## 3 Theory/Calculation
### Statistical Methods  
When modeling the dependence of selected variables, we used two basic statistical methods: the ANOVA method and multiple regression analysis. Statistical evaluation of the data is carried out in the R programming language. When testing statistical hypotheses, non-rejection (or rejection) of the hypothesis H0 is realized using the \(p\)-value. If the \(p\)-value \(\geq\alpha\), we do not reject the null hypothesis H0 (\(\alpha\) is the significance level). ANOVA is a method that allows you to compare the mean values of  
Figure 5: Cross-sectional view of the Ti layer: the angle of the columnar grains’ growth \(\alpha\); tube radius R = 110 mm; height from the evaporated target H = 335 mm. Figure 6: Cross-sectional view of the Ti layer: tube radius R = 110 mm, height of the evaporated target H = 305 mm. \(k\) independent basic sets. Its goal is to reveal whether the differences in the mean values of individual files are statistically significant or random. ANOVA examines the dependence of the quantitative variable Y on one or more qualitative variables (factors). Individual variants of the factor are referred to as factor levels. The basic assumptions for the ANOVA include normality and homoscedasticity. In practice, statistical tests are used to assess normality (e.g., the Shapiro-Wilk normality test, the Kolmogorov-Smirnov test, and others). If we want to compare the homogeneity (agreement) of variances of \(k\)-independent sets ([PERSON]'s test, [PERSON]'s test, [PERSON]'s test, and others). Let us consider two variables (factors) \(A\) and \(B\) and examine their influence on the variable \(Y\). If factor \(A\) has \(I\) levels and factor \(B\) has \(J\) levels (\(1\leq i\leq I\), \(1\leq j\leq J\)), then the number of all independent observations is \(I.J.\) In the case of a two-factor analysis of variance without interactions, we will start with a model in the following form  
\[y_{ij}=\mu+\tau_{i}+\theta_{j}+\varepsilon_{ij} \tag{1}\]  
for \(i=1\), \(2\), \(\cdots\), \(I\), \(j=1\), \(2\), \(\cdots\), \(J\), where \(\mu\) represents the overall mean of the monitored variable, \(\tau_{i}\) is the influence (effect) of factor A on the \(i\)-th level, the so-called line effects, \(\theta_{j}\) is the influence (effect) of factor \(B\) on the \(j\)-th level, so-called column effect and \(\varepsilon_{ij}\) is the implementation of random error, where \(\varepsilon_{ij}\sim N\left(0;\sigma^{2}\right)\). We test the influence of factor \(A\) with the hypothesis \(H_{0}^{A}:\tau_{1}=\tau_{2}=\cdots=\tau_{i}=0\) against the alternative hypothesis \(H_{1}^{A}:\)_at least one equality is not_\(\mathit{full}\)_. In an analogous way, we test the influence of factor \(B\), i.e., we test the hypothesis \(H_{0}^{B}:\theta_{1}=\theta_{2}=\cdots=\theta_{j}=0\) against \(H_{1}^{B}:\)_at least one equality is not_\(\mathit{full}\)_. As a measure of the size (magnitude, intensity) of the effect of Factor A, the coefficient \(\eta^{2}\) can be used in the dispersion analysis, which is similar to the coefficient of determination \(r^{2}\), which is used in the regression analysis. The following relation applies to the coefficient \(\eta^{2}\). \[\eta^{2}=\frac{SS_{A}}{SS_{T}} \tag{2}\]  
where \(SS_{A}\) represents the variability between the groups and \(SS_{T}\) the total variability. The closer the value of the coefficient is to \(1\), the greater the influence of the investigated factor A. When examining the output variable from the selected input variables, regression and correlation analysis are used. We will consider a classical linear regression model in the form  
\[Y=\vartheta_{0}+\sum_{j=1}^{k}\vartheta_{j}X_{j}+\varepsilon \tag{3}\]  
where \(\vartheta_{0}\) and \(\vartheta_{j}\) for \(j=1,\ 2,\cdots,\ k\) are model parameters, \(Y\) is the dependent variable, variables \(X_{j}\), \(j=1,\ 2,\cdots,\ k\) represent \(k\)-independent input variables and \(\varepsilon\) is a random error. We used the least squares method for parameter estimation of the selected regression model.