PERFORMANCES OF THE PRIMARY HEALTH-CARE FACILITIES IN BANGLADESH: DATA ENVELOPMENT ANALYSIS

Data Envelopment Analysis

In their 1978 publication, Charnes et al. noted that DEA, a nonparametric technique, develops an efficiency frontier by optimizing the ratio of weighted output to weighted input of firms; this calculation is subject to the condition that this ratio can equal, but never exceed, unity for any other firm. The technique helps in measuring the relative technical efficiency of the firms. The term decision-making unit (DMU) refers to any production entity under evaluation. The first attempt to develop the model was made by Farrell in 1957 (Farrell, 1957) and later the theoretical approach was developed by Charnes, Cooper, and Rhodes (CCR, 1978). The basic efficiency measure used in DEA is the ratio of total outputs to total inputs (Cooper et al., 2007). That is, let x_i represent the ith input and y_r represent the rth output of a DMU. Let the total number of inputs and outputs be represented by m and s, respectively, where m, s > 0 and v_i and u_r are the weights assigned to inputs and outputs, respectively (Rao, 2003).

The mathematical program, \(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\bf{\alpha}}\)\(\def\bupbeta{\bf{\beta}}\)\(\def\bupgamma{\bf{\gamma}}\)\(\def\bupdelta{\bf{\delta}}\)\(\def\bupvarepsilon{\bf{\varepsilon}}\)\(\def\bupzeta{\bf{\zeta}}\)\(\def\bupeta{\bf{\eta}}\)\(\def\buptheta{\bf{\theta}}\)\(\def\bupiota{\bf{\iota}}\)\(\def\bupkappa{\bf{\kappa}}\)\(\def\\buplambda{\bf{\lambda}}\)\(\def\bupmu{\bf{\mu}}\)\(\def\bupnu{\bf{\nu}}\)\(\def\bupxi{\bf{\xi}}\)\(\def\bupomicron{\bf{\micron}}\)\(\def\buppi{\bf{\pi}}\)\(\def\buprho{\bf{\rho}}\)\(\def\bupsigma{\bf{\sigma}}\)\(\def\buptau{\bf{\tau}}\)\(\def\bupupsilon{\bf{\upsilon}}\)\(\def\bupphi{\bf{\phi}}\)\(\def\bupchi{\bf{\chi}}\)\(\def\buppsy{\bf{\psy}}\)\(\def\bupomega{\bf{\omega}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\begin{equation}{\rm{max}}\mathop {\rm{\theta }}\limits_{{v_m}{u_s}} = {{\mathop \sum \nolimits_{r = 1}^s {u_r}{y_r}} \over {\mathop \sum \nolimits_{i = 1}^m {v_i}{x_i}}},\end{equation}is subject to \(\begin{equation}0 \le {{\mathop \sum \nolimits_{r = 1}^s {u_r}{y_r}} \over {\mathop \sum \nolimits_{i = 1}^m {v_i}{x_i}}} \le 1;\quad \quad {\rm{j}} = 1,2 \ldots .{\rm{n;\ }}{v_m},\;{u_s} \ge 0.\end{equation}\)

One constraint is that the ratio θ should not exceed 1 for any DMU. The objective is to obtain the weights that maximize the ratio. The fractional programs are then converted to the linear programming. A CCR-DEA linear program model is as follows (Cooper et al., 2007; Rao, 2003): \(\begin{equation}{\rm{max}}\mathop {\rm{\theta }}\limits_{v,u} = \mathop \sum \limits_{r = 1}^s {u_r}{y_r},\end{equation}\)subject to \(\begin{equation}\eqalign{&\mathop \sum \limits_{i = 1}^m {v_i}{x_i} = 1 \cr&\mathop \sum \limits_{r = 1}^s {u_r}{y_r} - \mathop \sum \limits_{i = 1}^m {v_i}{x_{i}} \le 0;\quad \quad {\rm{j}} = 1,2, \ldots .{\rm{n;\ }}{v_m},{u_s} \ge {\rm{\varepsilon }};{\rm{\ i}} = 1,2, \ldots .m{\rm{\ and\ r}} = 1,2, \ldots .s, \cr}\end{equation}\)where ε is an infinitesimal constant. DEA has the advantage of managing multiple inputs and outputs (Rowena, 2001). Several authors have studied the efficiency of the government hospitals, mainly at the primary levels (Akazili et al., 2008; Bahumroz, 1999; Christian & Simon, 2013; Marschall & Flessa, 2011; Osei et al., 2005; Osmani, 2012; Zere et al., 2006). These studies show that the efficiency score varies between 70% and 80%. DEA was found to be the most widely used measure of efficiency applied in both developed and developing countries.

Malmquist Productivity Index

A productivity index measures changes in a DMU's efficiency over time. The MPI was introduced by Caves et al. (1982) to calculate productivity changes among different DMUs. Specifically, it measures total factor productivity changes (TFPCs), which can be divided into technical efficiency changes (TECs) and technological changes (TCs). TEC can be further divided into pure technical efficiency changes (PTECs) and scale efficiency changes (SECs) (Charnes et al., 1997). \(\begin{equation}{\rm TEC} = {\rm PTEC} + {\rm SEC}\end{equation}\)\(\begin{equation}{\rm{TFPC}} = {\rm{TEC}} \times {\rm{TC}} = \left( {{\rm{PTEC}} \times {\rm{SEC}}} \right) \times {\rm{TC}}{\rm{.}}\end{equation}\)

Study Design and Data Collection Method

DEA is used to measure technical efficiency. It requires a large number of DMUs. The smaller the product of the number of inputs and outputs compared to the number of DMUs, the better the discrimination between efficient and inefficient DMUs (Shujie et al., 2010). A rough rule of thumb is to choose DMU equal to or greater than input times output (Charnes et al., 1978). Four inputs and two outputs were used in the study, and 16 UHCs were selected to collect primary data. The UHCs were selected using a crude performance ratio of provider:population. An envelopment input-oriented model with the assumption of a CCR model was used with the DEA Solver tool (Ozcan, 2008). Efficiency is very likely to vary by input combinations. The input variables used in the study were doctors, nurses, drugs, and equipment, and the output variables used were the number of outpatient visits and inpatient visits.

Sensitivity Analysis

Three models were considered for sensitivity analysis. In the first model, input variables were the doctor and the nurse, and the output variables were inpatient and outpatient visits. In the second model, the doctor was the only input variable, and in the third model, the nurse was the only input variable, with the same output variables as in all three models. The results are shown in Table 2.

TABLE 2 Sensitivity Analysis (N = 16)

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Results of the first model show that the same six facilities remained efficient, as in the original model. The scores of some of the inefficient facilities changed. These negligible changes suggest equipment and drug as inputs with a lesser effect on efficiency. The average efficiency score is now 67%. In the second model, where doctor is the only input, two of the efficient facilities fell below the frontier. The scores of the Dhamrai and Palash UHCs remained the same, while the scores of all other UHCs fell. This suggest that nurses substantially affect the efficiency, with an average score of 64%. In the third model, where nurse is the only input variable, only one UHC remained efficient with all other facilities having very low scores, indicating the importance of doctor on the efficiency score. There is a significant fall in the average efficiency score to 18%. The sensitivity analysis showed that doctor and nurse are crucial inputs in determining the levels of efficiency.

Results of MPI

The MPI helps to compare performances of the health-care facilities over time. It is used to detect efficiency gains and losses over time. The TFPC over period t and t + 1 is the product of TEC and TC. The technical efficiency change (getting closer to or further away from the efficiency frontier) measures the change in efficiency between these two periods (t and t + 1), while the technological change shows the shift in the technology or a shift in efficiency frontier over time. A value greater than 1 in both changes indicates growth in productivity, a positive factor value. The major sources of productivity gains and losses can be measured by comparing the values of TEC and TC. If TEC is greater than TC, productivity gains are due to technical efficiency. If TEC is less than TC, it is due to technological progress. TEC can further be divided into PTEC and SEC. If PTEC is greater than SEC, the technical efficiency is due to pure technical efficiency; and if PTEC is less than SEC, the major source of efficiency is an improvement in scale.

The results of the MPI of TFPCs, as summarized in Table 3, show that the MPI of the annual mean score of TFPC was 0.998. The value of TFP was less than 1 from 2011 to 2012; it indicates that all facilities on average experienced a loss of productivity during that period. The period from 2012 to 2013 shows the value of TFP as greater than 1—there was a productivity gain. The gain was due to increased technical efficiency.

TABLE 3 Malmquist Productivity Index Summary of Annual Means at the Surveyed UHCs

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The MPI of the facilities compares the change in productivity of the facilities. Only two facilities experienced both technical efficiency improvement and technological improvement. That is, both facilities experienced productivity growth. The facilities were Nabiganj and Sundarganj. Ruma UHC shows the highest productivity loss (79%). Twelve facilities experienced technical efficiency gain (TEC > TC), and the remaining four facilities experienced technological progress (TC > TEC). The findings indicate that facilities experiencing productivity gain were able to use their inputs in such a manner as to produce more output over time.