On the effect of prospective payment on local hospital competition in Germany

Abstract

The introduction of prospective hospital reimbursement based on diagnosis related groups (DRG) in 2004 has been a conspicuous attempt to increase hospital efficiency in the German health sector. As a consequence of the reform a rise of competition for (low cost) patients could be expected. In this paper the competition between hospitals, quantified as spatial spillover estimates of hospital efficiency, is analyzed for periods before and after the reform. We implement a two-stage efficiency model that allows for spatial interdependence among hospitals. Hospital efficiency is determined by means of non-parametric and parametric econometric frontier models. We diagnose a significant increase of negative spatial spillovers characterizing hospital performance in Germany, and thus, confirm the expected rise of competition.

Appendices

A Efficiency measurement

1.1 A.1 DEA efficiency scores

The efficiency score, $\theta^{D}_{i}$, is obtained under the assumption of variable returns to scale [4] by solving the following linear program

$$ \begin{array}{rll} \theta^{D}_{i}=\arg \min\limits_{\theta^{D}_{i}, \, \tau} \Big\{\theta^{D}_{i}>0 \Big| \sum\limits_{l} \tau_{l} q_{pl} \geq q_{pi} \; &\forall& \; p \in \{1,...,s\} \\ \theta^{D}_{i} x^D_{ki} \sum\limits_{l} \tau_{l} x^D_{kl} \; &\forall& \; k \in \{1,...,m^D\} \\ x^N_{ji} \geq \sum\limits_{l} \tau_{l} x^N_{jl}\; &\forall& \; j \in \{1,...,m^N\} \\ \sum\limits_{l} \tau_{l} =1, \; \tau_{l}>0 \; &\forall& \; l =1,...,N \Big\}, \end{array} $$

where q _ri , $x^N_{ji}$ and $x^D_{ki}$ denote output, non-discretionary and discretionary input variables of hospital i. The numbers of outputs, non- and discretionary inputs, and reference hospitals are s, m ^N, m ^D, and N, respectively.

1.2 A.2 SFA efficiency scores

Given Maximum Likelihood estimates of the model parameters in Eq. 1 the technical efficiency of the i-th hospital is obtained as [26]

$$ \begin{array}{rll} \theta^{S}_i &=& \exp\left\{-\mathrm{E}\left[ u_i|v_i-u_i \right] \right\} \quad \mathrm{with} \\ && {\kern6pt}\mathrm{E}\left[ u_i|v_i-u_i \right]= \hat{\mu}_i^* + \hat{\sigma}^* \frac{\phi\left(\hat{\mu}_i^*/\hat{\sigma}^*\right)}{\Phi\left(\hat{\mu}_i^*/\hat{\sigma}^*\right)}, \\ \hat{\mu}_i^*&=&\frac{\hat{\sigma}^2_v \hat{\gamma}'h_i - \hat{\sigma}^2_u \hat{\nu}_i}{\hat{\sigma}^2_v + \hat{\sigma}^2_u} \quad \mathrm{and} \quad \hat{\sigma}^{*}=\sqrt{\frac{\hat{\sigma}^2_v \hat{\sigma}^2_u}{\hat{\sigma}^2_v + \hat{\sigma}^2_u}}, \end{array} $$

where $\hat{\nu}_i\!=\!\ln q_i - \hat{\alpha}_0 - \sum\limits_{k} \hat{\alpha}_k \ln x_{ik} - \sum\limits_{k} \sum\limits_{k\geq l} \hat{\delta}_{kl} \ln x_{ik} \ln x_{il}$, and Φ and ϕ are the Gaussian cumulative and probability density function, respectively [29].

B Maximum Likelihood estimation

Model 2 can be written as

$$ {\textbf{\emph B}}{\textbf{\emph A}}y = {\textbf{\emph B}}{\textbf{\emph Z}}\beta + \epsilon, $$

where B = I _N  − ρ W and A = I _N  − λ W. Assuming a multivariate normal distribution of the error terms, the log likelihood function is given by

$$ \mathrm{ln}L = -\frac{N}{2}\mathrm{ln}(2\pi\sigma^2) + \ln|{\textbf{\emph A}}| + \ln|{\textbf{\emph B}}| - \frac{\epsilon'\epsilon}{2\sigma^2}, $$

(7)

where ϵ = B $\left(\boldsymbol{A} y - \boldsymbol{Z}\beta\right)$ and σ ² = ϵ′ ϵ/N. The ML estimator is

$$ \hat{\beta}_{ML} = \left(\boldsymbol{Z}'\widehat{\boldsymbol{B}}' \widehat{\boldsymbol{B}} \boldsymbol{Z}\right)^{-1} \boldsymbol{Z}'\widehat{\boldsymbol{B}}'\widehat{\boldsymbol{B}}\widehat{\boldsymbol{A}}y $$

where $\widehat{\boldsymbol{B}}= \boldsymbol{I}_N-\hat{\rho}_{ML} \boldsymbol{M}$ and $\widehat{\boldsymbol{A}}=\boldsymbol{I}_N-\hat{\lambda}_{ML} \boldsymbol{W}$.

C Construction of case mix weights

The more time the treatments of cases belonging to the j-th clinical department take relative to all other treatments, the higher the weight, π _j , of the respective cases. Let c _ij be the number of cases in the j-th clinical department of the i-th hospital. Then, the weighted cases of hospital i are calculated as

$$ wc_i= \sum\limits_{j=1}^J \pi_j \, c_{ij}, $$

where π _j  = los _j /los _G , $los_j=(\sum_{i=1}^N days_{ij}/c_{ij})/N$ is the mean length of stay for the cases belonging to the j-th clinical department over all hospitals and $los_G=(\sum_{j=1}^J los_j) /J$ is the mean length of stay over all clinical departments and all hospitals.

D Outlier detection

Table 5 shows the share of hospitals in the population and in the sample of the first and second stage over different subgroups characterized by the number of beds and ownership form. In the sample of both stages hospitals with less than 100 beds and profit oriented hospitals are under-represented.

Table 5 Distribution of hospitals pooled over all time periods

E First stage results

This appendix provides further details on the results of the parameter estimates of the SFA model and descriptive statistics of SFA and DEA efficiency scores.

1.1 E.1 SFA parameter estimates

The results of the SFA model are shown in Table 6. The parameter estimates are characterized by substantial heterogeneity over time. The estimated variance of the noise component, $\sigma_v^2$, decreases remarkably after 1996 and 2001 suggesting less measurement errors and a higher quality of the data. Similarly, the estimated variance of the inefficiency term, $\sigma_u^2$, shrinks after 2001. Moreover, the ratio of both variances, $\sigma_u^2/\sigma_v^2$, indicates a lower proportion of signal to noise after 2001.

Table 6 Estimates of the SFA translog model

To facilitate the interpretation of the estimated translog input coefficients output elasticities with respect to inputs are also reported. The estimated output elasticities are mostly significantly positive for the most years and, similar to other estimates, vary substantially over time. After 2001 the amount of material expenses and the number of non-medical employees appear to have a (theoretically implausible) negative effect on the number of weighted treated cases. However, these estimates are mostly insignificant. The exogenously fixed input variable beds obtains a negative estimate, indicating a positive effect of the number of beds on efficiency. The magnitude of the estimated impact is decreasing after 2001.⁶

The variation in the estimates over time might be explained by two reasons. Firstly, the definition of the costs has changed in 1996 and 2002. Secondly, the application of minimally invasive procedures has been expanded over the last years leading to shorter rehabilitation time and decreased hospital costs [50]. Both instances might have an effect on the stochastic noise in the hospital data and, thus, influence the parameter estimates.

1.2 E.2 DEA and SFA efficiency scores

Descriptive statistics of estimated DEA and SFA efficiency scores are shown in Table 7. For all subgroups mean SFA efficiency scores are markedly higher than their DEA counterparts. This could be seen as a reflection of the fact that in the DEA model all deviations from the frontier are assigned as inefficiency, while it is separated from stochastic noise in the SFA model. The difference between both efficiency measures varies over beds and ownership form. For instance, profit oriented hospitals are characterized by the highest DEA but lowest SFA efficiency scores. In total, 3% of the hospitals are DEA-efficient, i.e. $\theta^D_i=1$. The share of these hospitals varies over the groups. Almost 11% of small (<100 beds) and privately owned hospitals are clarified as efficient by means of DEA, while in the other groups the share varies between 0.87% (100–299 beds) to 3.97% (>500 beds). Similar to the SFA parameter estimates the Spearman rank correlation between the alternative efficiency measures vary over time.⁷ In the years 1995 and 1996, the correlation between both efficiency scores is negative. This might be explained by the poor quality of the data as suggested by the relative large variance of the noise component $\hat{\sigma}^2_{v}$ (Table 6). Measurement errors are in general more detrimental to DEA, since it exploits the noise as inefficiency (e.g. [5, 48]). In the SFA model an assumption about the distribution of the inefficiency term is required to separate inefficiency from noise. However, the decomposition of the error term could fail under large measurement errors [48]. Therefore, in the presence of large amounts of noise both methods may perform poorly. This might explain the remarkable differences between the efficiency scores. After 1996 the variance of the noise component drops to one third of its previous magnitude and the correlation between DEA and SFA efficiency scores reaches almost 78% in 1997. It decreases over time to 40% in 2006. The decline in the correlation is in line with the higher proportion of the noise component after 2001 as indicated by the decreased ratio of estimated variances $\hat{\sigma}^2_u/\hat{\sigma}^2_v$. Hospitals with more than 300 beds are characterized by the strongest similarity between both efficiency measures. In summary, the statistics document the heterogeneity between parametric and non-parametric efficiency measures as also documented in other comparative studies of hospital performance (e.g. [10, 16, 23, 24]).

Table 7 Summary statistics and correlation of DEA and SFA efficiency scores (in %)