Hip fracture surgery efficiency in Israeli hospitals via a network data envelopment analysis

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.

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

1. *Note that all variables are the originals we had, before any adjustment/transformation to DEA models was made
2. INTRCUPintracapsular percentage—the complement of the extracapsular rate we used, CHARL average Charlson comorbidity index (ranges between 1 and 30), WAIT2D percentage of patients waiting more than 2 days (48 h), DRAIN1D percentage of patients using drains for 1 day, INFEC and MORT percentage of patients infected or died within 365 days of surgery, SOESO average value of the socio-economic index (ranges between − 3 and 3) for patients undergoing hip surgery

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:

$$ \begin{aligned} & Max \;\emptyset \\ & {\text{s}} . {\text{t}} \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} X _{ij} \le x _{{ij_{0} }} ,} \quad \forall i = 1, \ldots ,m \\ & \quad \quad \O y_{{rj_{0} }} - \sum\limits_{j = 1}^{n} {\lambda_{j} y _{rj} \le 0 ,} \quad \forall r = 1, \ldots ,s \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \le 1} \\ & \quad \quad \lambda_{j} \ge 1,\quad \O \ge 1 \\ \end{aligned} $$

The above DEA dual problem formulation output-oriented fits the 4th model (PARTIAL) its efficiency is ϕ4. 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 ϕ1.

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

• 2. In model B the input vector is Z, and the output vector is Y; its efficiency is ϕ2.

  $$ \begin{aligned} & Max \;\emptyset \\ {\text{s}} . {\text{t}} \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} z _{dj} \le z _{{dj_{0} }} , } \quad \forall d = 1, \ldots ,D \\ & \quad \quad \O y_{{rj_{0} }} - \sum\limits_{j = 1}^{n} {\lambda_{j} y _{rj} \le 0 ,} \quad \forall r = 1, \ldots ,s \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \le 1} \\ & \quad \quad \lambda_{j} \ge 1, \quad \O \ge 1 \\ \end{aligned} $$

• 3. In model FULL the input vector is X and Z, and the output vector is Y; its efficiency is ϕ3.

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

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*

1. *Significant with P < .05

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