**4. The proposed model and the utilized data**

The data envelopment analysis (DEA) non‐parametric method of measuring the relative effi‐ ciency has been developed rapidly since 1978, when a novel article "Measuring the efficiency of decision making units" by A. Charnes, W.W. Cooper and E. Rhodes was published [25]. DEA is a data‐oriented approach to the evaluation of functioning of a set of peer entities called decision‐making units (DMUs), which transform multiple inputs to multiple outputs [26]. The definition of DMU is rather general to provide the flexibility to use it in a wide range of possible applications. DMU is generally regarded as an object responsible for converting inputs to outputs, the action of which is to be evaluated [27], which allows for the use of this method in many different contexts, both in manufacturing and in almost all public sectors.

The usefulness of the method stems from the possibility of assessing the relative efficiency of decision‐making units. It is used in the banking sector, health care, agriculture, transport or education for reasons that can be characterized as identifying sources of inefficiency, creat‐ ing DMU rankings, evaluation of management systems, assessment of the effectiveness of programmes or policies, creating a quantitative basis for the reallocation of resources, etc. [28]. The DEA method is used in testing the efficiency of health‐care systems at practically all levels, ranging from physicians (both primary and specialist care), through providers of medical services (hospitals, emergency assistance, etc.), to global, country‐level assessments.

Two basic radial models, CCR (with constant returns to scale) and BCC (with variable returns to scale), evaluate the radial (proportional) efficiency but do not account for the surpluses of inputs and shortages of outputs, thus allowing for detecting only the radial inefficiencies. According to the DEA definition of efficiency, the operation of DMU is fully (100%) efficient if and only if both the efficiency score equals one and the inputs and output slacks are zero. In the case where the efficiency score is equal to one and one or both slacks are different from zero, it can be said that DMU is weakly efficient [27, 29]. This is a drawback, as the efficiency result does not take into account the non‐zero slacks. This drawback is not present in the addi‐ tive model, which directly takes into account the slacks in the calculation of efficiency and can distinguish between efficient and inefficient DMUs—there is, however, no possibility to measure the size of inefficiency with a scalar measure similar to that used in the basic radial models. Drawing upon the additive model, a measure of the efficiency based on slacks was developed (slack‐based measure, SBM). This measure takes into account the non‐zero slacks of inputs and outputs, if they are present [27]. The DEA models can be focused on the inputs or outputs, depending on which variables the decision‐maker can control.

The calculations are based on the input‐oriented slack‐based model (SBM) under constant returns‐to‐scale assumption [30]. Since only the inputs are controllable by the decision‐mak‐ ers shaping the health policy, an input‐oriented model was adopted. In an input orientation, improvement of efficiency is possible through reduction of inputs. The SBM input efficiency score *ρ<sup>I</sup>* \* of DMU*<sup>o</sup>* (*<sup>o</sup>* <sup>=</sup> <sup>1</sup>, …,*n*) is calculated for given amounts of outputs *yrj* , *r* = 1, …,*s* and inputs *yij* , *i* = 1, …,*m*, where *j* = 1, …,*n* [26].

$$\rho\_l^\* = \min\_{\lambda^\*, s^\*, s^\*} 1 - \frac{1}{m} \sum\_{\mu=1}^m \frac{s\_i^\*}{x\_\mu} \tag{1}$$

subject to

$$\mathbf{x}\_{i\circ} = \sum\_{j=1}^{\mu} \mathbf{x}\_{ij} \boldsymbol{\lambda}\_{j} + \mathbf{s}\_{i}^{-} \quad \text{( $i = 1, \dots, m$ )}\tag{2}$$

$$\mathcal{Y}\_{ss} = \bigvee\_{j=1}^{n} y\_{\gamma^j} \Lambda\_j - s^+\_{\cdot} \quad \text{( $r = 1, \dots, s$ )}\tag{3}$$

$$
\lambda\_{\rangle} \ge 0 \text{ (\forall j)}, \quad s\_{\rangle}^- \ge 0 \text{ (\forall i)}, \quad s\_{\neq}^+ \ge 0 \text{ (\forall r)} \tag{4}
$$

where **λ** is the intensity vector and **s <sup>−</sup> , s <sup>+</sup>** are input and output slack vectors respectively. The aim of the study is to compare the health outcomes in selected countries. The statistical information available in the case of post‐communist countries is much more limited than e.g. in the case of OECD or European Union countries. The possibility of using variables in a model is determined by the consistency of measurement for post‐communist countries and the availability of the data [15]. Thus, the model used three variables treated as inputs, char‐ acterizing the structure of spending and the level of income inequality. The PR\_TE variable defines the share of private expenditure in total health expenditure. The OOP\_TE variable determines what is the total share of the out of pocket payment in the total health expendi‐ ture. It is assumed that the lower is the public's load of private health expenditure, the higher is the availability of medical services and thus the higher is the possibility of obtaining better health outcomes in the population. The third variable, GINI, is the value of the Gini index. The Gini index is a measurement of the income distribution of a country's residents. This number, which ranges between 0 and 1 and is based on residents' net income, helps define the gap between the rich and the poor, with 0 representing perfect equality and 1 representing perfect inequality. In this model it is expressed as a percentage.

[26]. The definition of DMU is rather general to provide the flexibility to use it in a wide range of possible applications. DMU is generally regarded as an object responsible for converting inputs to outputs, the action of which is to be evaluated [27], which allows for the use of this method in many different contexts, both in manufacturing and in almost all public sectors.

The usefulness of the method stems from the possibility of assessing the relative efficiency of decision‐making units. It is used in the banking sector, health care, agriculture, transport or education for reasons that can be characterized as identifying sources of inefficiency, creat‐ ing DMU rankings, evaluation of management systems, assessment of the effectiveness of programmes or policies, creating a quantitative basis for the reallocation of resources, etc. [28]. The DEA method is used in testing the efficiency of health‐care systems at practically all levels, ranging from physicians (both primary and specialist care), through providers of medical services (hospitals, emergency assistance, etc.), to global, country‐level assessments.

Two basic radial models, CCR (with constant returns to scale) and BCC (with variable returns to scale), evaluate the radial (proportional) efficiency but do not account for the surpluses of inputs and shortages of outputs, thus allowing for detecting only the radial inefficiencies. According to the DEA definition of efficiency, the operation of DMU is fully (100%) efficient if and only if both the efficiency score equals one and the inputs and output slacks are zero. In the case where the efficiency score is equal to one and one or both slacks are different from zero, it can be said that DMU is weakly efficient [27, 29]. This is a drawback, as the efficiency result does not take into account the non‐zero slacks. This drawback is not present in the addi‐ tive model, which directly takes into account the slacks in the calculation of efficiency and can distinguish between efficient and inefficient DMUs—there is, however, no possibility to measure the size of inefficiency with a scalar measure similar to that used in the basic radial models. Drawing upon the additive model, a measure of the efficiency based on slacks was developed (slack‐based measure, SBM). This measure takes into account the non‐zero slacks of inputs and outputs, if they are present [27]. The DEA models can be focused on the inputs

The calculations are based on the input‐oriented slack‐based model (SBM) under constant returns‐to‐scale assumption [30]. Since only the inputs are controllable by the decision‐mak‐ ers shaping the health policy, an input‐oriented model was adopted. In an input orientation, improvement of efficiency is possible through reduction of inputs. The SBM input efficiency

> ,*<sup>s</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> \_\_1 *<sup>m</sup>* ∑ *i*=1 *<sup>m</sup> s<sup>i</sup>* − \_\_ *xio*

<sup>−</sup> ≥ 0 (∀*i*), *sr*

, *r* = 1, …,*s* and inputs

<sup>−</sup> (*i* = 1, …,*m* ) (2)

+ (*r* = 1, …,*s* ) (3)

are input and output slack vectors respectively.

<sup>+</sup> ≥ 0 (∀*r* ) (4)

(1)

or outputs, depending on which variables the decision‐maker can control.

of DMU*<sup>o</sup>* (*<sup>o</sup>* <sup>=</sup> <sup>1</sup>, …,*n*) is calculated for given amounts of outputs *yrj*

*j*=1 *n*

*j*=1 *n*

**, s <sup>+</sup>**

\* = min *λ*, *s*<sup>−</sup>

*xi<sup>j</sup> λ<sup>j</sup>* + *s<sup>i</sup>*

*yrj λ<sup>j</sup>* − *sr*

score *ρ<sup>I</sup>* \*

100 Advances in Health Management

subject to

, *i* = 1, …,*m*, where *j* = 1, …,*n* [26].

*ρ<sup>I</sup>*

*xi<sup>o</sup>* = ∑

*yro* = ∑

where **λ** is the intensity vector and **s <sup>−</sup>**

*λ<sup>j</sup>* ≥ 0 (∀*j*), *s<sup>i</sup>*

*yij*

The overall health status of population is generally operationalized by indicators of longevity such as life expectancy or healthy life expectancy. So the outputs in this model are reflected by two variables: LE60—life expectancy at age 60 and HLE—healthy life expectancy at birth. The third output variable is ISR—infant survival rate, which is the opposite of infant mortal‐ ity rate (IMR is unwanted output and was included in the model as the difference 1000‐IMR).

Using the above‐described model, the 28 post‐communist countries<sup>1</sup> and the virtual unit (DMU) as an aggregate of average values for 16 developed countries of Western Europe<sup>2</sup> (DE16), which achieve very good health outcomes, were analysed. The virtual unit (DE16) consists of countries where the health system is organized according to Beveridge and Bismarck models. Data from the years 2000 and 2013 from the WHO database and The World Bank databases were used. In the case of missing data, the principle of using the nearest value was applied.

The calculations were carried out by means of the DEA‐Solver‐LV (3) software by Saitech.

The basic descriptive statistics of variables for years 2000 and 2013 are presented (**Table 1**).

The last row shows the difference between the mean values of the variables (2013–2000). The average share of private spending did not change; however, the share of patients' out‐of‐ pocket payment in the total expenditure decreased by 1.0 percentage point, which is a proof of weak development of the pre‐paid health insurance. The income inequalities in the coun‐ tries surveyed decreased slightly, by 0.5 p.p.; however, the span of this variable increased. All results improved: LE60 increased by about 9% and HLE by 6%. The infant mortality decreased significantly: in the year 2000, it was highest in Tajikistan and amounted to 74.7 infants per

<sup>1</sup> Albania, Armenia, Azerbaijan, Belarus, Bosnia and Herzegovina, Bulgaria, Croatia, Czech Republic, Estonia, Georgia, Hungary, Kazakhstan, Kyrgyzstan, Latvia, Lithuania, Montenegro, Poland, Republic of Macedonia, Republic of Mol‐ dova, Romania, Russian Federation, Serbia, Slovakia, Slovenia, Tajikistan, Turkmenistan Ukraine, Uzbekistan.

<sup>2</sup> Austria, Belgium, Denmark, Finland, France, Germany, Iceland, Ireland, Italy, Luxembourg, Netherlands, Norway, Spain, Sweden, Switzerland and United Kingdom.


**Table 1.** The basic descriptive statistics of variables for years 2000 and 2013.

1000 live births. In 2013, the highest infant mortality rate was reported in Turkmenistan—46.6 infants per 1000 live births. In 2000, it was 17 times higher and in 2013, 20 times higher than the lowest mortality observed in these years in Slovenia.
