Sources of data
The dataset used for this study is unbalanced panel data from the main public regional hospitals over the period 2001–2017 and comprised 41 observations. Disaggregated data at regional hospital level were available from 2001 to 2006. However, as disaggregated data were not available as from 2007, combined data for all the public hospitals were considered. Data were collected from various annual reports of the MoHW and the Ministry of Finance, Economic Planning and Development.
The health care system in Mauritius is a mix of public and private sector providers. Public healthcare delivery is served around a well delineated three tiers system, namely primary, secondary and tertiary. At the apex of the healthcare delivery system are national specialised hospitals and medical centres alongside regional hospitals. The regional hospitals, which are the focus of our study, act as referral centers for a decentralised network of Primary Health Care (PHC) facilities which comprises mediclinics, area and community health centres, within a defined demarcated area and population. The decentralised health care delivery endeavours to ensure minimal dichotomy in terms of access to health services across all regions.
The sampling criteria for selecting hospitals in the study are provision of common wide array of in-patient health care service and an average bed-capacity of at least 80 annually. The sample comprised all five regional public hospitals in Mauritius. The five regional hospitals selected had bed occupancy rate ranging between 62.8 and 80.2% whereas hospital admissions ranged between 82.4 and 86.1% over the period 2001–2017. Moreover, overall the public sector hospitals account for the bulk of bed capacity across Mauritius (85%) and leaving a meagre share to the private sector which comprise a network of 19 health facilities (15%) [26].
Output variables consisted of inpatients admitted and outpatients at the level of Accident, Emergency and unsorted outpatient departments. Inputs variables to the hospital production are measured in terms of capital and labour. Capital inputs are measured in terms of the total number of active hospital beds in all the five main regional hospital. Labour inputs are measured by the sum of personnel employed on a full-term basis in each hospital. To that effect, three categories of staff were considered for the efficiency analysis namely doctors (generalists and specialists), nurses and midwives, and non-medical which included staff other than nurses and doctors. The basis to adopt nomenclature for hospital staff in terms of doctors, nurses and non-medical is based on evidence that each one of them has a distinct role in patient care and deliver medical care at different service levels, especially from a quality and patient satisfaction perspective [27].
Stochastic frontier analysis (SFA) versus data envelopment analysis (DEA)
The two most commonly used approaches to measure providers’ efficiency are the parametric approach and the nonparametric or deterministic approach. The Stochastic Frontier Analysis (SFA) method is mostly used to evaluate parametric stochastic models, unlike the Data Envelopment Analysis (DEA) which is mostly employed to evaluate nonparametric or deterministic models. DEA is applied to analyse hospital efficiency as it is relatively easier to implement given its nonparametric basis, and the freedom provided on the specification of inputs and outputs. The DEA approach circumvents the need to measure output prices which are not available for transactions and services and fee-based outputs [28]. The rationale in favour of SFA developed initially by Aigner et al. [29] and expanded by Battesse and Coelli (1992; 1995) builds around the statistical weaknesses of DEA method [30,31,32]. The DEA method is non-stochastic. It fails to capture random noise such as dearth of resource inputs resulting from epidemics and strikes. Furthermore, DEA does not allow that statistical tests of the hypothesis regarding magnitude of inefficiencies be performed [33]. The robustness of SFA is based on that it makes a clear distinction between the two sources of error, due to inefficiency and random noise. SFA allows the decomposition of deviations from the efficient frontier into two components, inefficiency and noise [34]. DEA, which cannot distinguish between effects caused by inefficiency and a measurement error, attributes all these effects to inefficiency [35, 36].
SFA can be estimated using the two methods of Maximum Likelihood Estimation and/or Ordinary Least Square in panel data. SFA has the disadvantage that it builds on availability of structured information, including information about the production/cost technology and assumption around the distributional form of the inefficiency term. To that effect the analysis of inefficiencies could well be influenced by the model specification [31]. Albeit, these shortcomings the SFA remains the most reliable approach for measuring hospital inefficiencies. There are two distinct SFA modelling approaches of panel data. The first one generally assume a uniform variation for all production units, as inferred by Battese and Coelli [26, 31]. The others include three stochastic components respectively for efficiency, random noise, and time-invariant heterogeneity [15] and above all assume stochastic variation without any correlation over time [37]. Standard SFA models are limited to only one output and where technical inefficiency is denoted by difference between potential and observed output. Under such situation inpatient and outpatient workload are aggregated into one variable.
Modelling
Studies carried out to measure hospital efficiency are based mostly on two stochastic production function namely the Cobb-Douglas and the Translog functions. The SFA is a method to estimate a frontier production which assumes a given functional form for the relationship between inputs and an output (Coelli et al. [13]). The general form of the panel data version developed by Aigner, Lovell and Schmidt [29] and the production frontier stated by Coelli, Prasada and Battese (Coelli et al. [38]) is applied to the data as follows:
$$ {y}_i=f\ \left(\ {d}_i,{n}_{i,}{m}_i,{b}_i\right)+{e}_i $$
(1)
Equation 1 above can be specified as follows:
$$ {y}_i={\upbeta}_0+{\upbeta}_1d+{\upbeta}_2n+{\upbeta}_3\ m+{\upbeta}_4b+{e}_i $$
(2)
Where y is the output measure (in this case, number of outpatient and inpatient visit), d is a vector of doctors employed, n is a vector of nurses employed, m is vector of non-medical staff employed, b is a vector capital (proxied with number of hospital beds available), i is the decision making units (hospitals) and e represents errors decomposed as follows in eqs. 3 and 4:
$$ {e}_i={v}_i+{u}_i $$
(3)
$$ y_{i} = {\beta_0} + {\beta_1{d}} + {\beta_2{n}} + {\beta_3{m}} + {\beta_4{b}} + v_{i} + u_{i} $$
(4)
Where v is a random error term, normally distributed and uncorrelated with the explanatory variables; and u represents the hospital specific fixed effects or time invariant technical inefficiency.
For the purpose of this study in addition to the Cobb-Douglas and Translog production functions, a Multi-output distance function will be applied to the dataset.
The Cobb–Douglas form is built on assumptions of constant input elasticities and return to scale for all hospitals. The Cobb Douglas function is represented under eq. 5:
$$ \ln \left({y}_{it}\right)={\upbeta}_0+{\sum}_{i=1}^k{\upbeta}_{\mathrm{j}}\ln {x}_{j, it}+\left(\ {v}_{it}-{\mathrm{u}}_{it}\right) $$
(5)
Where j is the number of independent variables, i is the hospital), t is the time in years; ln represents the natural logarithm, yit represents the output of the i-th hospital at time t, xjit is the corresponding level of input j of the i-th hospital at time t, β is a vector of unknown parameters to be estimated. The vit is a symmetric random error, accounting for statistical noise with zero mean and unknown variance σv2. The uit is the non-negative random variable associated with technical inefficiency of hospital i, its mean is mi and its variance are σu2 [33].
The Cobb -Douglas function has been transformed to fit the data as illustrated in eq. 6.
$$ \ln \left( Outpatien{t}_{it}+ Inpatien{t}_{it}\right)=\beta 0+\beta 1\ln Be{d}_{it}+\beta 2\ln Docto{r}_{it}+\beta 3\ln Nurs{e}_{it}+\beta 4\ln Nonmedica{l}_{it} $$
(6)
The translog function relaxes the assumption of constant input elasticities and return to scale for all hospitals but is influenced by degrees of freedom and multicollinearity, as follows:
$$\mathrm{In} (y_{it}) = \beta_{o} + \sum^{k}_{j=1} \beta_{j} \mathrm{In}x_{j,it} +1/2 \sum^{k}_{j=1} \sum^{k}_{j=1} \mathrm{In}x_{h,it} (v_{it} - u_{it})$$
(7)
Where j, i, t, ln, yit, xjit are same as under eq. 5 and xjit times xhit is the interaction of the corresponding level of inputs j and h of the i-th hospital at time t, β is a vector of unknown parameters to be estimated. The vit is a symmetric random error, to account for statistical noise with zero mean and unknown variance σv2. The uit is the non-negative random variable associated with technical inefficiency of hospital i, its mean is mi and its variance is σu2 [33].
The Translog function form as defined in eq. 7 above is transformed as follows to suit the purpose of the current study.
$$ {\displaystyle \begin{array}{l}\ln \left({\mathrm{Outpatient}}_{\mathrm{it}}+{\mathrm{Inpatient}}_{\mathrm{it}}\right)=\upbeta 0+\upbeta 1\ {\mathrm{lnBed}}_{\mathrm{it}}+\upbeta 2\ {\mathrm{lnDoctor}}_{\mathrm{it}}+\upbeta 3\ {\mathrm{lnNurse}}_{\mathrm{it}}+\upbeta 4\ {\mathrm{lnNonmedical}}_{\mathrm{it}}\\ {}\kern12em +\upbeta 12\ \left({\mathrm{lnBed}}_{\mathrm{it}}\times {\mathrm{lnDoctor}}_{\mathrm{it}}\right)+\upbeta 13\ \left({\mathrm{lnBed}}_{\mathrm{it}}\times {\mathrm{lnNurse}}_{\mathrm{it}}\right)\\ {}\kern12em +\upbeta 14\ \left({\mathrm{lnBed}}_{\mathrm{it}}\times {\mathrm{lnNonmedical}}_{\mathrm{it}}\right)+\upbeta 23\ \left({\mathrm{lnDoctor}}_{\mathrm{it}}\times {\mathrm{lnNurse}}_{\mathrm{it}}\right)\\ {}\kern12em +\upbeta 24\ \left({\mathrm{lnDoctor}}_{\mathrm{it}}\times {\mathrm{lnNonmedical}}_{\mathrm{it}}\right)\\ {}\kern12em +\upbeta 34\left({\mathrm{lnNurse}}_{\mathrm{it}}\times {\mathrm{lmNonmedical}}_{\mathrm{it}}\right)\\ {}\kern12em +\upbeta 11\ 0.5\ \left({\mathrm{lnBed}}_{\mathrm{it}}\times {\mathrm{lnBed}}_{\mathrm{it}}\right)+\upbeta 22\ 0.5\left({\mathrm{lnDoctor}}_{\mathrm{it}}\times {\mathrm{lnDoctor}}_{\mathrm{it}}\right)\\ {}\kern12em +\upbeta 33\ 0.5\ \left({\mathrm{lnNurse}}_{\mathrm{it}}\times {\mathrm{lnNurse}}_{\mathrm{it}}\right)\\ {}\kern12em +\upbeta 44\ 0.5\ \left({\mathrm{lnNonmedical}}_{\mathrm{it}}\times {\mathrm{lnNonmedical}}_{\mathrm{it}}\right)\end{array}} $$
(8)
Where, β0 is the intercept of the constant term, β1, β2, β3, β4 are first order derivatives, β11, β22, β33, β44 are own second order derivatives and β12, β13, β14, β23, β24, β34, are cross second order derivatives. In view that the double log form model (with both the dependent and explanatory variables been in natural logs), the estimated coefficients show elasticities between dependent and explanatory variables [30, 33].
Finally, the rationale for using a Multi-output distance function is that the specified model of hospital production and inefficiency can be ran without aggregating inpatient and outpatient visits. The Multi-output distance function adapted to the current research is shown in eq. 9 below.
$$ {\displaystyle \begin{array}{l}\ln \left({\mathrm{Outpatient}}_{it}\right)=\upbeta 0+\upbeta 1\ {\mathrm{lnBed}}_{\mathrm{it}}+\upbeta 2\ {\mathrm{lnDoctor}}_{\mathrm{it}}+\upbeta 3\ {\mathrm{lnNurse}}_{\mathrm{it}}+\upbeta 4\ {\mathrm{lnNonmedical}}_{\mathrm{it}}\;\\ {}\kern7em +\upbeta 12\ \left({\mathrm{lnBed}}_{\mathrm{it}}\times {\mathrm{lnDoctor}}_{\mathrm{it}}\right)+\upbeta 13\ \left({\mathrm{lnBed}}_{\mathrm{it}}\times {\mathrm{lnNurse}}_{\mathrm{it}}\right)\\ {}\kern7em +\upbeta 14\ \left({\mathrm{lnBed}}_{\mathrm{it}}\times {\mathrm{lnNonmedical}}_{\mathrm{it}}\right)+\upbeta 23\ \left({\mathrm{lnDoctor}}_{\mathrm{it}}\times {\mathrm{lnNurse}}_{\mathrm{it}}\right)\\ {}\kern7em +\upbeta 24\ \left({\mathrm{lnDoctor}}_{\mathrm{it}}\times {\mathrm{lnNonmedical}}_{\mathrm{it}}\right)\\ {}\kern7em +\upbeta 34\left({\mathrm{lnNurse}}_{\mathrm{it}}\times {\mathrm{lmNonmedical}}_{\mathrm{it}}\right)\\ {}\kern7em +\upbeta 11\ 0.5\ \left({\mathrm{lnBed}}_{\mathrm{it}}\times {\mathrm{lnBed}}_{\mathrm{it}}\right)+\upbeta 22\ 0.5\left({\mathrm{lnDoctor}}_{\mathrm{it}}\times {\mathrm{lnDoctor}}_{\mathrm{it}}\right)\\ {}\kern7em +\upbeta 33\ 0.5\ \left({\mathrm{lnNurse}}_{\mathrm{it}}\times {\mathrm{lnNurse}}_{\mathrm{it}}\right)\\ {}\kern7em +\upbeta 44\ 0.5\ \left({\mathrm{lnNonmedical}}_{\mathrm{it}}\times {\mathrm{lnNonmedical}}_{\mathrm{it}}\right)\\ {}\kern7em +\upbeta 5\ {{\mathrm{lnY}}^{\ast}}_{\mathrm{it}}+\upbeta 51\ \left({\mathrm{lnBed}}_{\mathrm{it}}\times {{\mathrm{lnY}}^{\ast}}_{\mathrm{it}}\right)+\upbeta 52\ \left({\mathrm{lnDoctor}}_{\mathrm{it}}\times {{\mathrm{lnY}}^{\ast}}_{\mathrm{it}}\right)\\ {}\kern7em +\upbeta 53\ \left({\mathrm{lnNurse}}_{\mathrm{it}}\times {{\mathrm{lnY}}^{\ast}}_{\mathrm{it}}\right)+\upbeta 54\ \left({\mathrm{lnNonmedical}}_{\mathrm{it}}\times {{\mathrm{lnY}}^{\ast}}_{\mathrm{it}}\right)\\ {}\kern7em +\upbeta 55\ 0.5\ \left({{\mathrm{lnY}}^{\ast}}_{\mathrm{it}}\times \mathrm{lnY}{\ast}_{\mathrm{it}}\right)\end{array}} $$
(9)
where, Y* is the ratio of outpatient visits to inpatient admissions. β5, is the first order derivative while β55 represents own second order derivative. β51, β52, β53, β54, are cross second order derivatives [33].
The Cobb-Douglas, Translog and Multi-output distance function models were estimated using STATA 11.
Technical efficiency assessment
Adapting the common widely accepted definition for technical efficiency to the present study the ratio of the observed output (Yit) to the maximum feasible output (Ymax), defined by a certain level of inputs used by the hospital will determine the level of technical efficiency of hospital i at time t. Since both the Cobb Douglas and Translog functions are based on a normal-truncated normal maximum likelihood (ML) random model effect with time invariant efficiency developed by Battese and Coelli [38], the maximum feasible output is determined by the hospitals with inefficiency effect equal to 0 (vit = 0). Technical efficiency is derived building on the premises stated earlier under eqs. (1) and (3) where the general form of the panel data version is.
$$ \ln {y}_{it}=f\left({x}_{j, it},\beta \right)+{e}_i $$
(10)
Equation (10) can be formulated as:
$$ {y}_{it}=\exp \left(f\left({x}_{j, it},\kern0.5em \beta \right)\right)\ast \exp \left({v}_{it}\right)+\exp \left(-{u}_i\right) $$
(11)
Where f() is a suitable functional form of any of the model namely Cobb-Douglas, Translog and Multi-output distance, yit represents the output of the i-th hospital at time t, xj,it is the corresponding level of input j of the i-th DMU (hospital) at time t, and β is a vector of unknown parameters to be estimated [33].
The technical efficiency can be expressed as follows:
$$ T{E}_{it}=\frac{y_{it}}{e\mathrm{xp}\left(f\left({x}_{j, it},\kern0.75em \beta \right)\right)\ast \exp \left({v}_{it}\right)} $$
(12)
or
$$ T{E}_{it}=\frac{e\mathrm{xp}\left(f\left(\ {x}_{j, it},\kern0.75em \beta \right)\right)\ast \exp \left({v}_{it}\right)+\exp \left(-{u}_i\ \right)\ }{e\mathrm{xp}\left(f\left({x}_{j, it},\kern0.5em \beta \right)\right)\ast \exp \left({v}_{it}\right)} $$
(13)
$$ T{E}_{it}=E\ \left[e\left(-{u}_{it}\right)/\left({v}_{it}-{u}_{it}\right)\right] $$
(14)
Where uit represents hospital specific fixed effects or time invariant technical inefficiency and vit is a normally distributed random error term and is uncorrelated with the explanatory (independent) variables. Technical inefficiencies range between 0 and 1 as uit is a nonnegative random variable. A value of 0 infers that hospital is technically inefficient and, conversely, a value of unity implies perfect technical efficiency. Technical efficiency is calculated following Battese and Coelli, 1995, using the bc option available in STATA 11.
Quantification of potential gains through technical efficiency
As all public health facilities provides free health care services the sole source of revenue is annual government budgetary allocation. Potential gains from technical efficiency for all the five regional hospitals in this study are estimated by determining the share of annual government budgetary allocation received that could be saved potentially should technical inefficiency be avoided. The share of grants revenue that can be saved is defined in eq. (15).
$$ \mathit{\operatorname{Re}}{v}_i=\left( ef{f}_{max}-{eff}_i\right)\ast {G}_i $$
(15)
Where Rev i represents annual budgetary allocation of the i-th hospital that could be saved if inefficiencies were eliminated, eff max is maximum efficiency level (1.00 in this case), eff i is the current efficiency score of the i-th hospital estimated under the SFA specification above (translog function) and G I is the actual budget allocated by the government to the i-th hospital [41].
The savings realised in total hospital budgetary grant allocation (Rev i) is a proxy of the potential fiscal space available for the i-th hospital in case full efficiency is attained [22].