The aim of the research is to determine the relationship between technical efficiency and quality of hospital care in the conditions of the federal states of Germany and to approximate how the selected external factors influence their results.

Research questions QR1 and QR2 were formulated in connection with the aim of the research:

QR1: Is it possible to confirm a positive relationship between the level of technical efficiency of hospitals and the patient satisfaction with hospital care in the conditions of most federal states of Germany?

QR2: Is it possible to significantly explain the growth or decrease of technical efficiency of hospitals and the patient satisfaction with hospital care using selected external factors in the individual federal states of Germany?

The investigation comprises 64 systemic observations/homogeneous production units (N/DMUs = 64; 16 states × 4 years). It is therefore a macroeconomic view of the efficiency and quality of the hospitals within the context of external conditions.

There were 1887 hospitals in Germany in 2021, which can be classified according to their ownership, being the most frequently used classification of hospitals (Helmig 2005; Eichhorn 1976; Krolop et al. 2010). Based on ownership, hospitals fall into three main categories: for-profit or private hospitals; non-profit hospitals; and public hospitals. By law, public hospitals are part of the state, either wholly owned by the regional or local authorities, or as state-controlled organisations. Hospitals in non-profit ownership must use all their profits to support their organisational goals or missions and do not allocate surplus to the hospital’s controlling entity. By contrast, private hospitals are motivated by profit and have the right to distribute profits to their shareholders (Jeurissen et al. 2021; Lindlbauer and Winter 2016; Sonnentag et al. 2007).

For the sake of comparability, 353 psychiatric and university hospitals were not included in the research. The total number of hospitals included in the research is 1,534 general hospitals in Germany, of which 450 is in public ownership, 499 in non-profit ownership, and 585 in private ownership (GBE Health reporting of the federal government 2022; Federal Statistical Office, 2022). The numbers and types of hospitals in the individual federal states are introduced in Appendix 1.

2.1 Data

To calculate technical efficiency (TE), two universal inputs were selected, specifying the number of human resources of the hospitals (physicians and nurses) available to the state in proportion to the population attended (population of the state concerned, identified by lower-case letters x). Input × 1 – number of inhabitants per 1 physician in the federal state concerned; input × 2—number of inhabitants per 1 nurse in the federal state concerned. Professional medical staff is the essential and most important human resource, with the highest costs. Two outputs were selected in logical relationship to the inputs, referring to the hospital’s productivity in relation to the number of beds, identified by lower-case letters y. Output y1—number of hospitalised patients per 1 bed; output y2—number of treatment days per 1 bed. Data were obtained from public sources (Basic data of hospitals 2020; Gesundheits-Bercihtersstattung, 2023) (Federal Statistical Office 2022, 2023a). Due to the selected macro-view of the technical efficiency and the diversity of medical fields and the related expensiveness in the hospitals, the authors chose the input-oriented DEA model distinguishing the constant returns to scale (CRS) and variable returns to scale (VRS). Also, the relative concepts of inputs and outputs decreases the differences between the federal states implied by the composition of medical fields and their intensity of care as well as the size of the states. Table 1 shows the statistical description of the selected inputs and outputs.

Table 1 Statistical description of inputs and outputs

The lowest number of inhabitants per 1 physician is in the city states of Berlin, Bremen, and Hamburg, while the largest number of inhabitants per 1 physician is in the states of Rhineland-Palatinate, Lower Saxony, and Brandenburg. The number of inhabitants per 1 nurse is more varied than in the case of physicians. The lowest number of nurses is in the states of Bremen, Saarland, and Hamburg, while the largest number of inhabitants per 1 nurse is in Brandeburg, Schleswig–Holstein, Saxony, and Hesse.

The largest number of hospitalised patients per 1 bed is in the states of Bremen and Lower Saxony. By contrast, the lowest number of hospitalised patients per 1 bed is in the states of Brandeburg and Thuringia. The largest number of treatment days per 1 bed is in the states of Berlin, Hamburg, and Saarland, while the lowest number of treatment days per 1 bed is in the states of Rhineland-Palatinate, Saxony-Anhalt, and Thuringia.

The quality of hospital care is presented in the results of the regular nationwide survey of satisfaction of the hospital care patients. The assessment of the patient satisfaction scores was based on the data from the written survey among the insured persons with the patient experience survey (Patient Experience Questionnaires, PEQ) (Liste 2023), provided by Weisse Liste gGmbH for scientific purposes.

The general questionnaire—PEQ—was developed by Weisse Liste gGmbH and consists of 15 questions. The answers are measured using a five-point scale (1–5), where 1 is the best score and 5 is the worst score. The questions are divided into the following five topics (Q1–Q5): Q1—satisfaction with physician’s services (four questions); Q2—satisfaction with nursing services (four questions); Q3—satisfaction with the hospital organisation and services (five questions); Q4—patient’s recommendations (one question); and Q5—improvement of the health status after hospitalisation (one question). The data provided comprise aggregated results from the survey waves PEQ 32 (March 2018) to PEQ 56 (December 2021). Due to significant restrictions of normal hospital operation caused by the Covid-19 pandemic, which began in March 2020, the patient surveys with PEQ were suspended for three survey waves (April 2020, July 2020, and February 2021) (Liste 2023).

For the purposes of this survey, the data provided were used to calculate the overall patient satisfaction (TQ) using the method of five question mean, see Appendix 2.

Table 2 shows that in 2018–2021, the patients gave the best rating to the physicians and nurses, while they were less satisfied with the hospital services and organisation as well as improvement of their health status as a result of hospital care. Yet it may be argued that in general, the patient satisfaction results in the individual areas are nearly equal, oscillating around the value of 2, which can be translated as rather satisfied.

Table 2 Statistical description of patient satisfaction ratings (1–5)

Least satisfied, on average, were patients in the state of Bremen in all years surveyed and the state of North Rhine-Westphalia in 2018 and 2019. Most satisfied, on average, were patients in the states of Thuringia, Saxony-Anhalt, and Hamburg in 2021.

There are numerous external factors which may influence technical efficiency and perception of satisfaction. Nevertheless, with respect to the selected macro-analysis, selected inputs and outputs for the calculation of the DEA model, and previous research (Vrabková and Lee 2023; Vaňková and Vrabková 2022), the authors selected factors that reflect the population age and legal form of the hospital care providers (see factors X1–X5, Table 3). Data were acquired from public sources (Federal Statistical Office 2023b). In order to fine-tune the specificities of the 16 federal states, the relative value of the factors was selected in the form of proportions (%) of the respective population age in the state in the total population and the proportion of the legal forms of hospitals in the total number of hospitals in the given state. Population age is the crucial problem of modern Western societies; the population is ageing and is exposed to new social, economic, and environmental conditions, accentuated by a higher demand for medical and social care (Horecký et al. 2021). Economic orientation of the hospital care providers and their legal form is the classic precondition of allocational and technical efficiency (Ali et al. 2020). In contrast to non-profit hospitals supported by public administration, private hospitals, oriented towards economic profit, naturally tend to a more rational operation (Gavurova and Kocisova 2020; Vrabková and Vaňková 2021). Other economic external factors such as GDP, unemployment, and average income, were not included in the set due to the results of the multicollinearity testing according to the variance inflation factor (VIF) and the Pearson correlation coefficient (PCC), and population aged 15–64 was not included in the tobit regression model.

Table 3 Statistical description of external factors, in %

The total average values of the proportions of the three legal forms (see Table 3) in the total number of hospitals in the individual states show almost comparable “hospital market” share. However, there are significant differences in the states in terms of the numbers of hospitals falling within the individual legal forms. The highest proportion (44–41%) of public hospitals in the total number of hospitals in the state was found in the states of Bavaria, Bremen, and Saxony. On the contrary, the lowest proportions are in Berlin and Hamburg (3–5%). In the case of non-profit hospitals, the highest proportion was reported in the states of North Rhine-Westphalia, Rhineland-Palatinate, and Saarland (69–62%), while the lowest proportion was in Bavaria, Rhineland-Palatinate, and Saxony (11–17%).

The age structure of the population is very homogeneous, showing only minor differences in the representation of the two limit age groups in the total population in the state. Population up to the age of 15 accounts on the average for 14% proportion in the total population, and population aged 65+ accounts on the average for 23% proportion in the total population. The highest proportion of the youngest population is in the states of Berlin and Hamburg (15%), the lowest is in Saarland and Saxony-Anhalt (12%). Population aged 65+ accounts for the largest proportion in the states of Saxony-Anhalt and Thuringia (27–28%), while the lowest proportion of the oldest age group is in Hamburg and Berlin (18–19%).

2.2 Multi-criteria estimation of the technical efficiency according to the DEA model

Models for the assessment of production unit efficiency that take into account multiple variables—inputs and outputs—affecting the efficiency of the units fall within the category of multi-criteria decision-making, which also includes the data envelopment analysis model (DEA model).

In the case of multiple inputs expended to produce multiple outputs, the relative efficiency rate Uq is used, expressed by formula (1). The relative efficiency rate means that its value depends on the whole set of units. In other words, if the set of units assessed is increased by another unit that changes the effective limit, the efficiency rate of the remaining units changes as well. The prerequisite for (1) is a set of homogeneous production units (DMUs) U1, U2, ….Un, within which r outputs and m inputs are tracked. In the input–output matrix, inputs are identified as X = , and outputs are identified as Y = ,

$$_=\frac\text \frac_}}}_}}}_}}_}}}_}}}_}} ,$$

(1)

where vi, i equals 1, 2 …, m are weights given to the i input and uk, k equals to 1, 2, …, r are weights given to the j output (Cooper Seiford Joe 2011; Toloo 2014; Dlouhý et al. 2018).

The principle of the DEA model consists in dividing the objects surveyed to efficient and inefficient according to the extent of resources expended and the quantity of production or another type of output. The DEA model determines the empirical production function. The DEA model compares the units against the best units. The DEA models are based on the premise that for each problem, there is a production possibility set comprising all possible (permissible) combinations of inputs and outputs. The production frontier determines the optimal relationship between the input and outputs in order to maximise the output at the given the specific value of the input(s). The production possibility set is determined by the efficient frontier. To infer the efficient frontier, and therefore the set of permissible options, it is necessary to make assumptions about the nature of the returns to scale for the given problem.

DEA models have several modifications, with two types being the core ones—the CCR model (developed by Charnes, Cooper, and Rhodes) that expects constant returns to scale, and the BCC model (developed by Banker, Charnes, and Cooper) that expects variable returns to scale. In the case of constant returns to scale (CRS), the efficient frontier is conical in shape. By contrast, the expectation of variable returns to scale (VRS) leads to modifying the efficient frontier, convex in shape. The advantages and limitations of the basic DEA models were thoroughly considered with respect to the theoretical assumptions and verified application testing in research (Cook and Zhu 2005; Cooper et al. 2007). The main reason behind selecting the basic input-oriented models with extended returns to scale (CRS and VRS) is especially the research aim and the nature of the selected inputs and outputs, allowing an aggregated view of the homogeneous production units (all general hospitals within the given federal state). Two universal inputs (numbers of physicians and numbers of nurses) and equally universal outputs (numbers of hospitalised patients and bed occupancy rate in days) were used, and to compare the selection, it is possible to introduce the synthesis of Kohl et al. (Kohl et al. 2019), which shows that these are the most commonly used inputs and outputs in the DEA models. Other possible inputs like the number of beds, number of other employees, and the number of hospitals were not included in the calculation due to multicollinearity. On the other hand, subtler differentiation of inputs and outputs (e.g., in the form of absolute or relative expenditure and income) enables more accurate estimation of technical efficiency. This more likely applies to micro-economic conditions where homogeneous units are represented by university hospital, specialised clinics, etc., but may be a reason behind distorted results in macro-analyses (the hospitals are of different sizes, have specific requirements resulting from the specialisation of care provided, etc.). The calculation of technical efficiency also considers the model’s input orientation or output orientation.

If the calculation result is one, the unit Uq is efficient. Inefficient units have the efficiency rate less than one, i.e., z is less than 1. Mathematical notations of the calculations of input- (as well as output-) oriented models CRS and VRS are based on the formula (1) with the addition of the conditions of returns (constant/variable) and the maximisation/minimisation conditions. Input-oriented models are maximised. Calculations were made using the licensed programme DEAFrontier for MS Excel.

2.3 Regression analysis using the tobit model

Regression analysis is among the most frequently applied methods in econometrics. The regression models describe the relationship between two and more variables. Using regression analysis, it is possible to specify how an independent variable affects a dependent variable, i.e., how the dependent variable changes in response to the changes in the independent variable. In other words, this describes unidirectional dependence between economic quantities. In practice, most dependent variables cannot be explained by the influence of just one (independent) variable, so it is necessary to utilise multiple regression with several independent variables. The multiple regression model may be generally defined by formula (2):

$$_=_+__+__+ \dots + \beta }__+_,$$

(2)

where Y is the dependent (explained) variable, β0 is the constant, βn are estimated coefficients. Variable X is the independent or explanatory variable and μ is the random component (error term). It describes stochastic effects that affect the dependent variable Y, but were not explained by independent variables X (Soukup et al. 2023).

Tobit regression model, developed by James Tobin (1958), is used in the case of censored or otherwise limited dependent variable. Data may be censored from above (right-censored), from below (left-censored), or from both sides (interval-censored). In the standard tobit model, the dependent variable Y is censored by zero from below (Henningsen 2011). It can be described mathematically by formula (3)

$$y=\left\_^&\quad if\, _^>0\\ 0 &\quad if \,_^\le 0\end\right.$$

(3)

where yi is the latent (unobserved) dependent variable and xi is the observed independent variable. Coefficient β is the vector of unknown parameters and μi is the error term.

In practical tasks that use the DEA model (Fatih et al. 2020; Ayiko et al. 2020; Piubello Orsini et al. 2021) where the efficiency score varies between < 0, 1 > , the most used approach is the censored regression, most commonly the tobit regression. This is because the use of classic regression is likely to produce distorted results because the least square condition, used for estimating the regression model parameters, is not fulfilled.

In this research, tobit regression model has three variants, differentiated only by the dependent variable Y1, Y2 and Y3, see formulas (4a, 4b, 4c):

$$Y1(CRS)= _+_X1+_X2+_X3+_X4+_X5+\mu$$

(4a)

$$Y2 (VRS)= _+_X1+_X2+_X3+_X4+_X5+\mu$$

(4b)

$$Y3 (TQ)= _+_X1+_X2+_X3+_X4+_X5+\mu$$

(4c)

In tobit model (4a), the dependent variable Y1 is the total technical efficiency—CRS; censored < 0, 1 >. In tobit model (4b), the dependent variable Y2 is the net technical efficiency—VRS; censored < 0, 1 >. In tobit model (4c), the dependent variable Y3 is the overall satisfaction, censored < 1, 3 > .

Calculations were performed using the licensed statistical programme STATA 15.1.