- Email: [email protected]
Estimating hospital production functions through flexible regression models
Mathematical and Computer Modelling 54 (2011) 1760–1764
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Estimating hospital production functions through flexible regression models Francisco Reyes Santías a,c,∗ , Carmen Cadarso-Suárez b,d , María Xosé Rodríguez-Álvarez b,c,d a
Instituto Universitario de Ciencias Neurológicas, University of Santiago de Compostela, Spain
b
Unit of Biostatistics, Department of Statistics and Operations Research, University of Santiago de Compostela, Spain
c
Complexo Hospitalario Universitario de Santiago de Compostela (CHUS), Santiago de Compostela, Spain
d
Instituto de Investigación Sanitaria de Santiago de Compostela (IDIS), Santiago de Compostela, Spain
article
info
Article history: Received 5 October 2010 Received in revised form 29 November 2010 Accepted 30 November 2010 Keywords: Generalized Additive Models (GAMs) Production function Cobb–Douglas Translog
abstract The subject of discussion is health care production functions. The parametric two-factor Cobb–Douglas and transcendental logarithmic (translog) production functions are frequently used. However, empirical and theoretical work has often questioned the validity of the parametric Cobb–Douglas and translog as a model for the production of health care. The aim of this study is to propose a new flexible form of production function based on Generalized Additive Models (GAMs) and to compare it with the classical approaches, using data from public hospitals in the Spanish Region of Galicia during the period 2002–2008. © 2010 Elsevier Ltd. All rights reserved.
1. Background Two models are commonly used in the estimation of hospital production function [1]: the Cobb–Douglas model and the transcendental logarithmic (translog) model. Cobb–Douglas has long been popular among economists because it is easy to work with and can explain the substitution between health care inputs. However, empirical and theoretical work has often questioned the validity of the parametric Cobb–Douglas as a model for the production of health care [2,3]. In comparison with the Cobb–Douglas model, the translog function model has a number of advantages. This model adds the effects of interactions between inputs. The Cobb–Douglas model, in contrast, omits this effect, which is less realistic. Thus, most of the hospital production function studies have used this flexible translog function form [1,3]. Nevertheless, in some circumstances, parametric models like the Cobb–Douglas or Translog models can be very restrictive. Using these models for estimation and prediction, the functional shape of continuous inputs is ‘‘forced’’ to follow a linear parametric form, which frequently does not fit the data closely. The relative lack of flexibility of parametric models has led to the development of non-parametric regression techniques based on the broad family of generalized additive models (GAMs; see [4,5]). These techniques do not impose a parametric form on the effects of continuous inputs; instead, they assume only that these effects are additive and reasonably smooth, and can be estimated using a variety of non-parametric local smoothing methods. This paper studies the use of Additive Models (AMs) to calculate hospitals production functions. The results of the new approach have been compared with the two most popular production functions used in the health care sector, the Cobb–Douglas and the translog models.
∗
Corresponding author at: Instituto Universitario de Ciencias Neurológicas, University of Santiago de Compostela, Spain. Tel.: +34 678487167. E-mail address: [email protected] (F.R. Santías).
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.11.087
F.R. Santías et al. / Mathematical and Computer Modelling 54 (2011) 1760–1764
1761
2. Data description The variables we use consist of inputs to hospital production in the form of capital and labour, and outputs from production. We have chosen the output of Inpatient care measured as the number of admissions standardized by means of complexity, obtaining homogeneous units of production (UPHs), calculated by multiplying the number of admissions by the complexity (weight) obtained from the Diagnostic Related Groups (DRGs) [6]. Following Ferrier and Valmanis [7], hospital inputs are measured as follows: in terms of capital we use the average number of beds (Beds) in each hospital. Labour inputs are measured by the number of consultant full-time equivalents (FTEs) employed in each hospital. Workload statistics were collected by hospitals, as panel data, from the Regional Ministry’s Information System (Galicia, Spain), during the period 2002–2008. Hospitals have been classified within three clusters by Reyes [8] following their size: Cluster 1 (small; <200 beds), Cluster 2 (medium; 200–650 beds), and Cluster 3 (large; >650 beds). Of the total number of observations (n = 1139), 468 (41.1%) correspond to Cluster 1, 301 (26.4%) to Cluster 2, and 370 (32.5%) to Cluster 3. 3. The statistical models The Cobb–Douglas production function proposed by Charles W. Cobb and Paul H. Douglas [9], takes the following form for our model: nH −1
log UPHs = β0 + β1 log FTEs + β2 log Beds + β3 Year +
−
αk Hospitalk + ε.
(1)
k=1
A standard procedure for introducing the possibility of technical change is to include a time trend (Year). This captures observed changes in the technology. An alternative to the Cobb–Douglas production function is the translog production function [10]. The form of translog production function used is as follows. nH −1
log UPHs = β0 + β1 log FTEs + β2 log Beds + β3 log FTEs log Beds + β4 Year +
−
αk Hospitalk + ε.
(2)
k=1
The flexible model considered was the following AM, including a Beds-by-FTEs interaction. nH −1
log UPHs = β0 + β1 Year + f1 (log FTEs) + f2 (log Beds) + f3 (log FTEs, log Beds) +
−
αk Hospitalk + ε
(3)
k=1
where f1 and f2 are unknown smooth functions of the number of beds (log scale) and the number of physicians (log scale) respectively, and f3 is an unknown smooth function representing the possible interaction between the number of beds and the number of physicians (both in log scale). It should be noted that the categorical covariate ‘Hospital’ was also included in the previous models. In these models, nH denotes the number of hospitals (in our study 10), and Hospitalh is a dummy variable taking the value 1 for the hth hospital and 0 otherwise. With regard to the estimation of the model (3), penalized thin plate splines were used to represent the smooth functions f1 , f2 and f3 , and the optimal smoothing parameters were estimated via Restricted (or Residual) Maximum Likelihood (REML, see e.g. [11]). All the statistical analyses were performed using R software, version 2.9.1 [12]. AMs were fitted using mgcv package [5]. 4. Results In this section, we present in Table 1 the results of each estimated model, for the Regional Health Service hospitals as an overall and every hospital Cluster. We evaluate the models based on the AIC (Akaike Information Criterion [13]) and the economic interpretation for an output change due to changes in input factors. We also present for each model, the value of the corrected R2 . First of all, the models have been estimated for the Regional Health Service hospitals as an overall. The three variables (Beds, FTEs and Hospital) are all significant (p < 0.005) for both the translog and AM models. Even more, the interaction between Beds and FTEs is also significant for those models. However, Year variable, as a proxy of changes in technology of production is not significant (CD p = 0.476, translog p = 0.477, AM p = 0.690). This outcome indicates that technical change in production is neutral in relation with output. Related to goodness-of-fit for the models, both the R2 and AIC outcomes, indicate that the AM provides a better fit in comparison with the two classic models, CD and translog. Fig. 1 depicts productivity growth as a result of increments in the inputs, based on the flexible model. The figure includes three lines. The upper line represents a productivity change according to variations in inputs. The right lower line represents
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Table 1 Cobb–Douglas (CD), translog and AM models estimates for Global model, Cluster 1, 2 and 3. AIC = Akaike Information Criterion; df = degrees of freedom; s() = smoother using thin plate splines. p-value
R2 (100%)
AIC
1.00 1.00 1.00 9.00
<0.001 0.6839 0.476 <0.001
78.12
2610.092
1.030 0.147 −0.053 −0.011 –
1.00 1.00 1.00 1.00 9.00
<0.001 0.004 <0.001 0.477 <0.001
78.34
2599.273
s (Beds) s (FTEs) s (Beds, FTEs) Year Hospital
– – – −0.006 –
5.44 5.32 15.00 1.00 9.00
<0.001 <0.001 <0.001
82.90
2355.801
CD
Beds FTEs Year Hospital
0.816 0.018 −0.011 –
1.00 1.00 1.00 2.00
<0.001 0.706 0.654 0.849
63.41
1073.084
0.684 0.051 −0.010 –
1.00 1.00 1.00 1.00 2.00
<0.001
Translog
Beds FTEs Beds × FTEs Year Hospital
0.205 0.098 0.683 0.848
63.54
1072.302
AM
s (Beds) s (FTEs) s (Beds, FTEs) Year Hospital
– – – 0.015 –
4.38 5.42 15.16 1.00 2.00
0.193 0.018 0.041 0.453 0.970
78.50
875.337
Model
Effects
Coefficients
CD
Beds FTEs Year Hospital
0.917 0.014 −0.011 –
Translog
Beds FTEs Beds × FTEs Year Hospital
AM
df
Global
0.690 <0.001
Cluster 1
−0.128
Cluster 2
CD
Beds FTEs Year Hospital
1.021 0.087 −0.047 –
1.00 1.00 1.00 3.00
<0.001 0.343 0.158 <0.001
77.68
735.695
Translog
Beds FTEs Beds × FTEs Year Hospital
1.167 0.262 −0.107 −0.049 –
1.00 1.00 1.00 1.00 3.00
<0.001 0.015 0.002 0.140 <0.001
78.32
728.013
AM
s (Beds) s (FTEs) s (Beds, FTEs) Year Hospital
– – – −0.063 –
5.66 3.38 10.40 1.00 3.00
<0.001 0.638 <0.001 0.022 <0.001
85.30
627.366
0.930
1.00 1.00 1.00 2.00
<0.001 78.01
775.441
Cluster 3
CD
Beds FTEs Year Hospital
−0.030 0.023 –
0.595 0.359 0.417
(continued on next page)
variations in capital input (Beds) while the left side line shows variations in labour input (FTEs). According to the upper line, Fig. 1(a) shows a steady growth in productivity. This growth is dominated by the capital factor trend variable in the estimates while the contribution of the growing labour factor variable is slightly modest. Following the results for Cluster 1, unlike the previous findings, no significant effect of Beds variable has been detected in the AM (p = 0.193). Nevertheless, only in the AM, FTEs variable has a significant effect (AM p = 0.018, CD p = 0.706,
F.R. Santías et al. / Mathematical and Computer Modelling 54 (2011) 1760–1764
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Table 1 (continued) Model
Effects
Translog
Beds FTEs Beds × FTEs Year Hospital s (Beds) s (FTEs) s (Beds, FTEs) Year Hospital
AM
Coefficients 0.904
df
0.023 –
1.00 1.00 1.00 1.00 9.00
– – – 0.021 –
4.20 6.10 16.41 1.00 2.00
−0.057 −0.011
p-value
R2 (100%)
AIC
<0.001 0.545 0.772 0.363 0.418
<0.001 0.037 0.022 0.293 0.814
a
b
c
d
77.96
777.311
86.50
619.218
Fig. 1. Productivity growth as a result of increments in the inputs, based on the flexible model (AM). Variables (UPHs, FTEs, Beds) are expressed in logarithmic scale. Global sample, Clusters 1, 2 and 3.
translog p = 0.205). Even more, AM regression model is the only one able to detect a significant interaction between Beds and FTEs inputs (p = 0.041). No significant effects for both technical change (CD p = 0.654, translog p = 0.683, AM p = 0.453) and hospital (CD p = 0.849, translog p = 0.848, AM p = 0.970) variables were detected in the three regression models. Paying attention to AIC (CD = 1073.084, translog = 1072.302, AM = 875.337) and R2 values (CD = 63.41, translog = 63.54, AM = 78.50) we could observe a higher explanation power from the AM rather than for classic ones. The patterns of the output productivity across the behaviour of the two factors evaluated in the study are shown in Fig. 1(b). The changes in productivity follow a similar trend to FTEs which show improvements followed by decreased figures while Beds represents improvements in a log curve form. The outcomes seem to represent diseconomies of scale. Paying attention to the results for Cluster 2, it is interesting to observe that capital factor variable (Beds) is significant for the three models (CD p < 0.001, translog p < 0.001, AM p < 0.001), but labour factor (FTEs) is only significant for the translog model (p = 0.015). Furthermore, the effect of interaction between input factors is captured by models translog and flexible one (p < 0.003). Even more, the AM model, unlike the classic ones, is able to show that changes in production technology, captured by time trends, will affect the output (p = 0.022). The significant hospital effect for the three models
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(p < 0.001, in all cases) seems to reflect some variability related to the size of the hospitals included in Cluster 2. As in the previous results, not only the AIC estimates but also the R2 (CD = 77.684, translog = 78.32, AM = 85.30) seem to show the advantage of AM behind the classic ones. Fig. 1(c) displays an almost linear productivity growth related to increments in capital factor meanwhile, the additional resources in labour factor represents a comparatively smaller supply trend. The estimation results for Cluster 3 show significant effects of Beds for the three models (p < 0.001) while at the same time there are no significant effects related to FTEs, hospital and technical change (Years) variables for the CD and translog, but the AM detects effect for the FTEs (p = 0.037). However, there is a significant interaction between capital and labour factors for the AM (p = 0.022) whereas there is no significant interaction for the translog (p = 0.772). The goodness-of-fit of the AM measured by the R2 (CD = 78.01, translog = 77.96, AM = 86.50) as well as the AIC (CD = 775.441, translog = 777.311, AM = 619.218) have been more satisfactory for the AM compared with Cobb–Douglas and translog. Fig. 1(d) plots the gains in productivity due to additional resources in capital factor reflecting a linear relationship between these two variables. However, the labour factor patterns show a quadratic curve form with gains and reductions on the supply of this resource. 5. Discussion and conclusions The decision to measure production of hospitals by the AM made an attempt to improve flexibility for the functional form. The model proposed is certainly a simplified version of the complete econometric model specification (some other variables, in fact, can affect the analyzed phenomenon) but, also at this preliminary stage, the obtained results are really close to the desirable hypotheses. A selected set of simple indicators of production has been analyzed. These indicators have been compared across different hospital typologies. This comparative analysis gives important insights to the different variations among hospitals. As reported in Fig. 1, while medium size and small basic-care hospitals are almost homogeneous in terms of bed productivity, large size hospitals represent a more complex bed productivity trend. Among hospital typologies, the AM represents a large variability for consultants’ productivity. The interpretation of these results is surely an interesting instrument for decision makers in order to analyze the productive conditions of each hospital and the health care sector as an overall. Moreover, AMs may also be applied to check the classical models performance. Results in this study suggest that AM is a promising technique for the research and application areas on health economics. Moreover, results allow us to characterize the domains in which our approach may be effective like those related to demand, costs and utility functions in health care. Acknowledgements The authors would like to express their gratitude for the support received in the form of the Spanish MEC Grant MTM2008-01603 and the Galician Regional Authority (Xunta de Galicia) projects INCITE08PXIB208113PR and INCITE08CSA0311918PR. M.X. Rodríguez-Álvarez was supported by a grant [CA09/00539] from the Instituto de Salud Carlos III (Spanish Ministry of Science and Technology). References [1] M.D. Rosko, R.W. Broyles, The Economics of Health Care: A Reference Handbook, Greenwood Press, Inc., New York, Westport, CT, 1988. [2] G. Lopez Casasnovas, A. Wagstaff, La combinación de los factores productivos en el hospital: una aproximación a la función de producción, Investigaciones Económicas (Segunda época) 12 (1988) 305–327. [3] A. McGuire, The measurement of hospital efficiency, Social Science and Medicine 24 (1987) 719–724. [4] T.J. Hastie, R.J. Tibshirani, Generalized Additive Models, Chapman and Hall, London, 1990. [5] S.N. Wood, Generalized Additive Models: An Introduction with R, Chapman and Hall/CRC Press, 2006. [6] F.J. López Rois, R. Mateo, J.R. Gómez, C. Ramón, M. Pereiras, Methodological criteria for drawing up a contract-programme or singular sectorbased agreement of specialized care using HPUs, Secretara Xeral SERGAS, Consellera de Sanidade e Servicios Sociais, Xunta de Galicia, Santiago de Compostela, 1999. [7] G. Ferrier, V. Valmanis, Do mergers improve hospital productivity? Journal of the Operational Research Society 55 (2004) 1071–1080. [8] F. Reyes, Adopción, difusión y utilización de la Alta Tecnologa Médica en Galicia, in: Tomografía Computerizaday Resonancia Magnética, Universidade de A Coruña, Servizo de Publicacións, A Coruña, 2009. [9] C.W. Cobb, P.H. Douglas, A theory of production, The American Economic Review 18 (1928) 139–165. [10] L.R. Christensen, D.W. Jorgenson, L.J. Lau, Transcendental logarithmic production Frontiers, The Review of Economics and Statistics 55 (1973) 28–45. [11] D. Ruppert, M.P. Wand, R.J. Carroll, Semiparametric Regression, Cambridge University Press, 2003. [12] R Development Core Team. R: a language and environment for statistical computing, in: R Foundation for Statistical Computing, Vienna, Austria, 2009, URL http://www.R-project.org, ISBN 3-900051-07-0. [13] H. Akaike, A new look at the statistical model identification, IEEE Transactions on Automatic Control 19 (1974) 716–723.
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Estimating hospital production functions through flexible regression models Francisco Reyes Santías a,c,∗ , Carmen Cadarso-Suárez b,d , María Xosé Rodríguez-Álvarez b,c,d a
Instituto Universitario de Ciencias Neurológicas, University of Santiago de Compostela, Spain
b
Unit of Biostatistics, Department of Statistics and Operations Research, University of Santiago de Compostela, Spain
c
Complexo Hospitalario Universitario de Santiago de Compostela (CHUS), Santiago de Compostela, Spain
d
Instituto de Investigación Sanitaria de Santiago de Compostela (IDIS), Santiago de Compostela, Spain
article
info
Article history: Received 5 October 2010 Received in revised form 29 November 2010 Accepted 30 November 2010 Keywords: Generalized Additive Models (GAMs) Production function Cobb–Douglas Translog
abstract The subject of discussion is health care production functions. The parametric two-factor Cobb–Douglas and transcendental logarithmic (translog) production functions are frequently used. However, empirical and theoretical work has often questioned the validity of the parametric Cobb–Douglas and translog as a model for the production of health care. The aim of this study is to propose a new flexible form of production function based on Generalized Additive Models (GAMs) and to compare it with the classical approaches, using data from public hospitals in the Spanish Region of Galicia during the period 2002–2008. © 2010 Elsevier Ltd. All rights reserved.
1. Background Two models are commonly used in the estimation of hospital production function [1]: the Cobb–Douglas model and the transcendental logarithmic (translog) model. Cobb–Douglas has long been popular among economists because it is easy to work with and can explain the substitution between health care inputs. However, empirical and theoretical work has often questioned the validity of the parametric Cobb–Douglas as a model for the production of health care [2,3]. In comparison with the Cobb–Douglas model, the translog function model has a number of advantages. This model adds the effects of interactions between inputs. The Cobb–Douglas model, in contrast, omits this effect, which is less realistic. Thus, most of the hospital production function studies have used this flexible translog function form [1,3]. Nevertheless, in some circumstances, parametric models like the Cobb–Douglas or Translog models can be very restrictive. Using these models for estimation and prediction, the functional shape of continuous inputs is ‘‘forced’’ to follow a linear parametric form, which frequently does not fit the data closely. The relative lack of flexibility of parametric models has led to the development of non-parametric regression techniques based on the broad family of generalized additive models (GAMs; see [4,5]). These techniques do not impose a parametric form on the effects of continuous inputs; instead, they assume only that these effects are additive and reasonably smooth, and can be estimated using a variety of non-parametric local smoothing methods. This paper studies the use of Additive Models (AMs) to calculate hospitals production functions. The results of the new approach have been compared with the two most popular production functions used in the health care sector, the Cobb–Douglas and the translog models.
∗
Corresponding author at: Instituto Universitario de Ciencias Neurológicas, University of Santiago de Compostela, Spain. Tel.: +34 678487167. E-mail address: [email protected] (F.R. Santías).
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.11.087
F.R. Santías et al. / Mathematical and Computer Modelling 54 (2011) 1760–1764
1761
2. Data description The variables we use consist of inputs to hospital production in the form of capital and labour, and outputs from production. We have chosen the output of Inpatient care measured as the number of admissions standardized by means of complexity, obtaining homogeneous units of production (UPHs), calculated by multiplying the number of admissions by the complexity (weight) obtained from the Diagnostic Related Groups (DRGs) [6]. Following Ferrier and Valmanis [7], hospital inputs are measured as follows: in terms of capital we use the average number of beds (Beds) in each hospital. Labour inputs are measured by the number of consultant full-time equivalents (FTEs) employed in each hospital. Workload statistics were collected by hospitals, as panel data, from the Regional Ministry’s Information System (Galicia, Spain), during the period 2002–2008. Hospitals have been classified within three clusters by Reyes [8] following their size: Cluster 1 (small; <200 beds), Cluster 2 (medium; 200–650 beds), and Cluster 3 (large; >650 beds). Of the total number of observations (n = 1139), 468 (41.1%) correspond to Cluster 1, 301 (26.4%) to Cluster 2, and 370 (32.5%) to Cluster 3. 3. The statistical models The Cobb–Douglas production function proposed by Charles W. Cobb and Paul H. Douglas [9], takes the following form for our model: nH −1
log UPHs = β0 + β1 log FTEs + β2 log Beds + β3 Year +
−
αk Hospitalk + ε.
(1)
k=1
A standard procedure for introducing the possibility of technical change is to include a time trend (Year). This captures observed changes in the technology. An alternative to the Cobb–Douglas production function is the translog production function [10]. The form of translog production function used is as follows. nH −1
log UPHs = β0 + β1 log FTEs + β2 log Beds + β3 log FTEs log Beds + β4 Year +
−
αk Hospitalk + ε.
(2)
k=1
The flexible model considered was the following AM, including a Beds-by-FTEs interaction. nH −1
log UPHs = β0 + β1 Year + f1 (log FTEs) + f2 (log Beds) + f3 (log FTEs, log Beds) +
−
αk Hospitalk + ε
(3)
k=1
where f1 and f2 are unknown smooth functions of the number of beds (log scale) and the number of physicians (log scale) respectively, and f3 is an unknown smooth function representing the possible interaction between the number of beds and the number of physicians (both in log scale). It should be noted that the categorical covariate ‘Hospital’ was also included in the previous models. In these models, nH denotes the number of hospitals (in our study 10), and Hospitalh is a dummy variable taking the value 1 for the hth hospital and 0 otherwise. With regard to the estimation of the model (3), penalized thin plate splines were used to represent the smooth functions f1 , f2 and f3 , and the optimal smoothing parameters were estimated via Restricted (or Residual) Maximum Likelihood (REML, see e.g. [11]). All the statistical analyses were performed using R software, version 2.9.1 [12]. AMs were fitted using mgcv package [5]. 4. Results In this section, we present in Table 1 the results of each estimated model, for the Regional Health Service hospitals as an overall and every hospital Cluster. We evaluate the models based on the AIC (Akaike Information Criterion [13]) and the economic interpretation for an output change due to changes in input factors. We also present for each model, the value of the corrected R2 . First of all, the models have been estimated for the Regional Health Service hospitals as an overall. The three variables (Beds, FTEs and Hospital) are all significant (p < 0.005) for both the translog and AM models. Even more, the interaction between Beds and FTEs is also significant for those models. However, Year variable, as a proxy of changes in technology of production is not significant (CD p = 0.476, translog p = 0.477, AM p = 0.690). This outcome indicates that technical change in production is neutral in relation with output. Related to goodness-of-fit for the models, both the R2 and AIC outcomes, indicate that the AM provides a better fit in comparison with the two classic models, CD and translog. Fig. 1 depicts productivity growth as a result of increments in the inputs, based on the flexible model. The figure includes three lines. The upper line represents a productivity change according to variations in inputs. The right lower line represents
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Table 1 Cobb–Douglas (CD), translog and AM models estimates for Global model, Cluster 1, 2 and 3. AIC = Akaike Information Criterion; df = degrees of freedom; s() = smoother using thin plate splines. p-value
R2 (100%)
AIC
1.00 1.00 1.00 9.00
<0.001 0.6839 0.476 <0.001
78.12
2610.092
1.030 0.147 −0.053 −0.011 –
1.00 1.00 1.00 1.00 9.00
<0.001 0.004 <0.001 0.477 <0.001
78.34
2599.273
s (Beds) s (FTEs) s (Beds, FTEs) Year Hospital
– – – −0.006 –
5.44 5.32 15.00 1.00 9.00
<0.001 <0.001 <0.001
82.90
2355.801
CD
Beds FTEs Year Hospital
0.816 0.018 −0.011 –
1.00 1.00 1.00 2.00
<0.001 0.706 0.654 0.849
63.41
1073.084
0.684 0.051 −0.010 –
1.00 1.00 1.00 1.00 2.00
<0.001
Translog
Beds FTEs Beds × FTEs Year Hospital
0.205 0.098 0.683 0.848
63.54
1072.302
AM
s (Beds) s (FTEs) s (Beds, FTEs) Year Hospital
– – – 0.015 –
4.38 5.42 15.16 1.00 2.00
0.193 0.018 0.041 0.453 0.970
78.50
875.337
Model
Effects
Coefficients
CD
Beds FTEs Year Hospital
0.917 0.014 −0.011 –
Translog
Beds FTEs Beds × FTEs Year Hospital
AM
df
Global
0.690 <0.001
Cluster 1
−0.128
Cluster 2
CD
Beds FTEs Year Hospital
1.021 0.087 −0.047 –
1.00 1.00 1.00 3.00
<0.001 0.343 0.158 <0.001
77.68
735.695
Translog
Beds FTEs Beds × FTEs Year Hospital
1.167 0.262 −0.107 −0.049 –
1.00 1.00 1.00 1.00 3.00
<0.001 0.015 0.002 0.140 <0.001
78.32
728.013
AM
s (Beds) s (FTEs) s (Beds, FTEs) Year Hospital
– – – −0.063 –
5.66 3.38 10.40 1.00 3.00
<0.001 0.638 <0.001 0.022 <0.001
85.30
627.366
0.930
1.00 1.00 1.00 2.00
<0.001 78.01
775.441
Cluster 3
CD
Beds FTEs Year Hospital
−0.030 0.023 –
0.595 0.359 0.417
(continued on next page)
variations in capital input (Beds) while the left side line shows variations in labour input (FTEs). According to the upper line, Fig. 1(a) shows a steady growth in productivity. This growth is dominated by the capital factor trend variable in the estimates while the contribution of the growing labour factor variable is slightly modest. Following the results for Cluster 1, unlike the previous findings, no significant effect of Beds variable has been detected in the AM (p = 0.193). Nevertheless, only in the AM, FTEs variable has a significant effect (AM p = 0.018, CD p = 0.706,
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Table 1 (continued) Model
Effects
Translog
Beds FTEs Beds × FTEs Year Hospital s (Beds) s (FTEs) s (Beds, FTEs) Year Hospital
AM
Coefficients 0.904
df
0.023 –
1.00 1.00 1.00 1.00 9.00
– – – 0.021 –
4.20 6.10 16.41 1.00 2.00
−0.057 −0.011
p-value
R2 (100%)
AIC
<0.001 0.545 0.772 0.363 0.418
<0.001 0.037 0.022 0.293 0.814
a
b
c
d
77.96
777.311
86.50
619.218
Fig. 1. Productivity growth as a result of increments in the inputs, based on the flexible model (AM). Variables (UPHs, FTEs, Beds) are expressed in logarithmic scale. Global sample, Clusters 1, 2 and 3.
translog p = 0.205). Even more, AM regression model is the only one able to detect a significant interaction between Beds and FTEs inputs (p = 0.041). No significant effects for both technical change (CD p = 0.654, translog p = 0.683, AM p = 0.453) and hospital (CD p = 0.849, translog p = 0.848, AM p = 0.970) variables were detected in the three regression models. Paying attention to AIC (CD = 1073.084, translog = 1072.302, AM = 875.337) and R2 values (CD = 63.41, translog = 63.54, AM = 78.50) we could observe a higher explanation power from the AM rather than for classic ones. The patterns of the output productivity across the behaviour of the two factors evaluated in the study are shown in Fig. 1(b). The changes in productivity follow a similar trend to FTEs which show improvements followed by decreased figures while Beds represents improvements in a log curve form. The outcomes seem to represent diseconomies of scale. Paying attention to the results for Cluster 2, it is interesting to observe that capital factor variable (Beds) is significant for the three models (CD p < 0.001, translog p < 0.001, AM p < 0.001), but labour factor (FTEs) is only significant for the translog model (p = 0.015). Furthermore, the effect of interaction between input factors is captured by models translog and flexible one (p < 0.003). Even more, the AM model, unlike the classic ones, is able to show that changes in production technology, captured by time trends, will affect the output (p = 0.022). The significant hospital effect for the three models
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(p < 0.001, in all cases) seems to reflect some variability related to the size of the hospitals included in Cluster 2. As in the previous results, not only the AIC estimates but also the R2 (CD = 77.684, translog = 78.32, AM = 85.30) seem to show the advantage of AM behind the classic ones. Fig. 1(c) displays an almost linear productivity growth related to increments in capital factor meanwhile, the additional resources in labour factor represents a comparatively smaller supply trend. The estimation results for Cluster 3 show significant effects of Beds for the three models (p < 0.001) while at the same time there are no significant effects related to FTEs, hospital and technical change (Years) variables for the CD and translog, but the AM detects effect for the FTEs (p = 0.037). However, there is a significant interaction between capital and labour factors for the AM (p = 0.022) whereas there is no significant interaction for the translog (p = 0.772). The goodness-of-fit of the AM measured by the R2 (CD = 78.01, translog = 77.96, AM = 86.50) as well as the AIC (CD = 775.441, translog = 777.311, AM = 619.218) have been more satisfactory for the AM compared with Cobb–Douglas and translog. Fig. 1(d) plots the gains in productivity due to additional resources in capital factor reflecting a linear relationship between these two variables. However, the labour factor patterns show a quadratic curve form with gains and reductions on the supply of this resource. 5. Discussion and conclusions The decision to measure production of hospitals by the AM made an attempt to improve flexibility for the functional form. The model proposed is certainly a simplified version of the complete econometric model specification (some other variables, in fact, can affect the analyzed phenomenon) but, also at this preliminary stage, the obtained results are really close to the desirable hypotheses. A selected set of simple indicators of production has been analyzed. These indicators have been compared across different hospital typologies. This comparative analysis gives important insights to the different variations among hospitals. As reported in Fig. 1, while medium size and small basic-care hospitals are almost homogeneous in terms of bed productivity, large size hospitals represent a more complex bed productivity trend. Among hospital typologies, the AM represents a large variability for consultants’ productivity. The interpretation of these results is surely an interesting instrument for decision makers in order to analyze the productive conditions of each hospital and the health care sector as an overall. Moreover, AMs may also be applied to check the classical models performance. Results in this study suggest that AM is a promising technique for the research and application areas on health economics. Moreover, results allow us to characterize the domains in which our approach may be effective like those related to demand, costs and utility functions in health care. Acknowledgements The authors would like to express their gratitude for the support received in the form of the Spanish MEC Grant MTM2008-01603 and the Galician Regional Authority (Xunta de Galicia) projects INCITE08PXIB208113PR and INCITE08CSA0311918PR. M.X. Rodríguez-Álvarez was supported by a grant [CA09/00539] from the Instituto de Salud Carlos III (Spanish Ministry of Science and Technology). References [1] M.D. Rosko, R.W. Broyles, The Economics of Health Care: A Reference Handbook, Greenwood Press, Inc., New York, Westport, CT, 1988. [2] G. Lopez Casasnovas, A. Wagstaff, La combinación de los factores productivos en el hospital: una aproximación a la función de producción, Investigaciones Económicas (Segunda época) 12 (1988) 305–327. [3] A. 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