4.1 District efficiency assuming an aggregated production process

Following Fried et al. (Reference Fried, Lowell and Schmidt2006), we defined efficiency as the difference between the observed and optimal productivity, measured by ratios of output to input. We used SFA to compare the observed level of output to an optimal production frontier estimated based on the observed inputs and assumptions on the production function. We used a production function, rather than a cost function because information on prices was not available and because, due to central purchasing and national directives, the prices of the inputs measured and included in the analysis tend not vary across districts.

We used SFA rather than alternative non-parametric techniques because it allows accounting explicitly for measurement error and quantifying the effect of each inputs and of factors outside the DMU control on the production frontier (Kinfu, Reference Kinfu2013). Furthermore, when panel data are available, SFA models outperform non-parametric techniques if the assumed functional form for the production function is close to the underlying production technology (Giuffrida and Gravelle, Reference Giuffrida and Gravelle2001). We assumed a Cobb–Douglas functional form for f(.) accounting for inputs complementarities in health care delivery and assuming constant returns to scale.

We formulated the aggregated district SFA model as

(1) $$Y_{{it}} \,{\equals}\,f(X_{{it}}^{D} , \ {\rm }\beta ){\rm }E^{D} \left( {Y_{{it}} ,X_{{it}}^{D} } \right),$$

where Y _it is the output (outpatient consultations), delivered by district i at time t. $X_{{it}}^{D} $ includes health care inputs, here defined as total financial resources for outpatient care and HFs staff and equipment. The production frontier $f\left( {X_{{it}}^{D} ,{\rm }\beta } \right)$ defines the maximum level of output (Y _it ) attainable by district i at time t, for a given combination of inputs $\left( {X_{{it}}^{D} } \right)$ . β is the vector of parameters of the production function f (.). $E^{D} \left( {Y_{{it}} ,X_{{it}}^{D} } \right){\equals}{{Y_{{it}} } \over {f(X_{{it}}^{D} ,\ {\rm }\beta )}}$ is the district efficiency calculated as the ratio of observed production (Y _it ) to the maximum number of consultations deliverable with the inputs available and the technology in use, $f\left( {X_{{it}}^{D} ,{\rm }\beta } \right)$ .

We log transformed equation (1) to obtain a linear equation and we applied the Pitt and Lee (Reference Pitt and Lee1981) random effects model for panel data with time invariant inefficiency, to allow controlling for district characteristics fixed over time. Important efficiency changes would have been unlikely since the time period considered covered the same government mandate. We derived the empirical specification of the Pitt and Lee (Reference Pitt and Lee1981) SFA model following Greene (Reference Greene2008):

(2) $${\rm }y_{{it}} \,{\equals}\,\alpha {\plus}\beta x_{{it}}^{D} {\plus}\delta z_{{it}}^{D} {\plus}{\rm }{\varepsilon}_{{it}}^{D} {\rm ,}$$

where ${\varepsilon}_{{it}}^{D} \,{\equals}\,v_{{it}}^{D} {\minus}u_{i}^{D} $ with ${\rm }v_{{it}}^{D} \sim N(0,\sigma _{{v^{D} }}^{2} )$ , $u_{i}^{D}\sim N^{{\plus}} \left( {0,\sigma _{{u^{D} }}^{2} } \right)$ and $cov\left( {v_{{it}}^{D} ,u_{i}^{D} } \right)\,{\equals}\,0$ .

y _it is the natural log of the yearly number of outpatient consultations per capita.

$x_{{it}}^{D} $ is a vector of inputs that includes: (a) the (natural log of) annual recurrent expenditure for outpatient care per capita (see Anselmi et al., Reference Anselmi, Lagarde and Hanson2015a for details), a proxy for financial resources, which is the sum of the executed district, provincial and donor expenditure in district I; (b) the (natural log of) average HF staff and equipment, measured using an index which averages across HFs the availability of six items: working car, autoclave, motorbike, number of basic, medium- and high-level trained health cadres (see Anselmi et al., 2015b). While the index is not comprehensive of all inputs required in the HF, evidence suggests that the availability of HR and major equipment is correlated with that of drugs and other medical supplies (Wagenaar et al., Reference Wagenaar, Gimbel, Hoek, Pfeiffer, Michel, Manuel, Cuembelo, Quembo, Afonso, Gloyd and Sherr2014).

For each HF we calculated the average ratio of available resources to the minimum standard set by norms for each item (MISAU, 2002). Details on the calculation and further discussion of the implications are provided in Anselmi et al. (Reference Anselmi, Lagarde and Hanson2015b). We used existing norms as a benchmark to standardize availability of items across different types of HFs (MISAU, 2012a). We averaged across items to account for the complementarity between staff and equipment availability required to deliver effective health care and across HF within the same district to get a measure of availability in HFs rather than in the district.

$z_{{it}}^{D} $ is a vector of district characteristics that capture heterogeneity in the production technology and accommodates shifts of the production frontier. $z_{{it}}^{D} $ includes: (a) the (natural log of) number of HFs per type per 100,000 population; (b) the percentages of HFs with access to electricity and to running water (time variant); (c) the percentages of population that are economically active and illiterate (time invariant); and (d) provinces fixed effects to control for the influence of provincial management and other province characteristics. We included district characteristics and provincial dummies to control for differences in the epidemiological profile, for which specific data were not available, and for differences in demand for health care across geographic areas (Anselmi et al., Reference Anselmi, Lagarde and Hanson2015a, Reference Anselmi, Lagarde and Hanson2015b). Provincial dummies were also included to control for differences across provincial administrations that in a hierarchical system may influence both HAs and HFs activities. The observed error term $\left( {{\varepsilon}_{{it}}^{D} } \right)$ is a combination of the normally distributed stochastic error term $\left( { v_{{it}}^{D} } \right)$ and the non-negative half-normally distributed term $\left( {u_{i}^{D} } \right)$ . $u_{i}^{D} $ defines how far the i ^th district operates below the stochastic production frontier and measures the time invariant inefficiency in production (Kinfu, Reference Kinfu2013).

We finally obtained estimates of the district relative efficiency score ( $\widehat{{E_{i}^{D} }}$ ) from the Pitt and Lee (Reference Pitt and Lee1981) as follows (Greene, Reference Greene2008):

(3) $$\widehat{{E_{i}^{D} }}\,{\equals}\,e^{{{\minus}\widehat{{u_{i}{\vskip}^{D} }}}} \cong1\,{\minus}\,\widehat{{u_{i}^{D} }},$$

where $\widehat{{u_{i}^{D} }}$ is the input-oriented inefficiency scores obtained with the Jondrow, Lowell, Materov and Schmidt (JLMS) estimator (Jondrow et al., Reference Jondrow, Materov, Lovell and Schmidt1982). $u_{i}^{D} {\equals} {{{\minus}\lambda {\varepsilon}_{i} } \over \sigma }$ , where ɛ is half-normally distributed and can be estimated using the values provided by the SFA model estimates.

4.2 District efficiency assuming a two-step production process

4.2.1 First step: HA efficiency

In analogy with the SFA model used for the district aggregated production process, we derived the empirical specification of the district HAs SFA model as

(4) $${\rm }y_{{it}}^{A} \,{\equals}\,\alpha {\plus}\beta x_{{it}}^{A} {\plus}\delta z_{{it}}^{A} {\plus}{\varepsilon}_{{it}}^{A} ,$$

where ${\varepsilon}_{{it}}^{A} \,{\equals}\,v_{{it}}^{A} {\minus}u_{i}^{A} $ with ${\rm }v_{{it}}^{A} \sim N(0,\sigma _{{v^{A} }}^{2} )$ , $u_{i}^{A} \sim N^{{\plus}} \left( {0,\sigma _{{u^{A} }}^{2} } \right)$ and ${\rm cov}\left( {v_{{it}}^{A} ,u_{i}^{A} } \right)\,{\equals}\,0$ .

$ y_{{it}}^{A} $ is the natural log of the average HFs staffing and equipment index described in section 4.1.

$x_{{it}}^{A} $ includes the (natural log) of the recurrent expenditure per HF, for each source of funding. To standardize the availability of financial resource per HF across districts accounting for differences in number and type of existing HFs, we weighted each HF type based on the estimated cost of the minimum staff and equipment they should have according to norms (MISAU, 2012a).

$z_{{it}}^{A} $ includes the (natural log) all HFs number, the percentages of HFs with access to water and electricity, the average ratio of houses for personnel, available and in good conditions, to the minimum number established by norms (MISAU, 2012a, 2012d) as a proxy for HR retention capacity, and provinces fixed effects.

We estimated $\widehat{{E_{i}^{A} }}$ to the efficiency of HA i using the JLMS estimator as decribed in equation (3).

4.2.2 Second step: HF efficiency

In the second step we derived the empirical specification of the district HFs production process as

(5) $${\rm }y_{{it}} \,{\equals}\,\alpha {\plus}\beta y_{{it}}^{A} {\plus}\beta x_{{it}}^{F} {\plus}\delta z_{{it}}^{F} {\rm }{\plus}{\varepsilon}_{{it}}^{F} ,$$

where ${\varepsilon}_{{it}}^{F} \,{\equals}\,v_{{it}}^{F} {\minus}u_{i}^{F} $ with ${\rm }v_{{it}}^{F} \sim N(0,\sigma _{{v^{F} }}^{2} )$ , $u_{i}^{F} \sim N^{{\plus}} \left( {0,\sigma _{{u^{F} }}^{2} } \right)$ and ${\rm cov}\left( {v_{{it}}^{F} ,u_{i}^{F} } \right)\,{\equals}\,0$ .

As in section 4.1, y _it , is the natural log of the district total outpatient consultations per capita. $y_{{it}}^{A} $ includes the (natural log of) average availability of staffing and equipment with respect to norms, as in section 4.2.1. $x_{{it}}^{F} $ is the (natural log of the) number of DH, HC and clinics per 100,000 population. $z_{{it}}^{F} $ includes the percentages of the district population that are economically active and that are illiterate (time invariant) to control for catchment population characteristics, the percentages of HFs in the district with electricity and of those with access to running water (time variant), to control for HF conditions, and provinces fixed effects.

As before, using the JLMS estimator, we estimated $\widehat{{E_{i}^{F} }}$ , to which quantifies the average efficiency of HFs in district i.

Having estimated the district, HA and HFs efficiency scores ( $\widehat{{E_{i}^{D} }}$ , $\widehat{{E_{i}^{A} }}$ and $\widehat{{E_{i}^{F} }}$ ), we plotted them in a graph and calculated their correlation and rank correlation to compare the measured district performance under the assumption of an aggregated or two-step production process.

4.3 Testing for the effect of HA efficiency on district and HF efficiency

We tested for the effect of district administrative efficiency on district and HFs production (and efficiency) by including $\widehat{{E_{i}^{A} }}$ into the SFA models defined in equations (2) and (5). We looked at the significance of the coefficient associated with it, as done by Greene and Segal (Reference Greene and Segal2004) to test for the effect of efficiency on profitability.

We evaluated the goodness of fit of the district and HF models without and with inclusion of HA efficiency looking at the log likelihood and the Akaike information criterion (AIC), testing the hypothesis of a null coefficient associated with $\widehat{{E_{i}^{A} }}$ and performing a Likelihood-ratio (LR) test. Higher log likelihood, lower AIC and rejection of the LR test null hypothesis indicated that the SFA model including administrative efficiency fitted the data better.

We further compared the aggregated district SFA in equation (2) to a specification including $\widehat{{E_{i}^{A} }}$ but not HF staff and equipment. If the inclusion of staff and equipment levels, along with HA efficiency, did not improve the fit for the data, we could infer that the availability of staff and equipment depended on the financial resources available as well as the HA efficiency in managing them. A two-step model accounting for administrative efficiency would then be preferable.

We tested if accounting for heterogeneity in administrative efficiency affected the evaluation of district and HFs performance, by calculating the correlation and rank correlation of the efficiency scores estimated from equations (2) and (5) and from the relative counterparts including $\widehat{{E_{i}^{A} }}$ . High correlation would indicate a small effect of HA on district performance.

4.4 Data

Data on HFs type, staff and equipment for 2008–2011 were derived from the National Health Information System data, as provided by the MoH in June 2012 (MISAU, 2012d). After cross-checking the information with provincial and district annual reports and with local health authorities, we retained only information on equipment items that was found reliable. For 1.5% of the HFs we replaced staff and equipment figures in a specific year with the 2008–2011 average, when the discrepancy between the two was higher than 50%.

Figures for provincial and district recurrent expenditure for 2008–2011 were obtained from the Ministry of Finance (2012) and MoH budget execution reports (MISAU, 2012b). To get disaggregated district figures for all sources of expenditure, we assumed that donor and provincial expenditure were allocated to each district according to the number and type of HFs, as it used to be done before the decentralization (MISAU, 2012a), and that donor expenditure benefited exclusively primary and secondary care, which are managed at district level.

Population figures were based on annual projections from the 2007 Census (INE, 2010a, 2010b). District socio-economic indicators were estimated by the National Institute of Statistics from 2007 Census data (INE, 2008; INE, 2010a, 2010b), and were therefore time invariant over the period considered in the analysis.

We merged data at the district level to obtain a single 4-year panel database (2008–2011). We excluded Maputo City from the analysis, due to the presence of specialized and private health care providers, and eight districts, due to incomplete information. All models are estimated using Nlogit 5.

The descriptive statistics of the variables included in the analysis (Table 1), revealed considerable heterogeneity across districts in terms of service delivered, access to health care, expenditure and population characteristics, driven by unmeasured differences in population density and urbanization. Heterogeneity in expenditure by source reflected the undergoing decentralization process involving the gradual devolution of financial resources to districts, which explains the low but increasing district expenditure per capita.

Table 1

District descriptive statistics, Mozambique 2008–2011

Note: N=532 (133 districts over 4 years); HF=health facility; MZM=Mozambican Metical.