The relationship between the efficiency of orthopedic wards and the socio-economic status of their patients

Abstract

The purpose of this paper is to study the effect of the socio-economic status of patients on the efficiency of orthopedic wards in acute hospitals in Israel (20 hospitals), from the viewpoint of the regulator—Israel Ministry of Health. At the first stage, data envelopment analysis is used with two inputs, and three outputs, where one output is undesirable—“number of deaths”—which also reflects the quality of the health services. At the second stage, various nonparametric tests are utilized to test the relationship between the socio-economic status of patients and the efficiency. As by-product DEA provides benchmark analysis, which indicates the peers of each inefficient ward, and the I/O improvements are needed for achieving efficiency. Two versions of DEA were used: the output oriented version (variable returns to scale), and the non-oriented version (Additive). Further analysis provides comparison of the results with other simple efficiency measures. We also compare between the efficiency from the regulator viewpoint and the hospitals’ viewpoint.

Appendices

Appendix 1: The socio-economic index

The SEI includes 16 variables which compose SEI for each census tract:

• 1. The median age

• 2. Dependency ratio—the ratio between young (0–19) plus old (65+) populations and the working age population (20–64).

• 3. Average number of persons per household

• 4. Average years of schooling of aged 25–54

• 5. Percent of academic degree of age group 25–54

• 6. Percent of workers in academic or managerial occupations

• 7. Percent of wage and income earners of ages 15 and over

• 8. Percent of women aged 25–54 not in civilian labor force

• 9. Percent of wage and income earners above twice the average wage

• 10. Percent of sub-minimum wage earners

• 11. Percent of recipients of income support and income supplement to old-age pension

• 12. Monthly income per standard person

• 13. Average number of vehicles at household disposal per aged 18 and over

• 14. Average number of rooms per person in households

• 15. Average number of bathrooms per person in household

• 16. Percent of households with PC and internet access

SEI is composed of 16 variables, which are grouped into 4 main components:

• a. Demography—items 1–3

• b. Education—items 4–6

• c. Employment and pensions—items 7–11

• d. Standard of living—items 12–16

For more details see Burk et al. (2013).

Appendix 2: Health care data in Israel by district

See Tables 9, 10 and 11.

Table 9 Economic and health indicators in Israel by district

Table 10 Number of health resources per 1000 inhabitants in Israel in 2009–2011

Table 11 Population density by district in Israel

Appendix 3: Data envelopment analysis (DEA)

The basic DEA model was developed by Charnes, Cooper and Rhodes (CCR - 1978); it assumes Constant Returns to Scale (CRS). CRS version of DEA measures the total efficiency of n Decision Making Units (DMUs), where each has s outputs, sharing m inputs. Given, $\hbox {x}_{\mathrm{ij}}$, the past value of input i, of DMU j (for all i $=$ 1, ..., m and j $=$ 1, ..., n), and, $\hbox {y}_{\mathrm{rj}}$, the past value of output r of DMU j (for all r $=$ 1, ..., s), we solve n problems, one for each DMU. The problem of DMU k finds, $\hbox {v}_{\mathrm{ik}}$, the optimal weight of input i, and $\hbox {u}_{\mathrm{rk}}$, the optimal weight of output r of DMU k, which maximize its relative efficiency measure, $\hbox {h}_{\mathrm{kk}}$. The basic ratio used here is: $h_{kj} =\sum \nolimits _{r=1}^s {u_{rk}} y_{rj} /\sum \nolimits _{i=1}^m {v_{ik}} x_{ij}$

The problem of DMU k is: $\max h_{kk}$, subject to: $h_{kj} \le 1, j=1, \ldots , n, u_{rk} ,v_{ik} \ge 0,i=1, \ldots , m,r=, \ldots , s$

If with its ideal weights DMU k does not receive the maximal efficiency score 1 (100 %), then DMU k is not efficient (i.e., other DMUs or a combination of DMUs received the maximal score 1 in the ideal weights of DMU k). However, if DMU k receives the maximal efficiency rate 1 then, unit k is relatively efficient. This maximization problem is called: the output oriented version. The input oriented version minimizes the reciprocal of the above objective function.DEA provides the efficient frontier. Obviously, the input and output weights vary greatly from one DMU to another.

Variable Return to Scale version

Banker, Charnes and Cooper (BCC -1984) introduced the Variable Returns to Scale (VRS) version of DEA by adding a constant variable, $-\omega _k$, to the numerator of $h_{kj}$ in the above problem k. BCC provides technical efficiency (see Banker et al. 1984). The ratio between CCR and BCC efficiency, provides the Scale Efficiency, which is always less than or equal to 1.

The Linear Programming (LP) formulation of BCC

$$\begin{aligned}&\max h_{kk} =\sum \limits _{r=1}^s {u_{rk} y_{rk}} -\omega _k\\&\hbox {S.T.}\\&\begin{array}{lll} \sum \limits _{i=1}^m {v_{ik} x_{ik} =1} &{} &{} u_{rk} ,v_{ik} \ge 0 \\ &{} j=1\ldots n &{} i=1\ldots m \\ \sum \limits _{r=1}^s {u_{rk} y_{rj}} -\omega _k -\sum \limits _{i=1}^m {v_{ik} x_{ij} \le 0} &{} &{} r=1\ldots s \\ \end{array} \end{aligned}$$

This is the output oriented version. In the input oriented version, the denominator of the original ratio is minimized subject to the numerator equal to 1.