Comparing the Efficiency of Hospitals in Italy and Germany: Nonparametric Conditional Approach Based on Partial Frontier

Abstract

Traditional nonparametric frontier techniques to measure hospital efficiency have been criticized for their deterministic nature and the inability to incorporate external factors into the analysis. Moreover, efficiency estimates represent a relative measure meaning that the implications from a hospital efficiency analysis based on a single-country dataset are limited by the availability of suitable benchmarks. Our first objective is to demonstrate the application of advanced nonparametric methods that overcome the limitations of the traditional nonparametric frontier techniques. Our second objective is to provide guidance on how an international comparison of hospital efficiency can be conducted using the example of two countries: Italy and Germany. We rely on a partial frontier of order-m to obtain efficiency estimates robust to outliers and extreme values. We use the conditional approach to incorporate hospital and regional characteristics into the estimation of efficiency. The obtained conditional efficiency estimates may deviate from the traditional unconditional efficiency estimates, which do not account for the potential influence of operational environment on the production possibilities. We nonparametrically regress the ratios of conditional to unconditional efficiency estimates to examine the relation of hospital and regional characteristics with the efficiency performance. We show that the two countries can be compared against a common frontier when the challenges of international data compatibility are successfully overcome. The results indicate that there are significant differences in the production possibilities of Italian and German hospitals. Moreover, hospital characteristics, particularly bed-size category, ownership status, and specialization, are significantly related to differences in efficiency performance across the analyzed hospitals.

Appendix

1.1 The unconditional efficiency measure

The production technology is described by the vector of inputs $X\in {R}_{+}^p$ and the vector of outputs $Y\in {R}_{+}^q$. The production set Ψ includes all technically feasible combinations of inputs and outputs and is defined as follows:

$$ \varPsi =\left\{\left(x,y\right)\ |\ \mathrm{x}\ \mathrm{can}\ \mathrm{produce}\ y\right\}. $$

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Cazals et al. [12] and Daraio & Simar [13] proposed a probabilistic formulation to describe the production process using the joint probability function:

$$ {H}_{XY}H\left(x,y\right)= Prob\left(X\le x,\mathrm{Y}\ge y\right), $$

(2)

in which H _XY (x, y) is the probability that a decision-making unit (DMU) operating at (x, y) will be dominated.

In the output-oriented case, the joint probability function H _XY (x, y) can be decomposed as follows:

$$ \begin{array}{c}\hfill {H}_{XY}\left(x,y\right)= Prob\left(\ Y\ge y|X\le x\right) Prob\left(X\le x\right)\hfill \\ {}\hfill ={S}_{Y\Big|X}\left(y|x\right){F}_X(x),\hfill \end{array} $$

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in which F _X (x) is the cumulative distribution function of X and S _Y|X (y| x) is the conditional survival function of Y.

Under free disposability, the Farrell-Debreu output-oriented efficiency is given by:

$$ \lambda \left(x,y\right)=\mathit{\sup}\left\{\lambda |{S}_{Y\Big|X}\left(\lambda y|x\right)>0\right\}. $$

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The nonparametric estimator of λ(x, y) is obtained from the empirical version of the conditional survival function:

$$ {\widehat{S}}_{Y\Big|X,n}\left(y|x\right)=\frac{\sum_{i=1}^nI\left({x}_i\le x,{y}_i\ge y\right)}{\sum_{i=1}^nI\left({x}_i\le x\right)}, $$

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where I(∙) is an indicator function that takes the value of one if the argument is true and zero otherwise.

To obtain robust estimates, Cazals et al. [12] suggested estimating the partial efficiency measure (of order-m) by comparing a unit (x, y) to m randomly selected peers from the population of units producing more output than y. The order-m output-oriented efficiency measure is given by the following integral:

$$ {\widehat{\lambda}}_{m,n}\left(x,y\right)={\int}_0^{\infty}\left(1-{\left(1-{\widehat{S}}_{Y\Big|X,n}\left(uy|x\right)\right)}^m\right)du $$

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1.2 The conditional efficiency measure

Cazals et al. [12] and Daraio & Simar [13] demonstrated the conditional measures of efficiency by incorporating the set of environmental variables Z∈R ^r that might explain part of the production process.

The attainable conditional production set can be expressed by:

$$ {\varPsi}^Z=\left\{\left(x,y\right)\right|Z=z,x\ \mathrm{can}\ \mathrm{produce}\ y\Big\}. $$

The conditional measure of output-oriented efficiency must be adapted to the condition Z = z and is given by:

$$ \lambda \left(x,y|z\right)=\mathit{\sup}\left\{\lambda |{S}_{Y\Big|X,Z}\left(\lambda y|x,z\right)>0\right\}, $$

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in which S _Y|X , Z (y| x , z) = prob(Y ≥ y| X ≤ x, Z = z) is the conditional survival function.

Owing to the equality constraint Z = z, the nonparametric estimation of S _Y|X , Z (y| x , z) differs from the unconditional case because a smoothing technique in z is used. This technique requires using a kernel estimator:

$$ {\widehat{S}}_{Y\Big|X,Z,n}\left(y|x,\mathrm{z}\right)=\frac{\sum_{i=1}^nI\left({x}_i\le x,{y}_i\ge y\right)K\left(\left(z-{z}_i\right)/{h}_n\right)}{\sum_{i=1}^nI\left({x}_i\le x\right)K\left(\left(z-{z}_i\right)/{h}_n\right)}, $$

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in which K(∙) is some kernel function with compact support and h _n is the observation-specific bandwidth. De Witte & Kortelainen [43] suggested employing the kernel function of Li & Racine [44], which accommodates continuous, ordered, and unordered discrete environmental variables.

Similarly to the unconditional order-m efficiency, the conditional measure of output-oriented order-m efficiency is obtained by the following integral:

$$ {\widehat{\lambda}}_{m,n}\left(x,y|z\right)={\int}_0^{\infty}\left(1-{\left(1-{\widehat{S}}_{Y\Big|X,Z,n}\left(uy|x,z\right)\right)}^m\right)du $$

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