## 2 Materials and Methods
### Use of R Statistical Environment for Estimating Youngs Moduli
If adding some new predictor to the model leads to just a tiny increase in \(R^{2}\), then it can be dropped from the model [14]. Adding such predictors to the model also leads to overfitting. Therefore, it is useful to have more measures of goodness of model fit. One such measure is an improved version of \(R^{2}\), called adjusted \(R^{2}\), which adjusts thepreviously described \(R^{2}\) by taking into account the size of the dataset and number of parameters in the model. It is given by (7). \[adj\ R^{2}=1-\frac{RSS/(n-d-1)}{TSS/(n-1)} \tag{7}\]  
where \(n\) is a number of observations and \(d\) number of included parameters. \(RSS\) always decreases as the number of predictors in the model increases, but \(RSS\)/ (\(n-d-1\)) may increase or decrease, due to the presence of \(d\) in the denominator. Once all of the correct variables have been included in the model, adding additional \"needless\" variables will lead to a very small decrease in \(RSS\). There are some measures of model performance that are more computationally exhaustive. One of them is a root mean square error of cross-validation (\(RMSECV\)). It is based on a cross-validation. The cross-validation procedure splits the dataset into a training set and test set finitely many times (the number of splitting is set in advance). The leave-one-out cross-validation puts a single observation in the test set, while other observations make up the training set. A statistical method is repeated \(n\) times on \(n-1\) observations, and each time a prediction is made for one particular observation that is left out. In every repeat, the squared error is calculated. The estimate for the overall mean squared error is the average of these \(n\) errors, denoted by \(CV_{(n)}\) (9) [14]. \(RMSECV\) is given by (8). \[RMSECV=\sqrt{CV_{(n)}} \tag{8}\]  
\[CV_{(n)}=\frac{1}{n}\ \sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2} \tag{9}\]  
where \(n\) is the number of observations, \(y_{i}\) is the observed \(i\)th value and \(\hat{y}_{i}\) is the predicted value for \(y_{i}\) using \(x_{i}\) and a statistical method fitted on the remaining \(n-1\) observations. The main advantage of this model accuracy measure is its unbiasedness. After choosing a few representative linear models according to these two measures of model accuracy, it is good to check to see which one of them is the best. Assume that there are two models: the one with \(k\) predictors complete model (10) and the reduced model (11) with one predictor less than the complete model. \[Y=\beta_{0}+\beta_{1}X_{1}+\ldots+\beta_{k-1}X_{k-1}+\beta_{k}X_{k}+\epsilon \tag{10}\]  
\[Y=\beta_{0}+\beta_{1}X_{1}+\ldots+\beta_{k-1}X_{k-1}+\epsilon \tag{11}\]  
Models of that kind are called nested models. It is of interest whether the distinguished predictor, when it is added to the model, contributes to the quality of the model. Hypotheses are set as follows (12) and (13):  
\[Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\ldots+\beta_{p}X_{p}+\epsilon \tag{3}\]0  
\[Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\ldots+\beta_{p}X_{p}+\epsilon \tag{3}\]1  
For this purpose, the F-test for nested models is performed. The test statistic is given by (14). \[Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\ldots+\beta_{p}X_{p}+\epsilon \tag{3}\]2  
where \(RSS_{\mathrm{C}}\) and \(RSS_{R}\) are the residual sum of squares for the complete and reduced model, respectively. Further, \(p_{\mathrm{C}}\) and \(p_{R}\) are the numbers of predictors in the complete and reduced model and \(n\) is the length of the dataset. If the null hypothesis is true, the test statistic has an F distribution with \((n-p_{R})\) and \((n-p_{C})\) degrees of freedom. For performing this statistical test in R the _\"anova\"_ function is used. The level of significance is set to 5%. The same \(F\) statistic can be used for testing the hypothesis that more than one coefficient \(\beta_{i}\) equals 0. The regression tree method divides the data into smaller groups that are more homogenous with respect to the response variable.