## 2 Materials and Methods
### Data Obtained by Experiment and Their Formal Interpretation
#### 2.2.3 Correlation Analysis and Regression Dependencies Testing  
In terms of objectivity, validity, and interpretation of the regression dependencies and related point estimates obtained, it is necessary to test them with relevant hypotheses; it is also desirable to determine confidence intervals for these parameters. An important task of the regression model of the sought dependence is also the prediction of new values of the investigated quantity for the prediction interval. In general, the prediction interval is  
Figure 12: Graphical representation of the residuals for the pressure values against the actual obtained \(p\) values for the applied polynomial model of the third degree of dependence \(p=f(t)\). Figure 13: Applied polynomial regression models of the second degree for the dependencies \(\Delta\sigma_{t}=f(\Delta p)\) and \(\Delta\sigma_{m}=f(\Delta p)\) and marking of their 95% prediction interval. wider than the confidence interval. Figures 11 and 13 show the 95% prediction intervals of applied RD 1, 2 and 3. In general, hypothesis testing will be understood as a procedure in which, based on the relevant analysis, a decision is made whether, at the selected level of significance \(\alpha\), i.e., with the selected reliability 1-\(\alpha\), or 100-\(\alpha\) [%], the given hypothesis will or will not be rejected. In case of rejection, the opposite hypothesis is accepted. Thus, in testing, the principle of two opposing hypotheses is chosen, one of which asserts something, and the other rejects that assertion. Most of the time, the so-called null hypothesis \(H_{0}\) is chosen as the primary one, consisting of a statement that is assumed considering the circumstances; it is opposed by the so-called alternative hypothesis \(H_{1}\), which is accepted after \(H_{0}\) is rejected. The significance level \(\alpha=0.05\) is chosen if the risk of error is sufficiently small. In some cases, if incorrect acceptance of the alternative hypothesis \(H_{1}\) could have some serious consequences, a lower level of \(\alpha=0.01\) is chosen. To evaluate a certain RD, the selective residual variance, also called mean squared error _MSE_, obtained from _SSE_ is first applied. _MSE_ represents a point or unbiased estimate of the interval variance \(\sigma^{2}\); it expresses the measure of how, on average, the theoretically obtained value differs from its empirical equivalent, that is, how the values of the analysed variable are scattered around the regression function. The point estimate of the standard deviation \(\sigma\) is the root mean squared error _RMSE_ as the square root of the _MSE_[15, 16, 17, 18]. Using the _SSE_, _SSR_, and _SST_ values, it is then possible to express the \(R\)-squared or coefficient of determination \(R^{2}\). \(R^{2}\) reaches values in the range of 0 to 1. For values close to 1, especially with a sufficiently large set of values, it is possible to infer the mutual dependence of two quantities and a well-chosen regression function. On the contrary, values approaching zero indicate a weak interdependence of the two quantities or an inappropriately chosen regression function. The disadvantage of the \(R^{2}\) parameter is that it generally does not have the property of symmetry. This means that when measuring the closeness of the dependence of two quantities, the closeness of the first quantity to the second is or can be different from the closeness of the second quantity to the first. Another disadvantage of the \(R^{2}\) parameter is that for small sample ranges, this estimate is biased and overestimates the fit of the regression model. In such a case, an unbiased estimate of the coefficient of determination, the so-called modified or corrected coefficient of determination--the adjusted \(R\)-square \(R^{2}\)--is then applied. A number of sophisticated statistical tests are available to test the reliability and validity of the proposed regression models and specific RD in more detail, to investigate their specific properties, and to detect even less obvious shortcomings.