## 2 Methodology
### Bayes Factors
In other words, \(H_{0}:\ \theta_{1}=\theta_{2}\) and \(H_{1}:\ \theta_{1}\
eq\theta_{2}\), where \(\theta_{1}\) represents the affirmative responses to a Likert-scale question converted to binary and \(\theta_{2}\) is the ratio of affirmative responses in binary-scale surveys. This could be rewritten as \(H_{0}:\ \delta=0\) and \(H_{1}:\ \delta\
eq 0\) where \(\delta=\theta_{2}-\theta_{1}\). The a priori belief in both ratios is modeled by a vague prior distribution, such that \(\theta_{1}\sim Beta(1,1)\) and \(\theta_{2}\sim Beta(1,1)\). It is called vague in the sense that it reflects a belief that is very weak and easily molded by exposure to new information. In other words, the updating factor will have more strength than the prior distribution and will leave a greater role for the calculation of the a posteriori belief or distribution. Speaking of the updating or likelihood factor, the transformed Likert survey is modeled as \(s_{1}\sim Binomial(\theta_{1},\ n_{1})\) and the binary format survey as \(s_{2}\sim Binomial(\theta_{2},\ n_{2})\), where \(n\) represents the total number of responses and \(s\) the number of positive responses. Figure 4: Bayesian graphical model of the survey data. Typically, the use of Bayesian inference and the calculation of Bayes factors are associated with numerical computation methods. This is caused by the appearance of integrals when probability distributions are used to represent beliefs and likelihoods. However, comparisons similar to the present study have been performed by developing a simple equation for calculating the Bayes factor for each question asked [25,32], where \(n_{1}\) denotes the total number of responses collected in the transformed Likert survey, \(n_{2}\) denotes the total number of responses collected in the binary survey, \(s_{1}\) denotes the number of affirmative responses in the transformed Likert survey, and \(s_{2}\) denotes the total number of affirmative responses in the binary survey. This will be the expression used in our calculation due to its simplicity. \[BF_{01}=\frac{\binom{n_{1}}{s_{1}}\binom{n_{2}}{s_{2}}}{\binom{n_{1}+n_{2}}{ s_{1}+s_{2}}}\frac{\left(n_{1}+1\right)\left(n_{2}+1\right)}{n_{1}+n_{2}+1} \tag{3}\]  
In a general case, there is an easy way to estimate the Bayes factor. We will consider the hypotheses in a general manner as \(H_{0}:\phi=\phi_{0}\) and \(H_{1}:\phi\
eq\phi_{0}\). The Bayes factor can be obtained by only considering \(H_{1}\) (4) and dividing the height of the posterior for \(\delta\) by the height of the prior for \(\delta\) at the point of interest. This surprising method was first published by [33] and is called the Savage-Dickey density ratio method. The magnitude in this method has to be estimated using numerical methods. The revision of ([PERSON] et al., 2010) is recommended for a simple explanation with written examples of WinBUGS. Nevertheless, in the present work, thanks to the model used, Equation (3) was employed without the need for numerical methods. \[BF_{01}=\frac{p(\phi=\phi_{0}|D,H_{1})}{p(\phi=\phi_{0}|H_{1})} \tag{4}\]  
After ascertaining the Bayes factor for each question, the null hypothesis posterior probability was also calculated. It helps to measure the confidence in what the evidence says about the null hypothesis. In other words, how strongly we believe in \(H_{0}:\delta=0\). In order to calculate this, we use Equation (5), where \(\Pr(H_{0})\) represents the prior belief in \(H_{0}\) being true. In the equation, \(\Pr\) represents probability and \(p\) represents the probability distribution function. Since there are only two options and no prior leaning towards either the null or alternative hypothesis, the value of \(\Pr(H_{0})=0.5\) is chosen. Because there are only two hypotheses, \(\Pr(H_{1})+\Pr(H_{0})=1\), and therefore, \(\Pr(H_{1})=1-\Pr(H_{0})\). This equation can also be solved using probability distribution functions as the prior. In fact, Equation (3) is derived assuming a Beta(1,1) prior.