## 4 Results
### Effect of Factors on Monitored Variables (Two-Factor ANOVA)  
Let us consider two factors (variables) \(X_{1}\) and \(X_{2}\) and examine their influence on the observed result. Variable \(X_{1}\) is the height of the measured place above the target, which has five levels (\(X_{11}=305\) mm, \(X_{12}=335\) mm, \(X_{13}=365\) mm, \(X_{14}=395\) mm, \(X_{15}=425\) mm), factor \(X_{2}\) is the tube radius, which also has five levels (\(X_{21}=100\) mm, \(X_{22}=110\) mm, \(X_{23}=120\) mm, \(X_{24}=130\) mm, \(X_{25}=140\) mm). In the first step, we monitored whether the factors \(X_{1}\) and \(X_{2}\) affected the variable \(Y_{1}\), which represents the grain growth angle. The verification of the normality of the samples is realized using the Shapiro-Wilk normality test. For the case of the grain growth angle at the height of the measured place above the target of 305 mm, the \(p\)-value is 0.979. Testing results for each height of the measured location above the target, or for each tube radius, are listed in Table 4.  
Since the \(p\)-value \(>\alpha\) for each sample set, we do not reject the hypothesis H\({}_{0}\) about the normality of the individual base sets. We can state that all the samples exhibited a normal distribution. We will verify the condition of homogeneity of variances using the Bartlett test. The H\({}_{0}\) hypothesis is that \"the variances are equal across all groups\". In the case of factor A, the \(p\)-value is 0.896, and in the case of factor B, the \(p\)-value is 0.987. The results of the test show that we do not reject the hypothesis of a coincidence of variances (\(p\)-value \(>\alpha\)). In an analogous way, the fulfillment of the assumptions of variance analysis is verified for all samples. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline Height [mm] & \(X_{11}=305\) & \(X_{12}=335\) & \(X_{13}=365\) & \(X_{14}=395\) & \(X_{15}=425\) \\ \hline \(p\)-value & 0.979 & 0.776 & 0.881 & 0.899 & 0.833 \\ Decision & Not rejected H\({}_{0}\) & Not rejected H\({}_{0}\) & Not rejected H\({}_{0}\) & Not rejected H\({}_{0}\) & Not rejected H\({}_{0}\) \\ Tube radius [mm] & \(X_{21}=100\) & \(X_{22}=110\) & \(X_{23}=120\) & \(X_{24}=130\) & \(X_{25}=140\) \\ \hline \(p\)-value & 0.602 & 0.373 & 0.616 & 0.858 & 0.665 \\ Decision & Not rejected H\({}_{0}\) & Not rejected H\({}_{0}\) & Not rejected H\({}_{0}\) & Not rejected H\({}_{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of verification of normality assumptions—grain growth angle (\(\alpha=0.05\)). Figure 7: Boxplot—grain growth angle depending on the height of the measured place above the target or from the tube radius. The prerequisites for using analysis of variance are met; therefore, a two-factor analysis of variance will be performed. We can formulate two hypotheses for the given problem. The null hypothesis is H\({}_{0}\): the influence of factor X\({}_{1}\) (or X\({}_{2}\)) is insignificant on the value of variable Y\({}_{1}\), against H\({}_{1}\): there is an influence of factor X\({}_{1}\) (or X\({}_{2}\)) on the value of variable Y\({}_{1}\). The results of the variance analysis are in Table 5.  
Since the resulting \(p\)-values are significantly lower than the significance level (\(p\)-value (X\({}_{1}\)) = 4.3 \(\times\) 10\({}^{-12}\), \(p\)-value (X\({}_{1}\)) = 3.2 \(\times\) 10\({}^{-10}\), \(\alpha\) = 0.05), the hypothesis H\({}_{0}\) was rejected. The results of the analysis show that there is a demonstrable influence of both factors on the grain growth angle. We will use the coefficient \(\eta^{2}\) as a measure of the effect size of the factor. The influence of the height of the measured place above the target (factor X\({}_{1}\)) on the variability of the grain growth angle is 62.5%, and the influence of the tube radius (factor X\({}_{2}\)) is 35.6%. In an analogous way, the influence of factors X\({}_{1}\) and X\({}_{2}\) on other variables Y\({}_{2}\) to Y\({}_{5}\) was monitored.