How efficient are Greek hospitals? A case study using a double bootstrap DEA approach

Abstract

The purpose of this study was to measure Greek hospital performance using different input–output combinations, and to identify the factors that influence their efficiency thus providing policy makers with valuable input for the decision-making process. Using a unique dataset, we estimated the productive efficiency of each hospital through a bootstrapped data envelopment analysis (DEA) approach. In a second stage, we explored, using a bootstrapped truncated regression, the impact of environmental factors on hospitals’ technical and scale efficiency. Our results reveal that over 80 % of the examined hospitals appear to have a technical efficiency lower than 0.8, while the majority appear to be scale efficient. Moreover, efficiency performance differed with inclusion of medical examinations as an additional variable. On the other hand, bed occupancy ratio appeared to affect both technical and scale efficiency in a rather interesting way, while the adoption of advanced medical equipment and the type of hospital improves scale and technical efficiency, correspondingly. The findings of this study on Greek hospitals’ performance are not encouraging. Furthermore, our results raise questions regarding the number of hospitals that should operate, and which type of hospital is more efficient. Finally, the results indicate the role of medical equipment in performance, confirming its misallocation in healthcare expenditure.

Appendix: A double bootstrap estimator algorithm

The steps in this procedure borrow heavily from Simar and Wilson [25], and are quite simple to implement; the complete bootstrap algorithm can be illustrated as follows:

• 1. Obtain the efficiency scores $\widehat{{\theta_{1} }},\widehat{{\theta_{2} }}, \ldots ,\widehat{{\theta_{n} }}$ for each hospital $i = 1,2, \ldots ,n$ as in Eq. (1) by solving the linear programming model.

• 2. Use a smooth bootstrap to generate a random sample size of N $\theta_{{{\rm B}1}} ,\theta_{{{\rm B}2}} , \ldots ,\theta_{{{\rm B}n}}$ from $\widehat{{\theta_{1} }},\widehat{{\theta_{2} }}, \ldots ,\widehat{{\theta_{n} }}$ where:

$\theta_{i}^{*} = \overline{B} + \frac{{\widetilde{{\theta_{i} }}^{*} - \overline{B} }}{{\left(1 + {\raise0.7ex\hbox{${h^{2} }$} \!\mathord{\left/ {\vphantom {{h^{2} } {\widehat{{\sigma_{\theta }^{2} }}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\widehat{{\sigma_{\theta }^{2} }}}$}}\right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }}$ and $\widetilde{{\theta_{i} }}^{*} = \left\{ {_{{2 - \theta_{{{\rm B}i}} - h\varepsilon_{i}^{*} ,if \, \theta_{{{\rm B}i}} + h\varepsilon_{i}^{*} < 1}}^{{\theta_{{{\rm B}i}} + h\varepsilon_{i}^{*} ,if \, \theta_{{{\rm B}i}} + h\varepsilon_{i}^{*} \ge 1}} } \right\}$.

Denote that $\overline{B} = {\raise0.7ex\hbox{${\sum\limits_{i = 1}^{n} {\theta_{{{\rm B}i}} } }$} \!\mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^{n} {\theta_{{{\rm B}i}} } } n}}\right.\kern-0pt} \!\lower0.7ex\hbox{$n$}}$ and $\widehat{{\sigma_{\theta }^{2} }} = {\raise0.7ex\hbox{${\sum\limits_{i = 1}^{n} {(\widehat{{\theta_{i} }} - \widehat{{\overline{{\theta_{i} }} }})^{2} } }$} \!\mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^{n} {(\widehat{{\theta_{i} }} - \widehat{{\overline{{\theta_{i} }} }})^{2} } } n}}\right.\kern-0pt} \!\lower0.7ex\hbox{$n$}}$, h is the smoothing parameter of the kernel density estimate of original efficiency estimates, and $\varepsilon_{i}^{*} ,i = 1,2, \ldots ,n$ are random draws from the standard normal. Note that h is chosen via the maximizing the likelihood cross-validation function and using reflecting method described by Silverman [48].

• 3. Compute the bootstrap estimate of technical efficiency for each hospital $i = 1,2, \ldots ,n$ using the ratio ${\raise0.7ex\hbox{${\widehat{{\theta_{i} }}}$} \!\mathord{\left/ {\vphantom {{\widehat{{\theta_{i} }}} {\theta_{i}^{*} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\theta_{i}^{*} }$}}$.

• 4. Resolve the original DEA model using the adjusted outputs to obtain $\widetilde{{\theta_{k} }}^{*}$

• 5. Repeat steps (2)–(4) to obtain a set of estimates, i.e., each hospital will have B estimates of Θ. For this analysis 2,000 samples were generated for each hospital.

Bias corrected estimates of original technical efficiency scores are derived through $\widetilde{{\theta_{i} }} = \widehat{{\theta_{i} }} - \widehat{{{\text{bias}}_{i} }},\,\,\widehat{{{\text{bias}}_{i} }} = \frac{1}{B}\widehat{{\theta_{iB}^{*} }} - \widehat{{\theta_{i} }}$.

Truncated regression to account for environmental variables

The bootstrap algorithm examining the exogenous determinants of TE and SE efficiency is presenting as follows:

• 1. Use the original data to compute $\widehat{{\theta_{i} }}$ by the DEA method for each hospital $i = 1,2, \ldots ,n$.

• 2. Use the ML method to obtain the parameter estimates $\widehat{\beta },\widehat{\sigma }$ from the truncated regression.

• 3. Now loop over the next four steps ${\rm B}$ times to obtain a set of bootstrap estimates.

  • 3.1 For $i = 1,2, \ldots ,M$ draw $\varepsilon_{i}^{*}$ from $N(0,\widehat{\sigma }_{\varepsilon } )$ with left truncation $(1 - \widehat{{\beta^{'} }}z_{i} )$

  • 3.2 Then compute the following equation $\theta_{i}^{*} = \widehat{{\beta^{'} }}z_{i} + \varepsilon_{i}^{*} ,i = 1,2, \ldots ,M$

  • 3.3 Set $x_{i} ,y_{i}^{*} = {\raise0.7ex\hbox{${y_{i} \widehat{{\theta_{i} }}}$} \!\mathord{\left/ {\vphantom {{y_{i} \widehat{{\theta_{i} }}} {\theta_{i}^{*} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\theta_{i}^{*} }$}}, \, \forall i = 1,2, \ldots ,M$

  • 3.4 Compute $\widehat{{\theta_{i}^{*} }} = \theta_{i} (x_{i} ,y_{i} ),\quad \forall i = 1,2, \ldots ,M$ replacing $(x_{i} ,y_{i} ){\text{ with }}(x_{i}^{*} ,y_{i}^{*} )$.

• 4. Compute the bias-corrected estimator using the bootstrap estimates and the original estimates.

• 5. Estimate the truncated regression of $\widehat{{\widehat{{\theta_{i} }}}}$ on $z_{i}$ using the maximum likelihood (ML) method.

• 6. Repeat the next three steps ${\rm B}_{1}$ times to obtain a set of estimators.

  • 6.1 For $i = 1,2, \ldots ,M$, $\varepsilon_{i}$ is drawn from $N(0,\widehat{{\widehat{\sigma }}}_{\varepsilon } )$ with left truncation $(1 - \widehat{{\widehat{{\beta^{'} }}}}z_{i} )$.

  • 6.2 For $i = 1,2, \ldots ,M$ compute $\theta_{i}^{**} = \widehat{{\widehat{{\beta^{'} }}}}z_{i} + \varepsilon_{i}^{**} ,i = 1,2, \ldots ,M$

  • 6.3 The ML method is used again to estimate the truncated regression of $\theta_{i}^{**}$ on $z_{i}$ providing $\widehat{{\widehat{{\beta^{'} }}}},\widehat{{\widehat{\sigma }}}_{\varepsilon }$ estimates.

• 7. Construct the confidence intervals for the efficiency scores.