## Abstract

Data envelopment analysis (DEA) has been used previously for examining hospital efficiency, based on administrative data. Yet, previous DEA research devoted to quality assurance rarely considered medical processes or outcomes in efficiency studies. The goal of this study is to examine the relative efficiency of hip fracture surgery, based on clinical data reflecting medical process indicators and outcomes. To accomplish our goal, recent developments in DEA research were harnessed to model an output-oriented two-stage DEA network. The proposed DEA model has: two input variables reflecting the condition of the patient, fracture type and Charlson index; two intermediate variables reflecting clinical decisions, surgery within 48 h and usage of a drain for 1 day (rate); and two output variables reflecting the success of the surgery, survival rate after surgery and the rate of no infection. Using data from orthopedic wards in most of the acute Israeli hospitals (20 out of 22), no statistically significant correlation was found, either between the socio-economic index of patients who had hip fracture surgery and the relative efficiency scores produced by the two-stage network DEA model, or between efficiency and the geographical periphery status of the hospital. In addition to this, which points to a degree of social equality regarding hip fracture surgeries, we also compared the two-stage network model and related DEA models, providing several lemmas that represent the relationships between the various models mathematically.

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## Acknowledgements

We thank the Quality Assurance Unit at Israel Ministry of Health for the data. We also thank the anonymous referees for their helpful comments which improve the paper.

## Author information

## Appendices

### Appendix A: Descriptive statistics of the original* variables

INTRCUP | CHARL | WAIT2D | DRAIN1D | INFEC | MORT | SOECO | |
---|---|---|---|---|---|---|---|

Mean | 43.05 | 2.41 | 26.34 | 15.99 | 6.34 | 15.71 | .076 |

Median | 42.28 | 2.41 | 23.15 | 7.96 | 5.62 | 15.32 | .066 |

Std. deviation | 7.30 | .44 | 10.74 | 19.65 | 3.74 | 5.64 | .433 |

Minimum | 32.26 | 1.75 | 5.68 | 1.06 | 1.14 | 4.26 | − .705 |

Maximum | 61.05 | 3.39 | 45.26 | 77.17 | 12.43 | 23.96 | .903 |

### Appendix B: The basic DEA (VRS) version

The basic DEA variable return to scale (VRS) version (Banker et al. 1984) has only input set X and output set Y. This is a one-stage model, indicated with one arrow in Table 2: X → Y. Its dual formulation is:

The above DEA dual problem formulation output-oriented fits the 4th model (PARTIAL) its efficiency is *ϕ*4. The other three models are model 1 (A), model 2 (B), model 3 (FULL). All three models are one-stage systems like model 4: their formulation varies with respect to the vectors of inputs and outputs, as follows:

- 1.
In model A the input vector is X, and the output vector is Z; the efficiency score is

*ϕ*1.$$ \begin{aligned} & Max \;\emptyset \\ {\text{s}} . {\text{t}} \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} x _{ij} \le x _{{ij_{0} }} ,} \quad \forall i = 1, \ldots ,m \\ & \quad \quad \O z_{dj} - \mathop \sum \limits_{j = 1}^{n} \lambda_{j} z _{dj} \le 0 , \quad \forall d = 1, \ldots ,D \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \le 1} \\ & \quad \quad \lambda_{j} \ge 1, \quad \O \ge 1 \\ \end{aligned} $$ - 2.
In model B the input vector is Z, and the output vector is Y; its efficiency is

*ϕ*2.$$ \begin{aligned} & Max \;\emptyset \\ {\text{s}} . {\text{t}} \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} z _{dj} \le z _{{dj_{0} }} , } \quad \forall d = 1, \ldots ,D \\ & \quad \quad \O y_{{rj_{0} }} - \sum\limits_{j = 1}^{n} {\lambda_{j} y _{rj} \le 0 ,} \quad \forall r = 1, \ldots ,s \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \le 1} \\ & \quad \quad \lambda_{j} \ge 1, \quad \O \ge 1 \\ \end{aligned} $$ - 3.
In model FULL the input vector is X and Z, and the output vector is Y; its efficiency is

*ϕ*3.$$ \begin{aligned} & Max \;\emptyset \\ {\text{s}} . {\text{t}} \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} x _{ij} \le x _{{ij_{0} }} , } \quad \forall i = 1, \ldots ,m \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} z _{dj} \le z _{{dj_{0} }} , } \quad \forall d = 1, \ldots ,D \\ & \quad \quad \O z_{dj} - \sum\limits_{j = 1}^{n} {\lambda_{j} z _{dj} \le 0 , } \quad \forall d = 1, \ldots ,D \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \le 1} \\ & \quad \quad \lambda_{j} \ge 1, \quad \O \ge 1 \\ \end{aligned} $$

### Appendix C: Spearman correlations between the seven models

MODEL | A | B | A*B | AVE | FULL | PART |
---|---|---|---|---|---|---|

B | − .272 | |||||

A*B | .974* | − .114 | ||||

AVERAGE | .976* | − .132 | .998* | |||

FULL | − .289 | .681* | − .229 | − .229 | ||

PART | − .078 | .846* | .072 | .050 | .617* | |

NETWORK | − .175 | .843* | − .005 | − .028 | .502* | .854* |

### Appendix D: The frequency with which an efficient hospital is a peer for inefficient hospitals

Hospital no. | FULL | PARTIAL | NETWORK |
---|---|---|---|

s1 | 4 | 6 | 6 |

s2 | 4 | ||

s3 | 7 | 12 | 13 |

s4 | 1 | ||

s6 | 5 | ||

s9 | 1 | 1 | 7 |

s12 | 2 | 1 | |

s13 | 4 | 6 | |

s15 | 4 | 2 | 13 |

s16 | 9 | 13 | 25 |

s17 | 13 | ||

s19 | 2 | 5 | |

s20 | 1 | 1 |

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Cohen-Kadosh, S., Sinuany-Stern, Z. Hip fracture surgery efficiency in Israeli hospitals via a network data envelopment analysis.
*Cent Eur J Oper Res* **28, **251–277 (2020) doi:10.1007/s10100-018-0585-0

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### Keywords

- Efficiency
- Data envelopment analysis (DEA)
- Two-stage network DEA
- Socio-economic index (SEI)
- Geographical periphery status
- Hip fracture surgery
- Clinical quality assurance