudioside A and inulin [36; 44]. Pectin from apple puree, used as a substitute for sugar and fat, contributed to obtaining a hardness close to that of the control sample, similar to results obtained by [PERSON] et al. (2014), by using pectin as a substitute [10].