## 4 The Structure of the Hierarchy Model, the Questionnaire, and the Respondents
A questionnaire for ranking the sub-criteria by using the method of correlation was prepared by the authors. It was also translated into Russian language. An anonymous survey was carried out, with 52 questionnaires presented to passengers on the Vilnius-Moscow-Vilnius train. About 48% of the trip, which lasts for 14 h and 05 min (944 km), took place during the night. Respondent characteristics are presented in Table 1.  
The same passengers completed questionnaires and assessed the sub-criteria that determine the choice to travel by train as an alternative to aircraft. The results of this research were published in the article [11]. The number of respondents (52) was three times that of the criteria (sub-criteria) (15). Therefore, it was sufficient because \(m\geq n\). A description of 15 sub-criteria was presented in the questionnaire, and the respondents assigned different ranks to them (all the ranks had different assignable values). When applying expert research methods to assess the significance of criteria, there is a problem of determining the required (necessary) minimum number of experts. In practice, the mathematically unsound provision (principle) that the number of experts must be equal to or greater than the number of criteria is often observed. There is another common position that is often applied in practice, which maintains that the amount of data required for studies \(m\geq 30\) is also not substantiated, because in some cases the number m is sufficient (if the group range is small), while in other cases it is too small. The credibility of expert group assessments depends on the level of knowledge of individual experts and their number. Having assumed that the experts are accurate assessors, it can be stated that as their number increases, the reliability of the expertise of the whole group of experts \begin{table}
\begin{tabular}{c c} \hline \hline
**Items** & **Descriptions** \\ \hline Gender & 28 male respondents, 24 female respondents \\ Age & Average age is 46 years \\ Citizenship & Lithuanian (24), Ukrainian (1), Russian (26), passenger with \\  & dual Russian–Lithuanian citizenship took part (1) \\ Education & Most respondents (46) had higher education; 6 respondents had \\  & secondary education \\ Aim (goal) of trip & Business (21), tourism (15), visiting relatives and friends (10), \\  & medical treatment (5), research (1) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of 52 respondents who gave the judgement on ranks. (average of the opinion estimate) also increases. The minimum number of experts to be interviewed \(m_{\min}\) can be calculated according to the sample size formula [69]:  
\[m_{\min}=\frac{t^{2}\sigma_{R_{j}}^{2}}{\Delta_{j}^{2}}, \tag{10}\]  
where \(t\) is the value of \(t\) (Students) distribution, which depends on the probability taken to assess the importance of the criterion in deciding to go by train as an alternative to the bus. When the probability \(P=95\%\) (significance level \(\alpha=0.05\) for one-sided test), \(t=1.96\); \(\sigma_{R_{j}}\)--standard deviation of the ranks \(R_{j}\) of the evaluated \(j\)-th criterion; \(\Delta_{j}\) is the absolute error of passengers (respondents) rank values \(j\)-th criterion, indicating the accuracy of the survey results. The absolute error of the survey shows how much the average of the ranks \(\overline{R}_{j}\) calculated for the \(j\)-th criterion of \(m\) surveyed passengers may differ from the average of the population set \(\overline{R}_{jp}\) that would be determined by surveying all passengers. Due to the limited sample size \(m\), \(\overline{R}_{j}\) always differs from \(\overline{R}_{jp}\) by no more than plus or minus \(\Delta_{j}\). This difference is greater the smaller the m and the larger \(\sigma_{R_{j}}\).