Hospital efficiency under prospective reimbursement schemes: an empirical assessment for the case of Germany

Abstract

The introduction of prospective hospital reimbursement based on diagnosis-related groups (DRG) has been a conspicuous attempt to decelerate the steady increase of hospital expenditures in the German health sector. In this work, the effect of the financial reform on hospital efficiency is subjected to empirical testing by means of two complementary testing approaches. On the one hand, we apply a two-stage procedure based on non-parametric efficiency measurement. On the other hand, a stochastic frontier model is employed that allows a one-step estimation of both production frontier parameters and inefficiency effects. To identify efficiency gains as a consequence of changes in the hospital incentive structure, we account for technological progress, spatial dependence and hospital heterogeneity. The results of both approaches do not reveal any increase in overall efficiency after the DRG reform. In contrast, a significant decline in overall hospital efficiency over time is observed.

Appendices

Appendix 1: DEA efficiency scores

The efficiency score, θ ^v _i , is obtained under the assumption of variable returns to scale [3] by solving the following linear program

$$\begin{aligned} \theta_{i}^{v}=\arg \min_{\theta_{i}^{v}, \tau} \left\{\theta_{i}^{v}>0 | \sum_{l} \tau_{l} q_{pl} \geq q_{pi} \; \forall \; p \in \left\{1,\ldots ,s\right\}\right. \\ \quad\left.\theta_{i}^{v} x^D_{ki} \sum_{l} \tau_{l} x_{kl}^D \; \forall \; k \in \left\{1,\ldots ,m^D\right\}\right. \\ \quad\left.x_{ji}^N \geq \sum_{l} \tau_{l} x_{jl}^N\; \forall \; j \in \left\{1,\ldots ,m^N\right\}\right.\\ \quad\left.\sum_{l} \tau_{l} =1, \; \tau_{l}>0 \; \forall \; l =1,\ldots ,N \right\}, \\ \end{aligned} $$

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.

Appendix 2: Second-stage maximum likelihood estimation

The (unbalanced) model can be written in matrix notation as

$$y= \left( \begin{array}{lll} \lambda_1 &\cdots & 0 \\ \vdots &\ddots & \vdots\\ 0 &\cdots &{ \lambda_T} \\ \end{array}\right) \left( \begin{array}{lll} {\user2{W}}_1 &\cdots &0 \\ \vdots &\ddots &\vdots\\ 0 &\cdots &{\user2{W}}_T \\ \end{array}\right) y + \left(\begin{array}{lll} {\mathfrak{o}}_1 &\cdots &{\mathfrak{o}}_1\\ \iota_2 &\cdots & 0 \\ \vdots &\ddots &\vdots\\ 0 &\cdots & \iota_T \\ \end{array}\right) \delta + \omega + {\user2{Z}} \beta + e, \quad e = \left( \begin{array}{lll} \rho_1 &\cdots &0 \\ \vdots &\ddots &\vdots\\ 0 &\cdots &{ \rho_T} \\ \end{array}\right) \left( \begin{array}{lll} {\user2{M}}_1 &\cdots &0 \\ \vdots &\ddots &\vdots\\ 0 &\cdots &{\user2{M}}_T \\ \end{array}\right) e + \epsilon, $$

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where $y = \left(y_1^{\prime},\ldots, y_T^{\prime}\right)^{\prime}, {\user2{Z}}=\left({\user2{Z}}_1^{\prime},\ldots, {\user2{Z}}_T^{\prime}\right)^{\prime},\,e=\left(e_1^{\prime},\ldots, e_T^{\prime}\right)^{\prime}$ and $\epsilon=\left(\epsilon_1^{\prime},\ldots, \epsilon_T^{\prime}\right)^{\prime}$. The coefficients of the time dummy variables, δ _t , are collected in $\delta = \left(\delta_2 ,\ldots, \delta_T\right)^{\prime},$ where t = 1 is the benchmark, ${\mathfrak{o}_t}$ and $\iota_t$ is an N _t  × 1 vector of zeros and ones, respectively, where N _t is the number of hospitals sampled in time t. Fixed effects are summarized in $\omega = \left(\omega_1^{\prime},\ldots, \omega_T^{\prime} \right)^{\prime},$ where ω _t is an N _t  × 1 vector comprising the individual effects of the N _t hospitals. These are dropped out by means of the within transformation. The panel and cross-sectional models are estimated by means of a maximum likelihood (ML) approach.

Model (11) can be written as

$${\bf B}{\bf A}\widetilde{y} = {\bf B} \left(\begin{array}{ll} \widetilde{{{\bf 1}}} \& \widetilde{{\user2{Z}}} \end{array} \right) \left(\begin{array}{l} \delta \\ \beta\\ \end{array} \right) + \epsilon, $$

where $\widetilde{y},\,\widetilde{{\bf 1}}$ and $\widetilde{{\user2{Z}}}$ are the time demeaned variables of y, 1 and ${\user2{Z}},$ respectively, where

$${\bf 1}=\left(\begin{array}{lll} {\mathfrak{o}}_1 &\cdots &{\mathfrak{o}}_1\\ \iota_2 & \cdots &0 \\ \vdots &\ddots&\vdots\\ 0 &\cdots &\iota_T \\ \end{array}\right), \; {\user2{B}}=\left(\begin{array}{lll} {\user2{B}}_1 &\cdots &0 \\ \vdots &\ddots &\vdots \\ 0 &\cdots &{\user2{B}}_T \\ \end{array} \right), \; {\user2{A}}= \left( \begin{array}{lll} {\user2{A}}_1 &\cdots &0 \\ \vdots &\ddots &\vdots\\ 0 &\cdots &{\user2{A}}_T \\ \end{array}\right) $$

and ${\user2{B}}_t={\user2{I}}_{N_t}- {\rho_t} {\user2{M}}_t, \,{\user2{A}}_t={\user2{I}}_{N_t}-\lambda_t {\user2{W}}_t.$ Assuming a multivariate normal distribution of the error terms, the log likelihood function is given by

$$\hbox{ln}L = \sum\limits_{t=1}^T \left(-\frac{N_t}{2}\hbox{ln}(2\pi\sigma^2) + \hbox{ln}|{\user2{A}}_t| + \hbox{ln}|{\user2{B}}_t| -\frac{\epsilon_t^{\prime}\epsilon_t}{2\sigma^2}\right), $$

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where

$$\epsilon_t=\left\{\begin{array}{ll} {\user2{B}}_t \left({\user2{A}}_t \widetilde{y}_t - \widetilde{{\user2{Z}}}_t\beta\right) \forall \quad t=1\\ {\user2{B}}_t \left({\user2{A}}_t \widetilde{y}_t - \widetilde{1}_t \delta_t - \widetilde{{\user2{Z}}}_t\beta\right) \forall \quad t=2,\ldots,T \\ \end{array}\right. $$

and $\sigma^2=\sum_{t=1}^T \left(\epsilon_t^{\prime} \epsilon_t/N_t \right).$ The ML estimator is

$$\left(\begin{array}{l} \hat{\delta}_{{\text{ML}}} \\ \hat{\beta}_{{\text{ML}}} \end{array} \right) = \left[ \left(\begin{array}{l} \widetilde{{\bf{1}}^{\prime}} \\ \widetilde{{\user2{Z}}}^{\prime} \end{array} \right) \widehat{{\bf B}^{\prime}} \widehat{{\bf B}} \left(\begin{array}{ll} \widetilde{{\bf{1}}} \widetilde{{\user2{Z}}}\\ \end{array} \right)\right]^{-1} \left(\begin{array}{l} \widetilde{{\bf{1}}^{\prime}} \\ \widetilde{{\user2{Z}}}^{\prime} \end{array} \right) \widehat{{\bf B}}^{\prime} \widehat{{\bf B}}\widehat{{\bf A}} \widetilde{y}, $$

where $\widehat{{\bf B}}= \left(\begin{array}{lll} \widehat{{\user2{B}}}_1 &\cdots &0 \\ \vdots &\ddots &\vdots \\ 0 &\cdots &\widehat{{\user2{B}}}_T \\ \end{array} \right), \widehat{{\user2{B}}}_t={\user2{I}}_{N_t}-\hat{\rho}_{MLt} {\user2{M}}_t, \widehat{\user2{A}}= \left(\begin{array}{lll} \widehat{{\user2{A}}}_1 &\cdots &0 \\ \vdots & \ddots &\vdots \\ 0 &\cdots &\widehat{{\user2{A}}}_T \\ \end{array}\right)$ and $\widehat{{\user2{A}}}_t={\user2{I}}_{N_t}-\hat{\lambda}_{{\text{\it MLt}}} {\user2{W}}_t$.

Appendix 3: One-step simulated maximum likelihood estimation

The model after the first difference transformation reads as

$$\begin{aligned} \Updelta \ln q_{ij} = \Updelta \tau + \Updelta f(X_{ij},\alpha) + \Updelta v_{ij} - \Updelta u_{ij}, \\ \Updelta u_{ij} = \Updelta h_{ij} u_{ij}^{*}, \quad \Updelta \nu_{ij} \sim {{\mathcal{N}}}(0,\Upsigma_{\nu ij}), \end{aligned} $$

where $\Updelta \ln q_{ij} = \left(\begin{array}{l} \ln q_{ij2}-\ln q_{ij1}\\ \vdots \\ \ln q_{ijT_{ij}}-\ln q_{ijT_{ij}-1} \end{array} \right),\,\Updelta h_{ij}= \left(\begin{array}{l} h_{ij2}-h_{ij1}\\ \vdots \\ h_{ijT_{ij}}-h_{ijT_{ij}-1} \end{array} \right),\, {\Updelta} \nu_{ij} = \left(\begin{array}{l} \nu_{ij2}-\nu_{ij1}\\ \vdots \\ \nu_{ijT_{ij}}-\nu_{ijT_{ij}-1} \end{array} \right),\,\Updelta \tau = \left(\begin{array}{l} \tau_2-\tau_{1}\\ \vdots \\ \tau_{T_{ij}}-\tau_{T_{ij}-1} \end{array} \right)$ and $\Updelta f(X_{ij},\alpha) = \left(\begin{array}{l} \Updelta f(X_{ij2},\alpha_2) \\ \vdots \\ \Updelta f \left(X_{ijT_{ij}},\alpha_{T_{ij}}\right) \end{array} \right)$ are T _ij  × 1 vectors with $\Updelta f(X_{ijt},\alpha_t) = f(X_{ijt},\alpha_t) - f(X_{ijt-1},\alpha_{t-1}),\,f(X_{ijt},\alpha_t) = \ln x_{ijt}\alpha_{t} + \sum_{k} \sum_{k\geq l} \alpha_{tkl} \ln x_{ijtk} \ln x_{ijtl}.$ The error terms of successive time points of the i-th hospital, i.e., ν _ijt and ν _ijt−1, are correlated because of the first difference transformation. Thus, $\Updelta \nu_{ij}$ is multivariate normal distributed with covariance matrix $\Upsigma_{vij}$. The matrix has ${2\sigma_{\nu}^2}$ on the main diagonal. The off-diagonals contain either ${- \sigma_{\nu}^2}$ for successive correlated observations or zeros otherwise. For example, a hospital with data at time t = 1, 2, 3, 5, 6 obtains

$$\Updelta \nu_{ij} = \left(\begin{array}{l} \nu_{ij2}-\nu_{ij1} \\ \nu_{ij3}-\nu_{ij2}\\ \nu_{ij6}-\nu_{ij5} \end{array} \right) \quad \hbox{with} \quad \Upsigma_{vij}=\left(\begin{array}{lll} {2\sigma_{\nu}^2} &{- \sigma_{\nu}^2} & 0 \\ {- \sigma_{\nu}^2} & 2 {\sigma_{\nu}^2}& 0 \\ 0 & 0& {2\sigma_{\nu}^2} \\ \end{array} \right). $$

The estimated log-likelihood function is given by

$$\widehat{\ln L} = \sum\limits_{j=1}^J \ln \left[\frac{1}{S} \sum\limits_{s=1}^S \exp\left(\sum\limits_{i=1}^{N_j}\ln \tilde{f}\left(\varepsilon_{ij}|\eta_{j}^s\right)\right) \right], $$

where J is the number of regions, N _j is the number of hospitals in region j, η ^s _j is a (T × 1) vector of simulated random effects, and $\ln \tilde{f}(.)$ is an unbiased simulator for the conditional log-likelihood function of the i-th hospital (Wang and Ho [47])

$$\begin{aligned} \ln f(\varepsilon_{ij}|\eta_{j})= -\frac{1}{2}(T_{ij}-1)\ln(2\pi) -\frac{1}{2}|\Upsigma_{vij}| -\frac{1}{2}\Updelta\varepsilon_{ij}^{\prime} \Upsigma_{vij}^{-1}\Updelta\varepsilon_{ij} \\ \quad+\frac{1}{2}\left( \frac{\mu^2_{*}}{\sigma_{*}^2} - \frac{\mu^2}{\sigma_u^2}\right) + \ln\left( \sigma_{*} \Upphi \left( \frac{\mu_{*}}{\sigma_{*}}\right)\right) - \ln\left( \sigma_{u} \Upphi \left( \frac{\mu}{\sigma_{u}}\right)\right), \end{aligned} $$

where $\mu_{*} = {\frac{\mu/\sigma_u^2 - \Updelta\varepsilon_{ij}^{\prime} \Upsigma_{vij}^{-1}\Updelta h_{ij}}{ \Updelta h_{ij}^{\prime} \Upsigma_{vij}^{-1}\Updelta h_{ij} + 1/\sigma_u^2}},\,\sigma_{*}^2 = {\frac{1}{ \Updelta h_{ij}^{\prime} \Upsigma_{vij}^{-1}\Updelta h_{ij} + 1/\sigma_u^2}}$ and $\Updelta\varepsilon_{ij} =\Updelta \ln q_{ij} - \Updelta\tau - \Updelta f(X_{ij},\alpha)$.

Appendix 4: 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

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

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where π _j  = los _j /los _G , los _j  = (∑ ^N _i=1 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  = (∑ ^J _j=1 los _j ) /J is the mean length of stay over all clinical departments and all hospitals.

Appendix 5: Complementary empirical results

Table 3 shows the results of the estimated variance of the random regional effects of the SFA model and the spatial parameter estimates of the SARAR model under the district-based spatial weights matrices, ${\user2{W}}_d$ and ${\user2{M}}_d.$ Moreover, the Malmquist decomposition in pure efficiency change and the estimated time effects of both model specifications are provided. The spatial parameter estimates of λ,  ρ and σ ²_η are characterized by substantial heterogeneity over time. In the two-stage analysis of technical efficiency change, the spatial autocorrelation parameter, ρ, is mostly positive, while the spatial lag parameter, λ, is negative. This result is similar to the findings of [30], who applied a spatial two-stage analysis of estimated efficiency scores. However, the magnitudes of the estimated spatial effects are much smaller for efficiency change as for efficiency scores. And there is no hint for an increase in negative spatial spillovers of efficiency change due to the DRG reform as detected for the level of efficiency in Germany by [30]. The largest magnitude of both spatial parameters, λ and ρ, is obtained for periods with considerable overall efficiency improvements (1997) or deteriorations (2002). Thus, the interaction between nearby hospitals is particularly strong in periods that are characterized by an outstandingly large change in overall hospital performance.

Table 3 Complementary estimates