## 2 Materials and Methods
### Logistic Regression Model  
Logistic regression is a frequently used statistical method for classification problems when the variable is presented in a dichotomic scale form. It means that the predictive model of logistic regression determines the probability of one of two possible outcomes: Illness (number 1) or disease (number 2). For the given data, a logistic regression model was built to find different types of analyzed disease. The model coefficients were determined and were statistically significant at the level of \(\alpha=0.05\). The odds ratio values, defined as the ratio of the probability of success to the probability of failure, were then calculated. The main advantage of the odds ratio, as compared to conventional probability, is that the odds ratio assumes values in the range \((0,+\infty)\) for a \(p\) range from 0 to 1, and the logarithm value of the field \((-\infty,+\infty)\). This means that we can use regression methods not limited to a range (0-1), such as linear regression, to estimate the log of chance in a regression model [22; 23]. For the odds ratio, we assumed a 95% confidence interval; the span is based on the number of patients in the study group. The odds ratio can also be calculated taking into account the division of the respondents into two separate groups using Formula (1):  
\[OR_{AxB}=\frac{S(A)}{S(B)}=\frac{p(A)}{1-p(A)}\div\frac{p(B)}{1-p(B)}, \tag{1}\]  
where \(p(A)\) is the probability of an event (disease) \(A\) and \(p(B)\) is the probability of an event (disease) \(B\). We interpret this measure as follows [22; 23; 24]:  
* If \(OR>1\), then in the first group, the occurrence of the event is more likely. * If \(OR<1\), then in the second group, the occurrence of the event is more likely. * If \(OR=1\), then the event is equally likely in both groups. Transformation function on the logarithm of the probability of chance is called logit (Formula (2)):  
\[logit(P)=ln\frac{p}{1-p}=ln(p)-ln(1-p),\text{\ where\ }p=\frac{e^{logit(p)}}{1+e^{ logit(p)}}=\frac{1}{1+e^{logit(p)}}. \tag{2}\]  
The logistic model is based on the function \(f\) (\(z\)) (Formula (3)) [22]:  
\[f(z)=\ \frac{e^{z}}{1+e^{z}}=\frac{1}{1+e^{-z}}. \tag{3}\]  
The predictive model formulates the probability of malignancy with the probability of benign tumors. The constructed model also allows us to observe which of the tested independent variables influence the dependent variable explained on a dichotomous scale. Conditional probability for dependent variable Y assumes a value of 1 for the value of the independent variables \(x_{0},x_{1},\ldots,x_{k}\) and is described as Formula (4) [23]:  
\[P(Y=1|x_{0},x_{1},\ldots,x_{k})=\frac{e^{(a_{0}+\sum_{i=1}^{k}a_{i}x_{i})}}{1+ e^{(a_{0}+\sum_{i=1}^{k}a_{i}x_{i})}}, \tag{4}\]  
where \(a_{i},i=0\ldots k\) are regression coefficients and \(x_{1},x_{2},\ldots,x_{3}\) represent independent variables. Values of estimators are calculated using the most reliable method. The greater reliability of a model, the more likely it is that the variable will appear in the sample, which further means better matching the model to the data [22; 23]. In our work, we used the quasi-Newton method. The function is estimated at various points to estimate the first and second order derivatives. Then, the obtained information is used to minimize the loss function value. All variables which were taken into the study were subjected to significance tests. The Mann-Whitney test was used for comparison of the CD group with the UC group in cases where parameters were not shown to be in a normal distribution. We applied Student's \(t\)-test in cases of compatibility with normal distribution and homogeneity of variance, while the Cochran-Cox test was used in cases when compliance with the normal distribution had been shown, but there was no homogeneity of variance. In cases of comparison of data on the qualitative scale, a chi-square test was used. The Shapiro-Wilk test was applied to check compliance with the normal distribution, and the Leven test was used to test homogeneity of variance.