Considering endogeneity and heterogeneity–a hierarchical random parameters approach to measuring efficiency

Abstract

Modelling efficiency in healthcare with stochastic production frontiers (SPF) is complicated because of the immeasurable elements, quality of care and casemix as well as complex data structures. This analysis considers a SPF approach to estimating efficiencies for organizations in the Republic of Ireland that supply GP services outside of normal working hours. These organizations are run out of a number of primary care centres. The daily payroll for the centre is the output in the SPF and the services offered by these centres enter the production function as inputs. It is argued that these services are exogenous variables and are determined by patient characteristics and reported conditions and not the staff within the centre. A characteristic of the data used is a two-tier structure emanating from a centre lying within a co-op. To account for this tier structure the analysis considers a random parameters approach. The analysis also considers proxies for quality of care and casemix and incorporates them into the SPF. The sensitivity of efficiency values to the excluding the random parameters, quality of care and casemix variables is examined by estimating three reduced forms of the model which ignore each of these elements.

Acknowledgements

The author wishes to thank Professor William Greene for his input into this work. Thanks is also due to Professor Charles Normand and Mr. Kieran Collins who reviewed drafts of this work. All mistakes and/or omissions are owned by the author.

Notes

¹ For a complete review of SFA, the reader is referred to Kumbhaker and Lovell (2000).

² Within the former Eastern Health Board in ROI there are four co-ops; these co-ops are run slightly different to the other health boards’ co-ops and contain only one treatment centre. For the purpose of convenience and given that this does not affect our analysis (their data are not used), these four co-ops will be discussed as if they operated as one unit.

³ In this sample, four co-ops have a nurse triage unit and one co-op has a doctor triage unit.

⁴ For administrative, driving and nursing staff, this is straightforward as these staff are paid hourly. For medical staff, locum staff are paid hourly whereas GP's are paid a fee for the quantity of home visits and treatment centre visits that they provide. This fee differs for public and private patients.

⁵ One of the dummies in naturally excluded in the estimation to allow inclusion of the constant term.

⁶ Conditional estimators for the truncated-normal, normal-exponential and normal-gamma models may be found in the quoted seminal papers.

⁷ A public patient in ROI does not pay for services, whereas a private patient pays the fees as determined by the co-op.

⁸ The most common function forms in the stochastic frontier literature are Cobb–Douglas, translog and reduced translog. Cobb–Douglas was chosen for this analysis as the more flexible alternatives yielded marginal gains in terms of log-likelihood

⁹ Dummy m ₅ has been dropped to avoid multicollinearity.

¹⁰ Sensitivity testing was carried out by adding +2, +3, +4, +5, +10, +20, +40, +100 with robust results. The models were less robust when small fractions were added to the variables, 0.00001, 0.000001, 0.0000001. This is due to the large negative log values associated with these fractions.