### Data

The total number of public health centres in Ghana was 622 in 2004 [12] and using an expected inefficiency rate of the health centres to be 20% (based on the previous pilot study by Osei et al (2005) in Ghana), a precision of 8% (based upon a power calculation) at the 95% confidence level we calculated the required sample size as 84. Since this was a nation-wide survey, we expected about 10% missing/non-response in the collection of the data and the expected sample size came to 92 health centres. After cleaning and eliminating health centres with missing data, the sample size came to 89. The two researchers who were trained on how to collect the data visited each of the health centres in the sample in 2005 and reviewed their 2004 inputs and outputs records and a structured form was used to collect the inputs and output data. Inputs in the health centre production are classified as human resources (clinical and non-clinical staff), expenditure on drugs and other consumables and number of beds and cots. Outputs were categorized into outpatient visits, number of antenatal care visits, number of deliveries, number of children immunized, number of family planning visits. These inputs and outputs were use to estimate the technical efficiencies of the health centres. The instruments were pre-tested for consistency and accuracy before actual data collection. Data collection was preceded by a certification from the Ethical Review Committee of the Ghana Health Service. Consent was sought at each health facility before data collection. Supervision was conducted by the Principal Investigator to ensure that data were properly or scientifically collected. Data collected were entered using Epi Info™ 3.3, and the technical efficiency scores were computed using Data Envelopment Analysis programme, version 2.1 (DEAP 2.1).

### Selection of inputs and outputs data

The selection of inputs and outputs for a DEA study needs careful attention as it may affect the distribution of technical efficiency. Improved health status is the ultimate output of a health system. However, improved health status is influenced by a host of factors some of which are outside of the domain of the health sector. Furthermore, measuring improvements in health status accurately is fraught with difficulties. Health centres and other health care organizations rarely collect information on health outcomes routinely. Therefore output is measured by intermediate health services that ostensibly improve health status [11]. Health centres in Ghana deliver outpatient curative and preventive care. They have a strong bias towards health promotion and disease prevention [26]. The issue of case mix and variation in the quality of care is not expected to be a problem, as health centres are standardized in terms of their staffing and other resources and the types of curative and preventive programmes that they run. Inputs in health centre production can be classified as labour (clinical and non-clinical), capital (proxied by the number of beds used for emergency cases and child deliveries) and supplies including pharmaceuticals. The choice of inputs and outputs for the DEA analysis was guided in part by the previous DEA health care studies in the African Region and availability of data [23, 25, 26]. The inputs and output selected include the following.

#### Inputs

Input 1: Number of non clinical staff including labourers

Input 2: Number of clinical staff

Input 3: Number of beds and cots

Input 4: Expenditure (in local currency call cedi) on drugs and supplies. The inter-bank exchange rate of the cedi to the dollar was ¢8,500 to 1 US$ at the time of the study.

#### Output

Output 1: General outpatient visits

Output 2: Number of antenatal care visits

Output 3: Number of deliveries

Output 4: Number of children immunised

Output 5: Number of family planning visits

### Efficiency and DEA Analytical framework

The basic premise underlying the concept of efficiency is that no output can be produced without resources (inputs) and that these resources are limited in supply. From this, it also follows that there is a limit to the volume of output (commodities) that can be produced.

There are two basic measures of efficiency: allocative and technical efficiency. Allocative efficiency refers to how different resource inputs are combined to produce a mix of different outputs [27]. Technical efficiency on the other hand is concerned with achieving maximum outputs with the least cost. Overall efficiency measures the combined effect of allocative and technical efficiency [27].

In order to measure efficiency a norm must be specified. The norm set for measuring *technical efficiency* is that the minimum amount of resources should be used for a given level of output or, alternatively, the maximum amount of output that should be produced for a given level of resource use. If more resources than necessary are used to produce a given amount of output, this implies a waste of resources and therefore inefficiency. Equally, the difference in the amount of output that could have been produced from a given amount of resources and the amount of output that was actually produced can be used as a measure of technical inefficiency [28]. Technical inefficiency is thus a matter of degree depending upon how much unnecessary resources have been used. The size of a health centre may sometimes be a cause for inefficiency. A health centre may be too large for the volume of activities that it is conducting; and therefore may experience *inefficiencies of scale*. In the presence of inefficiencies of scale, a health centre is inefficiently large, unit costs increase as the scale of production increases. On the other hand, a health centre may be too small for its level of operation, and thus experience efficiencies of scale.

Until recently, the traditional methodology for measuring efficiency in economics (including health economics) has been the production frontier approach based on the principles of statistics and econometrics [28]. These functions, which are estimated to determine efficiency, are also known as stochastic frontier models (SFM). During the recent few decades, however, an alternative methodology to the stochastic frontier approach (SFA) has been developed and its application has grown rapidly over the years. This methodology has come to be known as the Data Envelopment Analysis (DEA) [28]. It has been found that there are several compelling methodological and practical advantages for using DEA over the stochastic frontier models. DEA accommodates multiple inputs and multiple outputs in a single measure of efficiency than the SFA and has become the dominant approach to efficiency measurement in health care and in many other sectors of the economy [16]. DEA does not impose a specified functional form to model and calculate the efficiency of a decision making unit (DMU). Unlike the parametric frontier models therefore, DEA does not suffer from the problem of model mis-specification, which has the potential of providing misleading results [28]. In addition, Unlike SFA, DEA does not suffer from the problems of multicollinearity and heteroscedasticity. DEA gives a measure of efficiency that is empirically obtainable in a given scenario (given available resources, institutional set-up, etc). Hence we can compare the efficiency of individual health centres realistic benchmarks.

On the other hand, DEA estimation can only tell how well a DMU or health centre (in our case) is doing compared to its peers but not compared to a "theoretical maximum". in other words since DEA gives a relative measure of efficiency it has the potential of justifying inefficiency i.e. even those that appear to be efficient in the sample might actually be inefficient in absolute terms. This problem can, however, be minimized by using a large sample data set. Another limitation or disadvantage is that since DEA is a non parametric technique, statistical hypothesis testing is difficult to do. Also since DEA is an extreme point technique, noise such as measurement errors can cause significant problem. Further overview of the DEA model is presented below

For assessing differences in the productive efficiency of health centres, we use DEA, a mathematical programming based method that converts multiple input and output measures into a single summary measure of productive efficiency. DEA is based on relative efficiency concepts proposed by Farrell but Charnes et al (1994) extended and developed Farrell's approach. DEA can be said to utilize an extended concept of Pareto efficiency [28].

Following Charnes et al (1978) the technical efficiency of health centres as the maximum of a ratio of weighted outputs to weighted inputs subject to the condition that the similar ratios for every health centre be less than or equal to unity. This is done by solving the following fractional programming problem

\begin{array}{l}Max{h}_{0}=\frac{{\displaystyle \sum _{r=1}^{s}{u}_{r}{y}_{rjo}}}{{\displaystyle \sum _{i=1}^{m}{v}_{i}{y}_{ijo}}}\hfill \\ \begin{array}{c}\text{Subjectto}\\ \begin{array}{cc}\frac{{\displaystyle \sum _{r=1}^{s}{u}_{r}{y}_{rj}}}{{\displaystyle \sum _{i=1}^{m}{v}_{i}{x}_{ij}}}\le 1,& j=1,\mathrm{...}{j}_{0},\mathrm{...}n\end{array}\\ \begin{array}{ccc}{u}_{r}\ge 0,r=1,\mathrm{...},s& \text{and}& {v}_{i}\ge 0,i=1,\mathrm{...},m\end{array}\end{array}\hfill \end{array}

(1)

The terms *y*
_{
rjo
}and *x*
_{
rjo
}represent the amount of output *r* and the amount of input *i* for the unit *j*
_{0}. Optimization is performed separately for each unit to compute an optimal set of weights (*u*
_{
r
}, *v*
_{
i
}) and efficiency measure *h*
_{0}. The method chooses values of *u*
_{
r
}and *v*
_{
r
}which are most favorable to the unit that is being studied. As a consequence, a unit that is superior to all others on any single output-input ratio will be rated efficient.

The standard DEA model, the relative efficiency of production unit is defined as the ratio of the sum of its weighted outputs to the sum of its weighted inputs. The weights have been determined so as to show the production unit at the maximum relative efficiency.

In the study we will adopt the input oriented-based approach because decision making units (Health centres) have better control over inputs than outputs hence our interest in the input based approach. This approach is also more popular in terms of usage than the output oriented approach [29, 30, 20, 21, 23, 25]. The model in (1) is a fractional programming model, which can be converted into the following linear forms (models 2 and 3) so that the methods of linear programming can be applied.

### Constant Returns to Scale (CRS) model

The constant returns to scale model assumes a production process in which the optimal mix of inputs and outputs is independent of the scale of operation. The following CRS model measures overall technical efficiency for each of the sample health centre. The objective function is to maximize the efficiency score *h*
_{0} for health centre *j*
_{0}, subject to the constraints that no health centre will be more than 100% efficient and the coefficient values are positive and non-zero, when the same set of *u* and *v* coefficients (weights) are applied to all other health centres being compared.

\begin{array}{l}Max{h}_{0}={\displaystyle \sum _{r=1}^{s}{u}_{r}{y}_{r{j}_{0}}}\hfill \\ \begin{array}{ll}\text{Subjectto}\hfill & {\displaystyle \sum _{i=1}^{m}{v}_{i}{x}_{i{j}_{0}}=1}\hfill \\ \begin{array}{cc}{\displaystyle \sum _{r=1}^{s}{u}_{r}{y}_{rj}}-{\displaystyle \sum _{i=1}^{m}{v}_{i}{x}_{ij}\le 0}& j=1,\mathrm{...},n\end{array}\hfill \\ {u}_{r},{v}_{i}\ge 0\hfill \end{array}\hfill \end{array}

(2)

### Variable Returns to Scale (VRS) model

The VRS model, though similar to the CRS model, measures pure technical efficiency and returns to scale for each of the sample health centres. Scale efficiency can be measured by dividing the CRS efficiency score by the VRS efficiency score. From the VRS model, it is possible to analyze whether a health centre's production indicates increasing return to scale, constant return to scale, or decreasing return to scale by the sign of the variable *z*
_{
jo
}. Increasing returns to scale exists if the value of *z*
_{
jo
}is greater than zero (*z*
_{
jo
}> 0), constant returns to scale if the value of *z*
_{
jo
}is equal to zero (*z*
_{
jo
}= 0), and decreasing returns to scale if the value of *z*
_{
jo
}is less than zero (*z*
_{
jo
}< 0). Thus, we can analogize the existence of efficiencies of scale similar, confirm the most productive scale size (minimum efficient scale) of a health centre and estimate the number of health centres operating at the efficient scale.

\begin{array}{l}Max{h}_{0}={\displaystyle \sum _{r=1}^{s}{u}_{r}{y}_{r{j}_{0}}+{z}_{j0}}\hfill \\ \begin{array}{ll}\text{Subjectto}\hfill & {\displaystyle \sum _{i=1}^{m}{v}_{i}{x}_{i{j}_{0}}+{z}_{j0}=1}\hfill \\ {\displaystyle \sum _{r=1}^{s}{u}_{r}{y}_{rj}}-{\displaystyle \sum _{i=1}^{m}{v}_{i}{x}_{ij}+{z}_{j0}\le 0}\hfill & j=1,\mathrm{...},n\hfill \\ {u}_{r},{v}_{i}\ge 0\hfill \end{array}\hfill \end{array}

(3)

The paper concentrated on the VRS model. This is so because the VRS model isolates the pure technical efficiency component and scale efficiency which related to the size or structure of the decision making unit (DMU). Health centres that are overall efficient exhibit constant returns to scale. The size of a Health centre may sometimes be a cause for inefficiency. A health centre may be too large for the volume of activities that it is conducting; and therefore may experience *inefficiencies of scale*. On the other hand, a health centre may be too small for its level of operation, and thus experience efficiencies of scale. Inefficiency due to congestion refers to too many inputs (staff, funds, drugs, etc) leading to decreased output or what is commonly known as inefficiencies of scale which to some extend are realistic assumption for a developing country like Ghana where political and other irrational reasons affect the establishment of facilities such health centres, schools etc. It is important to point out that this study does not attempt to address allocative efficiency in the paper, as it was difficult to get accurate input prices. The study also does not address issues of productivity, due to lack of appropriate panel data.