# Estimation of Association between Healthcare System Efficiency and Policy Factors for Public Health
## 2 Materials and Methods
### Mann-Whitney U Test and Tobit Regression Model  
We assumed that the difference mainly originated from policy factors (microscopic) that are highly influenced by its healthcare system (macroscopic). Therefore, we introduced ten healthcare policy indicators categorized by [PERSON] et al. (2018) and used them to determine the microscopic causes of differences in efficiency. DEA is a representative non-parametric methodology to estimate the relative efficiency score. Therefore, we could not apply the t-test since this method does not comply with normal distribution. In this sense, the Mann-Whitney U test is conducted to verify that a statistically significant difference existed between the two independent groups. In this study, countries were divided into two groups by the median efficiency, and then the Mann-Whitney U test was used to confirm whether there was a difference in the score of the policy indicators between the two groups. Next, we chose three policy indicators that showed a significant difference between the two groups. After that, policy indicators with significant differences were selected as an independent variable, and the DEA efficiency score was selected as a dependent variable, and the influence of the corresponding policy index was analyzed through regression analysis. Since the DEA efficiency score was truncated from zero to one, the ordinary least square (OLS) regression resulted in biased and inconsistent results ([PERSON] et al., 2018). Additionally, the Tobit Regression Model was used when the dependent variables (efficiency score of each DMU) were bounded from above, below, or both. Therefore, the Tobit regression model was introduced in this study.  
The formula of the general Tobit regression model is as follows:  
\[\begin{cases}\text{Y}_{\text{t}}=\text{X}_{\text{t}}\beta+\mu_{\text{t}}&\text {if}\,\text{X}_{\text{t}}\beta+\mu_{\text{t}}>0\\ 0&\text{if}\,\text{X}_{\text{t}}\beta+\mu_{\text{t}}\leq 0,\\ \text{t}=1,\,2,\,\ldots,\,\text{N},\end{cases} \tag{1}\]  
where Yt is the dependent variable; Xt is the independent variable vector; \(\beta\) is the unknown coefficients; and \(\mu_{\text{t}}\) is the independent distribution error assumed to be a normal distribution with an average of zero.  
The efficiency score in this study can be defined as follows:  
\[\text{E}_{\text{i}}=\alpha+\beta_{1}\text{UCI}+\beta_{2}\text{UI}+\beta_{3} \text{DC}+\mu_{\text{i}}, \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Regulation** & **Financing** & **Provision** & **Countries** \\ \hline NHS & State & State & State & Denmark, Finland, Iceland, Norway, Sweden, Portugal, Spain, UK \\ NHI & State & State & Private & Australia, Canada, Ireland, New Zealand, Italy \\ SHI & Societal & Societal & Private & Austria, Germany, Luxembourg, Switzerland \\ PHS & Private & Private & Private & USA \\ ESHI & State & Societal & Private & Belgium, Estonia, France, Czech Republic, Hungary, the Netherlands, Poland, Slovakia, Israel, Japan, Korea \\ \hline \hline \end{tabular}
\end{table}
Table 1: Classification of the OECD healthcare systems (Chile, Greece, Latvia, Mexico, Slovenia, and Turkey are missing due to lack of data). Source: [PERSON] et al. (2018).  
where E\({}_{i}\) represents the efficiency score; UCI, UI, DC are the three policy factors used as independent variables (each abbreviation represents user choice of insurers, user information, and decentralization); i represents the number of countries; \(\alpha\) represents a constant; and \(\mu_{i}\) represents an error term. We chose these three factors (UCI, UI, and DC) to find the effect of policy factors on the efficiency of the healthcare system. First, UCI was selected for the basic insurance coverage and UI was selected for the quality on prices of various healthcare systems. Finally, DC was chosen to estimate the effect of decentralization on sub-national governments. More specific information about how these three policy factors were selected are described in Section 3.