Regions and the selected technical parameters of hospital care
The provision of inpatient hospital care in the Czech Republic was compared to the Slovak Republic in terms of the aggregated data according to NUTS III (regions). The states were selected for a number of reasons. First, both were part of a single country in the past, a fact that determines certain common systemic elements for this international evaluation. Both countries have the same healthcare system based on the public health insurance, so-called Bismarck healthcare model. Moreover, these neighbouring Central European states use the same administrative division of public administration – state, regions, municipalities. This means, the regions of the Czech Republic and the regions of the Slovak Republic fall within NUTS III category of the European Nomenclature of Territorial Units for Statistics. An indicator which is not the same is the population size – the Czech Republic has roughly twice the population. The selected input and output parameters were therefore converted to 10,000 citizens.
Data for the Czech Republic were primarily taken from the Czech Health Statistics Yearbook for 2009–2018 [
30], published by the Institute of Health Information and Statistics of the Czech Republic (IHIS CR), with data for hospital care, i.e., both outpatient and inpatient. Data from the same area for the Slovak Republic come from the statistical documents Health Statistics Yearbook for 2009–2018 [
31] and Bed Fund 2009–2018 [
32], published by the National Health Information Centre of the Slovak Republic (NHIC SR). To make the data comparable and increase their explanatory power, they were converted using the methodology of inpatient care reporting utilised in the Czech Republic. The reference period was the last decade before the COVID-19 pandemic.
Until 1990, the structure of the healthcare facility network was identical in the Czech Republic and in the Slovak Republic, a result of the previous common existence in a single country [
33]. Following the split, both countries implemented fundamental restructuring as well as privatisation of healthcare facilities. The regulated decrease in the number of acute inpatient beds started in 1997 in favour of the aftercare beds, the need of which was growing due to population ageing. The implementation of changes in the area of provision of healthcare, the funding of healthcare (especially the introduction of the DRG classification system), and the availability of healthcare influence the average duration of hospitalisations, the use of hospital beds, as well as the overall equipment of the hospitals.
The total of 32,065 healthcare facilities (including field offices) were registered in the Czech Republic as of the end of 2018. Inpatient care was provided by 314 facilities, thereof 194 hospitals, with the total capacity of 60,633 beds. The number of hospitals did not change significantly compared to 2009 (increase by 3 hospitals), but the role of the hospital promoters and owners of the hospital assets changed, caused by the reform in the public administration as a whole. As of the end of 2018, the total of 12,902 healthcare facilities were registered in the Slovak Republic. Of this number, 180 facilities provide their services in the form of inpatient care. The inpatient care network comprises general hospitals, specialised hospitals, spa centres, treatment facilities, hospices, and nursing homes. The number of hospitals in the Slovak Republic remained almost stable within the reference decade. As of 31 December 2018, there were 114 hospitals with the capacity of 29,863 beds; the number decreased by 4 hospitals against 2009.
The set of units examined are regions as higher territorial self-governing units of the selected countries. The data comprise all hospital care providers regardless of the type or legal form. The modelling of the technical efficiency was performed using the output-oriented model, which is based on the assumption of constant returns to scale. This model should reduce the output parameters in order to attain the target, that is to make the given homogeneous production unit efficient with respect to the defined input parameters. The number of beds, the number of physicians, and the number of general nurses were chosen as input parameters. The number of beds of a healthcare facility specifies the capacity of the inpatient care and is an important regional-level input indicator, as documented by the results of the systematic scoping review [
34]. Human resources in healthcare implement the medical care, bringing new and innovative medical treatments that influence the condition and quality of health of the patients, as introduced by Vrabková, Vaňková [
35]. Likewise, the articles by Trebble et al. [
36], Winkelmann et al. [
37] point out these key regional-level parameters and emphasise the need to optimise the staffing resources in healthcare within the region. The number of hospitalised patients and the number of treatment days were chosen as output parameters. The output parameters were chosen in a manner to correspond to the logic of the selected inputs while allowing their monitoring in an aggregated form on the level of the individual regions. The productivity and structure of inpatient units is usually referred to the number of treatment days and the number of hospitalised patients, as also documented by the results of Jia, Yuan, 2017 [
26] and Bouckaert et al., 2018 [
38]. Financial parameters were deliberately not included in the input and output parameters, as the research focuses on key personnel and technical capacities allocated in the regions in order to ensure hospital care and guaranteed by the respective regional governments. Financial parameters could also distort the results of technical efficiency of the capacities because hospital care in the regions is implemented in varying proportions by both private and public providers and with mixed cash flows. The definition and description of the individual parameters is introduced in Table
1 below.
Table 1
Definition of variables
×1 | Number of physicians | Professionally competent physicians under professional supervision; professionally competent physicians without professional supervision; physicians with specialised competence.Data as of 31 December, converted number of workloads per 10,000 citizens. |
×2 | Number of nurses | General nurses; specialist assistants; paediatric nurses. Data as of 31 December, converted number of workloads per 10,000 citizens. |
×3 | Number of beds | Specified number of beds as of the last day of the reference period, i.e., 31 December, per 10,000 citizens. |
y1 | Hospitalised patients | Non-additive data. Calculated as the average number of admitted and discharged patients in the reference period. Converted number per 10,000 citizens. |
y2 | Number of treatment days | Whole day during which the patient is provided with medical services offered by the healthcare facility, i.e., including accommodation and food. The first and the last calendar day spent in the healthcare facility count as full treatment days. Converted number per 10,000 citizens. |
The statistical characteristics of the selected parameters (inputs and output) are documented in Table
2, showing minimum, maximum, and mean values. The last two lines express the absolute average increase/decrease and the mean coefficient of increase of the inputs and output per the regions in the given country between 2009 and 2018. From the point of view of mean values, it is evident that the regions of the Czech Republic show a significantly higher number of physicians (× 1) and nurses (× 2) and a slightly higher number of beds compared to the regions of the Slovak Republic. Likewise, the outputs (y1, y2) are higher in the Czech regions. The dynamics values
\(\left(\overline{d},\overline{k}\right)\) report, in the period of view (2009–2018), an increase in the number of physicians (× 1) and in the number of hospitalised patients (y1) in both countries, more significantly in the Czech regions. The number of nurses (× 2) increased in the Czech regions, while a slight decrease was reported in the Slovak regions. The number of beds (× 3) decreased in both countries during the reference period, but this decrease was much slighter in the Czech regions (mean decrease by 72 beds per region) than in Slovakia (mean decrease by 404 beds per region).
Table 2
Statistical characteristics and basic dynamics of inputs and outputs between 2009 and 2018
Min. | CZ | 321 | 1055 | 1187 | 51,830 | 300,071 |
SK | 344 | 981 | 1931 | 65,452 | 116,360 |
Max. | CZ | 4650 | 11,994 | 10,233 | 345,827 | 2,560,682 |
SK | 1491 | 3195 | 5740 | 170,911 | 1,356,790 |
Mean | CZ | 1445 | 4190.5 | 4256 | 157,182 | 1,077,624 |
SK | 760 | 2064.6 | 3870 | 122,989 | 920,674 |
\(\overline{d}\) | CZ | 359.1 | 498.9 | −71.8 | 6629.2 | − 156,199.2 |
SK | 143.7 | −98.3 | − 403.9 | 4150.7 | − 106,730.8 |
\(\overline{k}\) | CZ | 1.02 | 1.01 | 1.00 | 1.00 | 0.99 |
SK | 1.02 | 0.99 | 0.99 | 1.00 | 0.99 |
Specific results of the above statistical description and basic dynamics for the individual regions are introduced in Additional file
1, Table I.
The individual regions in both countries (CZ: 14 regions; SK: 8 regions) have been differentiated in terms of population, with the contributing factor of settlement structure, i.e., presence of large cities (regional capitals) and capitals of the country. In the Czech Republic, such regions include the Capital City of Prague CZ010, the South Moravian Region CZ064, and the Moravian-Silesian Region CZ080; and in the Slovak Republic, these are the Bratislava Region SK010, the Prešov Region SK041, and the Košice Region SK042.
Methods: DEA CCR and window analysis
Evaluation of production units was performed in three sequential steps. The output-oriented DEA model with constant returns to scale, so-called CCR model, was chosen for the evaluation of efficiency of inpatient care in the individual regions. The mathematical expression of this model is as follows (1):
$$\text{minimise under conditions}\hspace{0.12em}\begin{array}{lc}g=\sum\nolimits_j^mv_jx_{\mathit{jq},}&\\\sum\nolimits_i^ru_iy_{\mathit{ik}}\leq\sum\nolimits_j^mv_jx_{\mathit{jk},}&k=1,2,\dots,n,\\{\textstyle\sum_j^r}u_iy_{iq}=1&\\u_I\geq\varepsilon&I=1,2,\dots,r,\\v_I\geq\varepsilon,&j=1,2,\dots,m\end{array}$$
(1)
where: u
i is the weight given to output
i,
yiq is the amount of output
i produced by DMU
q, vj is the weight given to input
j, xjq is the amount of input
i produced by DMU
q.
The optimal value of the purpose function is Uq ≥ 1. The degree of technical efficiency is given by the ratio of the weighted sum of inputs to the weighted sum of outputs, but weights are sought such that the value of the efficiency measure is equal to or greater than one. A value of 1 is therefore assigned to effective units, a value greater than 1 to inefficient units.
The subsequent step of the evaluation was calculation of the window analysis (WA) for the reference period (2008–2019). It is fundamental for the purpose of analysis to determine the time window duration [
39]. A three-year time window was chosen, as the authors deem it very important to detect the time trends within the structure of healthcare provided and to attain statistical stability of the estimates obtained.
The total number of windows
w in the solved problem can be expressed by the following relationship:
The following applies:
$$\mathrm{number}\ \mathrm{of}\ \mathrm{DMUs}\ \mathrm{in}\ \mathrm{each}\ \mathrm{window}:\mathrm{np}/2$$
(3)
$$\mathrm{number}\ \mathrm{of}\ \mathrm{different}\ \mathrm{DMUs}:\mathrm{npw}$$
(4)
where: w = number of windows; n = number of DMUs; k = number of periods; p = duration of window (p ≤ k).
The total of 66 production units were analysed in each of the eight windows for
n = 22 production units, in T = 10 consecutive periods with the defined window width w = 3. This means 24 efficiency rates were calculated for each of the 22 production units. For the purposes of the final calculation, the arithmetic mean of all values determined was calculated using the formula (
5).
The total efficiency rate for the reference period is given by the relationship:
$${E}_q=\frac{\sum_{i=1}^z\sum_{t=1}^w{E}_{iq}^t}{z.v},\kern4.5em q=1,2,\dots, n$$
(5)
The last step in the evaluation of the production units in time is the Malmquist Index (MI) and its breakdown. When evaluating the changes of efficiency in time (dynamic approach to technical efficiency), the MI allows its breakdown into two components: (i.) changes in the relative efficiency of the units against the set of the remaining units, and (ii.) technology-induced frontier shift of the production possibilities [
40,
41].
The construction of the MI is based on the assumption that evaluation focuses on the production units of a certain branch over the period of time
t = 1, 2, ..., T. For each period, technology
St is known, through which inputs
xt are transformed into outputs
yt. The function
Dqt (xt, yt) characterises the technology in time
t and allocates the efficiency rate to the production unit evaluated
Uq. Efficient units define the frontier of production possibilities. The function
\({D}_q^{t+1}\) (
xt, yt) correlates the inputs and outputs from the period
t with the technology from the period
t + 1, while the function
\({D}_q^{t+1}\) (
xt + 1, yt + 1) correlates the inputs and outputs from the period
t + 1 the technology from the period
t. However, a situation may occur where (
xt + 1,
yt + 1) does not belong to the technology
St, there can be a case
\({D}_q^t\) (xt, yt) >
1, i.e., the unit evaluated attains efficiency which is higher than the frontier of production possibilities in the previous period. Also, opposite situation may arise where
\({D}_q^t\) (
xt, yt) <
1 if the course of production possibilities decreases compared to the previous period [
42].
The mathematical expression of the Malmquist Index is as follows (6):
$${M}^Q\left({x}^t+1,{y}^t+,{x}^t,{y}^t\right)$$
(6)
where
Eq is the change of the unit’s relative efficiency
q relative to other units between the periods
t and
t + 1,
Pq describes the change of the frontier of production possibilities due to the technology development between the periods
t and
t + 1. Mathematical representation of the components
Eq and
Pq is (7) and (8):
$${E}_q=\frac{D_q^{t+1}\left({x}^{t+1},{y}^{t+1}\right)}{D_q^t\left({x}^t,{y}^t\right)}$$
(7)
$${P}_q=\left[\frac{D_q^t\left({x}^{t+1},{y}^{t+1}\right){D}_q^t\left({x}^t,\kern0.5em {y}^t\right)}{D_q^{t+1}\left({x}^{t+1},{y}^{t+1}\right){D}_q^{t+1}\left({x}^t,{y}^t\right)}\right]\frac{1}{2}$$
(8)
For the purposes of the MI, where there are tasks with multiple inputs and outputs, it is necessary to use a certain DEA model, for instance, the CCR model specified above, which envisages constant returns to scale. The breakdown of the MI allows expressing its two components (efficiency change and frontier shift), where MI = efficiency change (Eq) x frontier shift (Pq).
In case of the output-oriented MI, the results are interpreted as: