Managing for efficiency in health care: the case of Greek public hospitals

Abstract

This paper evaluates the efficiency of public hospitals with two alternative conceptual models. One model targets resource usage directly to assess production efficiency, while the other model incorporates financial results to assess economic efficiency. Performance analysis of these models was conducted in two stages. In stage one, we utilized data envelopment analysis to obtain the efficiency score of each hospital, while in stage two we took into account the influence of the operational environment on efficiency by regressing those scores on explanatory variables that concern the performance of hospital services. We applied these methods to evaluate 96 general hospitals in the Greek national health system. The results indicate that, although the average efficiency scores in both models have remained relatively stable compared to past assessments, internal changes in hospital performances do exist. This study provides a clear framework for policy implications to increase the overall efficiency of general hospitals.

Appendix

The DEA bootstrap procedure

VRS and CRS efficiency measures are estimated in each bootstrap replication according to the following steps:

• 1. Calculate the DEA input-orientated efficiency score θ _i for each hospital i = 1, 2, … ,n using DEA with either a CRS or VRS specification.

• 2. Using a smooth bootstrap generate a random sample size of n: θ_1b , θ_2b , … , θ _nb , b = 1, … ,B where b is the bth iteration of the bootstrap from θ₁, θ₂, … ,θ _n where $\theta_{i}^{*} = \bar{B} + \frac{{\bar{\theta }_{i}^{*} - \bar{B}}}{{\sqrt {1 + h^{2} /\sigma_{\theta }^{2} } }},\;\bar{\theta }_{i}^{*} = \left\{ {\begin{array}{*{20}c} {\theta_{\text{ib}} + h\varepsilon_{i}^{*} \quad if\quad \theta_{\text{ib}} + h\varepsilon_{i}^{*} \le 1} \\ {2 - \theta_{\text{ib}} - h\varepsilon_{i}^{*} \quad {\text{otherwise }}} \\ \end{array} } \right\}$, Denote that $\bar{B} = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {\theta_{\text{ib}} }$ and $\sigma_{\theta }^{2} = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {(\theta_{i} - \bar{\theta }_{\iota } )^{2} }$, h is the smoothing parameter of the Kernel density of the original efficiency estimates, $\varepsilon_{i}^{*}$, i = 1, 2, … ,n are random draws for the standard normal distribution. Note that we obtain h for our bootstrapping application by maximizing the likelihood cross-validation function in the Gaussian kernel estimation. The Kernel density estimation is a nonparametric technique for density estimation in which a known density function (the kernel) is averaged across the observed data points to create a smooth approximation.

• 3. Calculate bootstrapped input based on bootstrap efficiency $x_{ib} = (\theta_{\iota } /\theta_{i}^{*} )x_{i}$.

• 4. Resolve the original DEA model using the adjusted inputs to obtain $\theta_{\text{ib}}^{*}$, b = 1, … ,B.

• 5. Repeat steps (2)–(4), B times to provide for B sets of θ estimates and compute the estimated confidence intervals for the efficiency scores. For this analysis, 2,000 samples were generated for each hospital.

Bootstrapped truncated regression procedure

The bootstrap algorithm for the second stage analysis is described briefly in the following steps:

• 1. Calculate the DEA input-orientated inefficiency score $(\delta_{i} = 1/\theta_{i})$ for each hospital i = 1, 2, … n using the DEA method. Note that in this step, we use only δ _i  ≥ 1.

• 2. Maximum likelihood is used in the truncated regression to obtain the parameter estimates β, σ_ε.

• 3. For each hospital i = 1, … ,n the next four steps (1–4) are repeated B times to yield a set of bootstrap estimates.

  • 3.1 Draw $\varepsilon_{i}^{*}$ from N (0, $\sigma_{\varepsilon }^{2}$) with left-truncation at (1 − βz _i )

  • 3.2 Compute $\delta_{i}^{*} = \beta z_{i} + \varepsilon_{i}^{*}$.

  • 3.3 Construct a pseudo data set ($x_{i}^{*} ,\;y_{i}^{*}$), where $x_{i}^{*} = x_{i}$ and $y_{i}^{*} = y_{i} \delta_{i} /\delta_{i}^{*}$.

  • 3.4 Compute a new DEA estimate $\delta_{\iota }^{*} = \delta_{\iota } (x_{i} ,\;y_{i} )$ replacing ($x_{i} ,\;y_{i}$) 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 δ _i on z _i using the maximum likelihood method.

• 6. Repeat the next three steps B _i times to obtain a set of estimators.

  • 6.1 For i = 1, 2, … ,n, ε _ι is drawn from N (0, $\sigma_{\varepsilon }^{2}$) with left truncation (1 − βz _i ).

  • 6.2 For i = 1, 2, … ,n compute $\delta_{i}^{**} = \beta z_{i} + \varepsilon_{i}^{**}$.

  • 6.3 The maximum likelihood method is used again to estimate the truncated regression of $\delta_{i}^{**}$ on z _i providing β, σ_ε estimates.

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