## 2 Methodology
### Bayes Factors  
People update their knowledge as they have new experiences or obtain new data. In childhood, it takes a few attempts to understand that, unlike cartoons, cats cannot talk. A child does not undertake such an investigation without context: the child carries a degree of prior information based on previous experiences. This prior information helps to interpret the new data so that, weighing the new and old, the child can develop his/her updated information (posterior information). The child continues to refine this information while gathering new evidence. Allowing the posterior to balance prior knowledge and data is critical to the knowledge-building process. As new data are collected, the prior information  
Figure 3: Threshold selection for transformation from Likert to dichotomous scale. loses weight. So even if two people had different prior knowledge, they would come to the same conclusion after collecting sufficient data. In the context of hypothesis testing, at the beginning, the observer has a set of two rival hypotheses. In this case, the two hypotheses are the null hypothesis \(H_{0}\) and the alternative hypothesis \(H_{1}\). The null hypothesis states that there is no difference between two sets of data or no effect between before and after treatment. These two datasets are usually associated with the subjects of a treatment and a control group. In the case of this paper, the datasets are the binary survey success ratio \(\theta_{bin}\) and the Likert-transformed survey \(\theta_{likert}\). Therefore, the null hypothesis is stated as \(H_{0}:\ \theta_{bin}=\theta_{likert}\), and the alternative as \(H_{1}:\ \theta_{bin}\
eq\ \theta_{likert}\). These can be rewritten as \(H_{0}:\ \theta_{bin}-\theta_{likert}=0\) and \(H_{1}:\ \theta_{bin}-\theta_{likert}\
eq 0\). Both options can be assumed to be equally plausible, with \(p(H_{0})=p(H_{1})=0.5\). By using information from the data, the probabilities are updated: the hypothesis that best describes the data increases in credibility and the one that worst reflects the data suffers a decrease [17]. This is a process similar to reinforced learning. This updating is performed using [PERSON] rule (1). Below, we describe this rule for one of the hypotheses. In it, \(p(H_{0})\) is the a priori belief in \(H_{0}\), \(p(H_{0}|data)\), the a posteriori belief given the data, and finally, \(p(data|H_{0})/p(data)\) is the predictive update factor. The a posteriori belief reflects the probability that the observed data conform to that hypothesis. \[p(H_{0}|data)=p(H_{0})\ \frac{p(data|H_{0})}{p(data)} \tag{1}\]  
To compare two hypotheses, [PERSON] rule can be written as an odds ratio [22; 29]. This equation shows that the change from the prior hypothesis odds \(p(H_{1})/p(H_{0})\) to the posterior hypothesis odds \(p(H_{1}|data)/p(H_{0}|data)\) occurs due to the ratio of the predictive update factors \(p(data|H_{1})/p(data|H_{0})\), commonly known as the Bayes factor. The Bayes factor is abbreviated as \(BF_{10}\), where the subscripts indicate the hypotheses placed in the numerator and denominator positions, respectively. \[BF_{10}=\frac{p(data|H_{1})}{p(data|H_{0})}=\frac{p(H_{0})\ p(H_{1}|data)}{p( H_{1})\ p(H_{0}|data)} \tag{2}\]  
For example, suppose that the rival hypotheses are equally plausible beforehand; that is, \(p(H_{0})=p(H_{1})=0.5\). Then, the odds ratio of the prior hypotheses is equal to one. If the observed data are 10 times more likely under \(H_{0}\) than under \(H_{1}\), the assumption would be that the update factor is 10 times larger for \(H_{0}\). This would cause the posterior probabilities of \(H_{0}\) also to be 10 times greater. Since the hypotheses are mutually exclusive, with \(p(H_{0})+p(H_{1})=1\), this means that the data have increased the probability of \(H_{0}\) from 0.5 (prior probability of \(H_{0}\)) to \(10/11\ \approx\ 0.91\) (posterior probability of \(H_{0}\)).