 Research article
 Open Access
 Open Peer Review
Inefficiency, heterogeneity and spillover effects in maternal care in India: a spatial stochastic frontier analysis
 Yohannes Kinfu^{1}Email author and
 Monika Sawhney^{2}
https://doi.org/10.1186/s129130150763x
© Kinfu and Sawhney; licensee BioMed Central. 2015
 Received: 13 April 2014
 Accepted: 23 February 2015
 Published: 25 March 2015
Abstract
Background
Institutional delivery is one of the key and proven strategies to reduce maternal deaths. Since the 1990s, the government of India has made substantial investment on maternal care to reduce the huge burden of maternal deaths in the country. However, despite the effort access to institutional delivery in India remains below the global average. In addition, even in places where health investments have been comparable, inter and intrastate difference in access to maternal care services remain wide and substantial. This raises a fundamental question on whether the subnational units themselves differ in terms of the efficiency with which they use available resources, and if so, why?
Methods
Data obtained from round 3 of the country’s District Level Health and Facility Survey was analyzed to measure the level and determinants of inefficiency of institutional delivery in the country. Analysis was conducted using spatial stochastic frontier models that correct for heterogeneity and spatial interactions between subnational units.
Results
Inefficiency differences in maternal care services between and within states are substantial. The top one third of districts in the country has a mean efficiency score of 90 per cent or more, while the bottom 10 per cent of districts exhibit mean inefficiency score of as high as over 75 per cent or more. Overall mean inefficiency is about 30 per cent. The result also reveals the existence of both heterogeneity and spatial correlation in institutional delivery in the country.
Conclusions
Given the high level of inefficiency in the system, further progress in improving coverage of institutional delivery in the country should focus both on improving the efficiency of resource utilization—especially where inefficiency levels are extremely high—and on bringing new resources in to the system. The additional investment should specifically focus on those parts of the country where coverage rates are still low but efficiency levels are already at a high level. In addition, given that inefficiency was also associated inversely with literacy and urbanization and positively related with proportion of households belonging to poor households, investment in these areas can also improve coverage of institutional delivery in the country.
Keywords
 Institutional delivery
 Efficiency of institutional delivery in India
 Health efficiency analysis
 Stochastic frontier model
 Spatial stochastic frontier analysis
Background
Maternal mortality—the death of women during pregnancy, childbirth, or in the 42 days after delivery—remains one of the greatest public health challenges of our time.The Fifth Goal of the United Nations Millennium Declaration (MDG 5) of 2000 calls for a reduction in maternal mortality ratio (MMR) in all countries so that by 2015 it is one quarter of its 1990 level [1]. However, the progress recorded so far has been relatively slow so much so that maternal mortality is often described as the most seriously “off track” of all the healthrelated MDGs [25].
Recent estimates show that, globally, more than aquarterof a million women die each year because of childbirth and pregnancy complications [6]. Some 99 per cent of these deaths occur in the developing world and about half of these total come from just six countries—including India, Nigeria, Democratic Republic of Congo, Pakistan, Indonesia and Sudan—which make up not more than a quarter of the world population [6,7]. India alone accounts for 19% of the global total—the highest for any country in the world—with some twothird of which coming from just nine of its 35 states and federal territories [8].
Maternal death is a great tragedy because most of the deaths associated with pregnancy, childbirth, or in the 42 days after delivery are preventable through effective interventions, such as by promoting institutional delivery that ensures women access to skill birth attendants [914]. The National Population Policy of India stipulates a similar strategy to curb the high level of maternal mortality that prevails in the country [15]. The Child Survival and Safe Motherhood Program of 1992–1997 and the Phase1 of the Reproductive and Child Health Program (RCH1) implemented during 1997–2004 also constitute part of the same effort to improve maternal and newborn health in India [16,17]. Following the adoption of the MDGs, the Indian government further reinforced its efforts through introducing a system of conditional cash assistance to mothers as they attend delivery and postdelivery care. These interventions were particularly noteworthy given their emphasis on reaching rural communities and women belonging to lower socioeconomic status in the country [18]. The country also instituted a districtbased decentralized approach that ensures follow up and program ownership at the grass root level.
However, despite successive initiatives and efforts on the part of the government, interstate and intrastate variations in institutional delivery remain wide in India [19]. The same is the case with respect to other critical inputs required for improving maternal health in the country. For instance, while over 70 per cent of health facilities in relatively wellresourced districts of the country had highly trained practitioners (such as ladymedicalofficer, an obstetrician, or a gynecologist) in some districts located in less developed parts of the country this proportion was fewer than 2 per cent. Similarly, annual per capita public expenditure on health varies considerably across the different parts of the country [20]. What is even more concerning is the fact that the returns from past interventions also seem to be more uneven, with some districts achieving limited outcome than others even when they had comparable health inputs. This raises a fundamental question on whether the districts in the country differ in terms of the level of efficiency with which they use the resources available to them, and if so, why?
To address this question, the efficiency of institutional delivery was analyzed using a stochastic frontier approach. While there are a few previous studies on aspects of health care efficiency, the present paper introduces several novel dimensions [2125]. First, the analysis covers the entire country, uses data that are more recent and focuses on districts, which constitute the basic unit of the country’s health system. Second, given the expectation that districts located in close proximity are able to interact with each other and influence each other’s output and efficiency levels, through competition and/or learning effects, spatial dependence and spillover effects have been explicitly introduced into the present analysis. This is the first time that such a model is introduced in the health care efficiency literature in any part of the world. Third, given India is a heterogeneous country with respect to level of development, governance and models of social service provision that could significantly influence and distort inefficiency estimates, the study also controls for heterogeneity in the analysis. Finally, to the best of our knowledge, the present paper is also the only study that has so far looked into inefficiencies in institutional delivery care in India. Hence, by doing so, the present study not only provides new evidence on India but also introduces alternative analytical dimensions that can be applied to other settings.
The remaining part of the article is organized as follows: The following section reviews the stochastic frontier model and introduces the methodology and data used in the study. The empirical results and discussion are presented next followed by some concluding remarks in the final part of the paper.
Methods
Classical stochastic health frontier Model
Estimation of a stochastic frontier model requires a range of assumptions on the error components. Primarily, the model assumes that both the symmetric error term (v) and the nonnegative inefficiency component (u) are independent of each other and iid across observations [26,28]. In practice, v is also usually assumed to be normal N (0; σ_{v}^{2}) while the distribution of u (η) can be selected from halfnormal [28,35], exponential [29], truncated normal [36], gamma [37] or lognormal [38] distributions.
Model (2) and (3) are essentially identical, and in moderately sized samples, these respecifications represent a minor reformulation of the familiar classical stochastic frontier model [31].
While Greene’s respecifications improve the performance of the classical frontier model by controlling for area level heterogeneities in the data, in many instances, particularly at a lower level of geography, contiguous geographic units also tend to interact with each other more directly which leads to various forms of spatial interdependence or interaction between health production units. For instance, in India, as in many other parts of the world, people cross boundaries to seek care and services in neighboring districts. Moreover, there is also a tendency among neighboring districts to compete for scares health resources (such as human resources for health or health budget from state authorities) or emulate each other’s way of doing things (in a good or bad way) as they go about in their day to day business of providing care to their respective population. Interactions such as these are nontrivial because they can potentially affect both the production function and the mean efficiency distribution of the familiar stochastic health frontier model. Besides, the correlation itself lends to violation of the conventional assumption of independence of observational units [40]. There is, therefore, both a theoretical as well as a statistical reason for us to look beyond the standard approach—which views the atomistic agent (in our case the district) as a decision maker acting in isolation—and capture the interactions between health production agents in the system more directly.
Methodologically, this requires us to use spatial models that are capable of identifying and measuring spillover effects (or spatial correlation) in the system. A number of studies in other disciplines [4046] have applied such models, but the approach has not been previously developed and tested for subnational health care inefficiency analysis despite the fact that spillover effects and spatial externalities seem to be quite a common place in national health systems around the globe.
Stochastic health frontier model with spillover effects

y is an n x 1 vector of observations on the dependent variable (in this paper this represents observations on institutional delivery rate);

X is an n x k matrix of observations on input variables that directly influence the production function, and β is the corresponding k x 1 parameter vector.

Z is an n x k matrix of exogenous variables that affect inefficiency (but not the production frontier) and δ is the corresponding k x 1 parameter vector.

W is an n x n spatialweighting matrix (with 0 diagonal elements) usually specified in terms of firstorder continuity relations or as functions of distance. In many applications, the weightingmatrix for all lag variables is assumed to be the same, but there is no methodological restriction to apply a different set, when and if required.

Wy, Wx, Wû, and Wz,are n x 1 vectors representing spatial lags and ρ, τ, φ and ω are the corresponding scalar parameters measuring degrees of spatial interactions with respect to output, inputs, inefficiency level and the exogenous variables affecting inefficiency, respectively. Hence, ρ, for example, measures the degree to which access to level of institutional delivery in neighboring districts influence the level of service coverage in the district of interest.

Wv are spatial errors (with a coefficient λ)
Note that as before v represents measurement and specification error and is assumed to be normal N (0; σ_{v}^{2}), while the distribution of u (η) can be selected from halfnormal [28,35], exponential [29], truncated normal [36], gamma [37] or lognormal [38] distributions.
In this paper, we fit the classical stochastic frontier model described in equation (1) and compare the results with the outcome from equation (2) that corrects for heterogeneity as well as with results from equations (4) and (4), which capture the effects of output and efficiency lags, respectively. In addition, to assess the effects of heterogeneity and spatial correlation corrected models on efficiency estimates we also generate and compare efficiency scores for each district from each of these models. As is common with the standard practice [51,54], the scores generated in this fashion would allow us to examine how well each decisionmaking agent, represented by a district in our analysis, was performing its function compared to the maximum possible potential, given current resources at its disposal. A district is generally classified as inefficient, if it is observed to have a coverage rate below the maximum level that can be attained from a given set of inputs. Theoretically, inefficiency scores range from 0 (the most efficient) to 1 (the least efficient), with values in between representing a shortfall of observed output from maximum feasible output.
In our analysis, we chose to focus on districts, because they are responsible for the allocation and management of health inputs in their respective jurisdictions. This in turn means that their success or otherwise is a good reflection of the success or failure of the country’s health system or more specifically the progress toward meeting the target for MDG 5 for the country.
Data and variable description
The proposed analysis requires five sets of information: data on the output of interest; on input variables that directly affect the production function; on exogenous variables that affect the inefficiency distribution (but not the production function); on a variable (or set of variables) that capture(s) spatial heterogeneity in the data and finally a spatialweighting matrix. The spatialweighting matrix is required to generate spatially lagged variables that we will use in measuring the spatial correlation and spillover effects in the system.
Output is measured using institutional delivery rate reported for each district. We focused on institutional delivery because improving maternal health through promoting access to institutional delivery is an integral part of India’s primary health care agenda. Second, institutional delivery itself is one of the key interventions known to have the greatest impact on improving maternal health in the developing world [1,1114]. Finally yet importantly, institutional delivery is also one of the indicators used for monitoring progress towards MDG5, a goal to which India is also a signatory.
In measuring the inputs to the production process, three indicators were selected: namely, the density of health facilities per 1000 square kilometer, proportion of facilities that received ‘untied funding’ in previous financial year and proportion of facilities in a district that had highly trained practitioners (i.e. a general surgeon or obstetrician /gynecologist/ladymedicalofficer). These variables represent some of the key inputs required for providing safe delivery care. To these, we have also added a composite variable called service readiness index to capture the effects of availability of selected basic amenities, such as communication equipment, operating theater and a labour room in the facilitates reported in each area. This index was considered useful because even if facilities are provided with the required human and financial resources (and are made available within accessible distance), they will still not be fully ready to provide services unless they are also equipped with amenities that are vital for their function [53].
Furthermore, three exogenous variables were added into the analyses to look more closely into the determinants of inefficiency in the country. These included literacy rate, urbanization and proportion of households in the lowest wealth quintile. It should be noted that these variables as such do not constitute direct inputs to the production of institutional delivery, but we assume that they are part of the environment within which districts had to maximize their outputs, and, therefore, can exert an influence on the performance of the country’s health system. For instance, districts with high illiteracy rate may face resistance in promoting institutional delivery because of cultural barrier, and, as a result, may need to divert resources away from service provision to health promotion and advocacy purposes. Similarly, in communities where over all living standard is low women may still be unable to access services (even if they are available) due to lack of transport and associated transactional costs. This means that the presence (or absence) of such contextual variables, could individually or collectively facilitate (or hinder) the efficiency with which districts can achieve their stated health goal, but on their own the exogenous variables do not affect output levels as such.
List of variables and corresponding descriptive statistics, India, 200708
Variable:  Descriptive statistics  

Description  Type  Code  Mean  Standard deviation 
Institutional delivery rate (%)  Output  Y  50.34  23.70 
Number of facilities per 1000 sq km  Input  x _{1}  0.016  0.023 
Facilities having Medical Officer/Obstetrician/Gynecologist (%)  Input  x _{2}  28.62  20.82 
Facilities received untied funding in previous financial year (%)  Input  x _{3}  74.87  22.54 
Facilities with selected basic amenities (%)  Input  x _{4}  35.01  17.03 
Population in lowest wealth quintile (%)  Exogenous  z _{1}  18.92  17.33 
Population residing in urban areas (%)  Exogenous  z _{2}  25.25  17.55 
Population literate age 7+ years (%)  Exogenous  z _{3}  70.63  10.56 
Districts in ‘backward state’ (%)  Heterogeneity  H _{1}  41.58  49.33 
Number of districts covered*  499  
Contiguitymatrix used for creating lag variables**  
Total links (number)  3222  
Minimum links (number)  0  
Average links (number)  5.42  
Maximum links (number)  10 
Moreover, we constructed two separate spatial lag variables for output and efficiency level of districts. Spatial lags are a weighted average of the values of the variables of interest for neighboring areas with the weight determined based on some measure of connectivity. In the present case, a contiguitybased weighting scheme, which involve assigning a weight of one for contiguous areas and a value of zero for noncontiguous districts, was applied [40,45,46,55]. This is one of the most common approaches for generating spatial lag variables [40,55]. Keeping with the practice, we also used identical spatial weighting matrix for both of our lag variables [45,46].
As shown in Table 1, the weightingmatrix produced over 3000 links (or contiguous units which are also known as neighbors), with each district having, on average, about five neighbors while some districts sharing border with as many as 10. The spatial data needed for creating the contiguitymatrix was obtained from open source shape files (or ‘coordinate files) for India. On the other hand, the remaining data used in the paper were extracted from India’s latest round district level household and facility survey (DLHS3, 2007–08) [19]. These data were normalised around their respective means and transformed to a logscale to control for the effects of different units of measurement. Analyses of data were performed using version 13 of the STATA software.
Results and discussion
Maximum likelihood estimates of classical and spatial stochastic frontier models, 2007–08, India
MODEL 1  MODEL 2  MODEL 3  MODEL 4  

Production function  
Constant  0.8616 (0.0705)***  1.2178 (0.0652)***  1.7013 (0.0799)***  1.6790 (0.0794)*** 
x_{1}  0.0822 (0.0282)***  0.0481 (0.0237)**  0.0746 (0.0238)***  0.0804 (0.0244)*** 
x_{2}  0.1080 (0.0346)***  0.1532 (0.0281)***  0.1044 (0.0268)***  0.1201 (0.0266)*** 
x_{3}  0.1587 (0.0378)***  0.0426 (0.0343)  −0.0207 (0.0344)  −0.0180 (0.0343) 
x_{4}  0.1557 (0.0351)***  0.0651 (0.0292)**  0.0814 (0.0274) ***  0.0857 (0.0280) *** 
γ (Heterogeneity index)  −0.8317 (0.0566)***  −0.6972 (0.0573)***  −0.8623 (0.0559)***  
ρ (Lagged output)  0.4588 (0.0493) ***  
φ (Inefficiency lag)  0.4465 (0.0462) ***  
Inefficiency  
Constant  −0.2091 (0.2024)  −0.2193 (0.1780)  −0.7763 (0.2226)***  −0.9486 (0.2570)*** 
z_{1}  0.6349 (0.1175)***  0.4457 (0.1014)***  0.3972 (0.1087)***  0.3827 (0.1134)*** 
z_{2}  −0.6651 (0.1432)***  −0.6417 (0.1236)***  −0.9082 (0.1868)***  −0.9945 (0.2186)*** 
z_{3}  −0.0874 (0.1114)  −0.0305 (0.1011)  −0.1132 (0.1123)  −0.1043 (0.1170) 
Predicted mean inefficiency  0.4709  0.4592  0.3950  0.3756 
Distributions of u and v  
δ_{u}  1.0664  0.9989  0.7995  0.7437 
δ_{v}  0.4536  0.3088  0.3626  0.3852 
λ  2.35  3.24  2.2049  1.9307 
Log likelihood  −549.2938***  −468.0001***  −426.2494***  −423.7609*** 
N  499  499  499  499 
Results for the classical model indicate that the coefficients for inputs not only have the correct sign but also that they are all statistically significant meaning that ‘untied funding’, more health facilities per square kilometres and more qualified health workers per facilities as well as facilities that are well equipped with basic amenities (such as labour room, operation theatre and communication equipment) will lead to significantly high rate of institutional deliveries in a district. These results remain consistent across the remaining three models except for ‘untied funding’ which tended to loose significance once heterogeneity and spatial interactions were controlled for probably because existing rules governing untied funding favours marginalised parts of the country. The magnitudes of the coefficients for the other variables also decline as we progressively introduce heterogeneity and spatial correlations into the analysis.
In all the four models, the asymmetry parameter, λ, is well above unity, which is an indication of inefficiency in the provision of maternal care service in the country. The predicted mean inefficiency from the classical stochastic frontier model (Model 1) is around 47 per cent but once heterogeneity and spatial interactions are incorporated both the mean inefficiency score and the estimated underlying standard deviation of u (δ_{u}) change significantly. These are consistent with Greene’s [31,39] conjecture that unaccounted for heterogeneity was indeed showing up as inefficiency in the original model. The statistically significant coefficient for the heterogeneity indicator variable as well as the loglikelihood values for Model 1 and Model 2 also confirms that the model with a heterogeneity component handles the data notably better.
However, as discussed earlier Model 2 only corrects for heterogeneity and does not address the potential effects of spatial interactions between neighbouring districts on their respective output and efficiency level. As can be seen from the loglikelihood values for Model 3 and Model 4, correcting for spatial correlations significantly improve the model fit. Similarly, once spatial dependence in output and efficiency are controlled for the estimated underlying distribution of u (δ_{u}) falls almost by 25 per cent (i.e. from 1 to between 0.75 and 0.8) and the predicted mean inefficiency almost by 20 per cent (i.e. from 0.48 to between 0.38 and 0.40). In both cases, the observed changes in the efficiency function were not accompanied by significant changes in the residual distribution (δ_{v}) of the two models. This, in turn, reinforces our belief that the spatial stochastic frontier models add something important to specifying the inefficiency distribution of institutional delivery, beyond what we would expect from a heterogeneity corrected frontier model.
Further evidence on the existence of spill over effects comes from the coefficient for the lagged output variable (ρ). This estimate is both large and positive and highly statistically significant by standard criteria. This provides support for the conjecture that the coverage of institutional delivery in a district covaries with the level observed for its geographical neighbours. The corresponding lag coefficient for inefficiency (φ), which is equally large and highly significant, also shows clearly the impact of efficiency levels of neighbouring districts on the output of the districts of interest in the country.
Impact of spatial interactions on efficiency levels of districts, 2007–08, India
Magnitude of effect on efficiency level  Model 3  Model 4  

[Output lag corrected]  [Efficiency lag corrected]  
Number of districts affected  %  Number of districts affected  %  
Negative impact on efficiency  96  19.2  69  13.8 
Efficiency increase of up to 9.99%  157  31.5  149  29.9 
Efficiency increase between 10.0 – 29.99%  161  32.3  171  34.3 
Efficiency increase of 30% or more  85  17.0  110  22.0 
Total  499  100.0  499  100.0 
Inefficiency distribution by state and federal territories, 2007–08, India
State/Territory  INEFFICIENCY DISTRIBUTION for FINAL MODEL [MODEL 4 [i.e. Corrected for heterogeneity and spatial correlation for efficiency]  Corrected ONLY FOR heterogeneity: [Model 2]  Gains or loss in efficiency due to spatial interaction (%)  

Number of districts with inefficiency level:  Total  Mean inefficiency score  Rank  Mean inefficiency score  Rank  
<10%  > = 10% and less than 25%  > = 25% and less than 70%  > = 70%  
Daman & Diu  1  0  0  0  1  0.0713  1  0.2027  6  184.4 
Goa  1  1  0  0  2  0.0929  2  0.1605  3  72.8 
Puducherry  1  1  0  0  2  0.1067  3  0.1492  2  39.8 
Kerala  4  9  1  0  14  0.1360  4  0.1390  1  2.2 
Tamil Nadu  2  18  3  0  23  0.1631  5  0.1772  5  8.7 
Lakshadweep  0  1  0  0  1  0.1661  6  0.1661  4  0.0 
Andhra Pradesh  1  11  10  0  22  0.2499  7  0.3110  8  24.5 
Gujarat  2  7  14  0  23  0.2749  8  0.3554  10  29.2 
Tripura  0  2  2  0  4  0.2767  9  0.4712  16  70.3 
Madhya Pradesh  2  20  20  1  43  0.2894  10  0.3014  7  4.2 
Punjab  0  3  14  0  17  0.3120  11  0.4211  14  35.0 
Uttaranchal  0  2  6  0  8  0.3164  12  0.5291  21  67.2 
Karnataka  0  8  15  1  24  0.3174  13  0.3584  11  12.9 
West Bengal  0  6  8  0  14  0.3225  14  0.4947  18  53.4 
Andaman & Nicobar  0  1  1  0  2  0.3288  15  0.3288  9  0.0 
Rajasthan  0  9  22  1  32  0.3435  16  0.3735  12  8.7 
Maharashtra  1  8  21  2  32  0.3440  17  0.4153  13  20.7 
Haryana  0  5  14  0  19  0.3468  18  0.4975  19  43.4 
Arunachal Pradesh  0  2  11  1  14  0.3876  19  0.5343  22  37.9 
Assam  0  5  16  1  22  0.3884  20  0.4677  15  20.4 
Jammu & Kashmir  1  4  5  3  13  0.4104  21  0.5482  23  33.6 
Orissa  0  10  10  5  25  0.4135  22  0.4882  17  18.1 
Himachal Pradesh  0  0  10  0  10  0.4582  23  0.5196  20  13.4 
Manipur  0  2  4  2  8  0.5110  24  0.6363  26  24.5 
Jharkhand  0  1  2  1  4  0.5110  25  0.8064  27  57.8 
Uttar Pradesh  1  8  41  15  65  0.5182  26  0.6201  24  19.7 
Bihar  0  4  22  8  34  0.5310  27  0.6333  25  19.3 
Meghalaya  0  1  1  5  7  0.6692  28  0.8321  28  24.3 
Chhattisgarh  0  0  8  6  14  0.6745  29  0.8386  29  24.3 
Total  17  149  281  52  499  0.3756  0.4592  22.3 
Conclusion
Efficiency analysis provides several benefits to health providers, planners and policy makers alike. First, the resulting analyses help stakeholders in identifying geographic units that may be able to attain better outcome without increased allocation of resources. Second, the evidence from the analysis can also provide information on those exogenous factors whose presence (or absence) affects the performance of services and ultimately health outcomes in the country. The paper thus combines models from stochastic frontier analysis and spatial econometric literature to assess the level of inefficiency of maternal health care provision in India. The focus on India is relevant because it alone accounts for about a fifth (19%) of total maternal deaths in the world, which means that the target for MDG 5 at a global level cannot be achieved without significant progress in reducing maternal mortality in that country..
The available data was modelled using three variants of stochastic frontier model, including the standard stochastic model, a model corrected for heterogeneity [39] as well as a spatial stochastic frontier model developed and tested in this paper. This confirmed the existence of both inefficiency, heterogeneity and spillover effects in the delivery of maternal care in India. Spill over effects were captured through spatial lag variables with respect to outputs and efficiency distribution.
Consistent with previous work by Greene [31,39] for the case of unaccounted for heterogeneity where the effects show up as inefficiency, the resulting estimates from the spatial stochastic frontier model revealed that the same problem also arises when the model is not corrected for spill over effects. Results showed that the predicted mean efficiency score and the estimated underlying distribution of u (δ_{u}) were significantly lower for the spatial stochastic frontier models than those of the classical model and the model with only heterogeneous effects. The range of statistical tests undertaken in the analysis also confirmed that the spatial health frontier models fit the data notably better than both the classical and heterogeneity corrected models. These suggest that in health systems where interactions are a common place failing to correct for spatial externalities may lead to these unaccounted for effects to show up as inefficiency in the analysis. However, this does not necessarily imply that the rankings of spatial units with respect to inefficiency resulting from the spatial stochastic frontier model should be different from that of either the classical or the heterogeneity correct model. But in the data used in the present analysis, the rankings also changed, considerably (see Table 4). In sum, the fact that the spatial lag coefficients are large and significant may suggest the emergence of collective behavior and aggregate patterns in the delivery of maternal care services in the country, which may have been brought about by peer group effects (possibly operating through yardstick competition).
Regarding the role of inputs in the production process, all the three models showed the correct sign and with one exception (untied funding) were all significantly linked to levels of institutional delivery rate. In other words, more health facilities per square kilometres and more qualified health workers per facilities as well as facilities that were well equipped with key amenities lead to significantly high rate of institutional delivery. On the other hand, rresults from the inefficiency analysis showed that most districts in the country were operating well below their maximum capacity. The overall mean inefficiency score in the country was about 38%, while as many as twothird of districts in the country had mean efficiency level of about 75 per cent or less. This suggests that further progress in improving maternal care in India should focus not only on putting new resources but also in ensuring that existing resources are utilised efficiently, especially in parts of the country where inefficiency is extremely low. Finally, we should note that the reported efficiency estimates refer to the efficiency of an output, not the absolute level of the output itself. Thus, in those districts where efficiency level is already high but the rate of institutional coverage is still low further progress can only come by way of putting new resources in these areas. The fact that urbanization, literacy and low proportion of population in lowest income quintile (a proxy measure for income) had an enabling effect on the efficiency of institutional delivery mean that changes in any or all of these can also be expected to improve the system.
Declarations
Acknowledgments
The authors would like to thank Professor William Greene for reviewing the manuscript and providing valuable comments and suggestions that have greatly improved the final manuscript.
Authors’ Affiliations
References
 Nations U. Millennium Development Goal. New York: United Nations; 1990.Google Scholar
 Nations U. The Millennium Development Goals Report2013. New York: United Nations; 2013.Google Scholar
 Hogan MC, Foreman K, Naghavi M, Ahn S, Wang M, Makela S, et al. Maternal mortality for 181 countries, 1980—2008: a systematic analysis of progress towards Millennium Development Goal 5. Lancet. 2010;375(97726):1609–23.View ArticlePubMedGoogle Scholar
 Countdown Coverage Writing Group (on behalf of the Countdown to 2015 Core Group). Countdown to 2015 for maternal, newborn, and child survival: the 2008 report on tracking coverage of interventions. Lancet. 2008;371:1247–58.View ArticleGoogle Scholar
 Hill K, on behalf of the Maternal Mortality Working Group, Thomas K, AbouZahr C. Estimates of maternal mortality worldwide between 1990 and 2005: an assessment of available data. Lancet. 2007;370:1311–9.View ArticlePubMedGoogle Scholar
 World Health Organization. Trends in maternal mortality: 1990 to 2010, WHO, UNICEF, UNFPA and The World Bank estimates. Geneva: World Health Organization; 2012.Google Scholar
 United Nations: World Population Prospects: The 2012 Revision. [http://esa.un.org/wpp/unpp/panel_population.htm]
 Registrar General of India. Special Bulletin on Maternal Mortality in India 2007–09. New Delhi: Government of India; 2011.Google Scholar
 Paxton A, Maine D, Freedman L, Fry D, Lobis S. Averting Maternal Death and Disability: The evidence for emergency obstetric care. Int J Gyna Obst. 2005;88:181–93.View ArticleGoogle Scholar
 Weil O, Fernandez H. Is safemotherhoodanorphaninitiative? Lancet. 1999;354:940–3.View ArticlePubMedGoogle Scholar
 Adam T, Lim S, Mehta S, Bhutta Z, Fogstad H, Mathi M, et al. Cost effectiveness analysis of strategies for maternal and neonatal health in developing countries. BMJ. 2005;331:1107.View ArticlePubMedPubMed CentralGoogle Scholar
 Campbell OMR, Graham W, on behalf of The Lancet Maternal Survival Series steering group. Strategies for reducing maternal mortality: getting on with what works. Lancet. 2006;368:1284–99.View ArticlePubMedGoogle Scholar
 Khan K, Wojdyla D, Say L, Gulmezoglu AM, van Look PF. WHO analysis of causes of maternal death: a systematic review. Lancet. 2006;367:1066–74.View ArticlePubMedGoogle Scholar
 Burchett E, Helen and Mayhew, Susannah H. Maternal mortality in lowincome countries: What interventions have been evaluated and how should the evidence base be developed further? Int J Gyna Obst. 2009;105:78–81.View ArticleGoogle Scholar
 Ministry of Health and Family Welfare (Government of India). National Population Policy. New Delhi: National Commission on Population; 2000.Google Scholar
 Bank W. India Child Survival and Safe Motherhood Project. Washington DC: World Bank; 1997 [http://documents.worldbank.org/curated/en/1997/03/731996/indiachildsurvivalsafemotherhoodproject].Google Scholar
 Ministry of Health and Family welfare (2005) Reproductive and Child Health programme (RCH) II Document 2, The Principles and Evidence Base for state RCH II Programme Implementation Plan (PIPs). Government of India, New Delhi.Google Scholar
 Ministry of Health and Family Welfare. National Rural Health Mission. New Delhi: Government of India; 2005.Google Scholar
 International Institute for Population Sciences (IIPS). District Level Household and Facility Survey (DLHS3), 2007–08: India: Key Indicators: States and Districts. Mumbai: IIPS; 2010.Google Scholar
 Bhat R, Jain M. Analysis of public expenditure on health using state level data. Indian Institute of Management: Ahmedabad (India); 2004.Google Scholar
 De P, Dhar A, Bhattacharya BN. Efficiency of Health Care System in India: An Interstate Analysis Using DEA Approach. Soc work pub health. 2012;27(5):482–506.View ArticleGoogle Scholar
 Prachitha J, Shanmugan R. Efficiency of Raising Health Outcomes in Indian States. In: Working Paper 70. Chennai, India: Madras School of Economics; 2012.Google Scholar
 Purohit BC. Efficiency of Health Care Sector at SubState Level in India: A Case of Punjab. Online J Health Allied Scs. 2009;8(3):2.Google Scholar
 Purohit BC. Efficiency of health care system at the substate level in Madhya Pradesh, India. Soc work pub health. 2010;25:42–58.View ArticleGoogle Scholar
 Vinish K, Deepa S. InterState Disparities in Health Outcomes in Rural India: An Analysis Using a Stochastic Production Frontier Approach. Dev Pol Rev. 2005;23(2):145–63.View ArticleGoogle Scholar
 Jacobs R, Smith P, Street A. Measuring Efficiency in health Care: Analytic Techniques and Health Policy. Cambridge: Cambridge University Press; 2006.View ArticleGoogle Scholar
 Kumbhakar S, Lovell K. Stochastic Frontier Analysis. Cambridge: Cambridge University Press; 2000.View ArticleGoogle Scholar
 Aigner D, Lovell CAK, Schmidt P. Formulation and Estimation of Stochastic Frontier Production Function Models. J Econometrics. 1977;6:21–7.View ArticleGoogle Scholar
 Meeusen W, Jan van den Broeck. Efficiency estimation from CobbDouglas production functions with composed error. Int Econ Rev. 1977;18(2):435–44.Google Scholar
 Kinfu Y. The efficiency of the health system in South Africa: evidence from stochastic frontier analysis. App Econ. 2013;45(8):1003–10.View ArticleGoogle Scholar
 Greene W. Reconsidering heterogeneity in panel data estimators of the stochastic frontier model. J Econometrics. 2005;126:269–303.View ArticleGoogle Scholar
 Hollingsworth B, Wildman J. The efficiency of health production: reestimating the WHO panel data using parametric and nonparametric approaches to provide additional information. Health Econ. 2003;12(6):493–504.View ArticlePubMedGoogle Scholar
 Evans DB, Tandon A, Murray CJL, Lauer J. Comparative efficiency of national health systems: cross national econometric analysis. BMJ. 2001;323:307.View ArticlePubMedPubMed CentralGoogle Scholar
 Coelli T, Rao P, Battese G. An introduction to efficiency and productivity analysis. London: Kluwer Academic Publishers; 1998.View ArticleGoogle Scholar
 Jondrow J, Materov I, Lovell K, Schmidt P. On the Estimation of Technical Inefficiency in the Stochastic Frontier Production Function Model. J Econometrics. 1982;19(2/3):233–8.View ArticleGoogle Scholar
 Stevenson R. Likelihood functions for generalized stochastic frontier estimation. J Econometrics. 1980;13:57–66.View ArticleGoogle Scholar
 Greene W. A gamma distributed stochastic frontier model. J Econometrics. 1990;46:141–64.View ArticleGoogle Scholar
 Migon HS, Medici EV. Bayesian hierarchical models for stochastic production frontier. Estuar Coast Shelf Sci. 2005;57:27–52.Google Scholar
 Greene W. Distinguishing between heterogeneity and inefficiency: stochastic frontier analysis of the World health Organization’s panel data on national health care systems. Health Econ. 2004;13:959–80.View ArticlePubMedGoogle Scholar
 Hanining R. Spatial data analysis: Theory and practice. Cambridge: Cambridge University Press; 2003.View ArticleGoogle Scholar
 Keleljan HH, Pruda IR. Specification and estimation of spatial autoregressive models with autoregressive and hetroskedastic disturbances. J Econometrics. 2010;157:53–67.View ArticleGoogle Scholar
 LeSage J, Pace KR. Introduction to Spatial Econometrics. New York: Taylor & Francis Group; 2009.View ArticleGoogle Scholar
 Schmidt M, Shmidt A, Moreira ARB, Helfand S, Fonseca TCO. Spatial stochastic frontier models: accounting for unobserved local determinants of inefficiency. J Prod Anal. 2009;31:101–12.View ArticleGoogle Scholar
 Hollingsworth B. The measurement of efficiency and productivity of health care delivery. Health Econ. 2008;17:1107–28.View ArticlePubMedGoogle Scholar
 Anselin L. Spatial externalities, spatial multipliers and spatial econometrics. Int Reg Sci Rev. 2003;26:153–66.View ArticleGoogle Scholar
 Anselin L. Spatial Econometric. Richardson: University of Texas; 1999.Google Scholar
 Arbia G. Spatial Econometrics: Statistical foundations and applications to regional convergence. Berlin: Springer; 2006.Google Scholar
 Cressie NAC. Statistics for sptial data, Revised ed. New York: Wiley; 1993.Google Scholar
 Anselin L, Florax RJGM. Small sample properties of tests for spatial dependence in regression models: Some further results. In: Anselin L, RJGM F, editors. New Directions in Spatial Econometrics. Berlin: Springer; 1995. p. 21–74.View ArticleGoogle Scholar
 Keleljan HH, Pruda IR. Estimation of simultaneous systems of spatially interrelated crosssectional equations. J Econometrics. 2004;118:27–50.View ArticleGoogle Scholar
 Kumbhakar SC, Ghosh S, McGuckik JT. A generalized production frontier approach for estimating determinants of inefficiency in U.S. diary farms. J Bus Econ Stat. 1991;9:279–86.Google Scholar
 Huang CJ, Liu JT. Estimation of nonneutral stochastic frontier production function. J Prod Anal. 1994;5:171–80.View ArticleGoogle Scholar
 World Health Organization (WHO). Service availability and readiness assessment (SARA): Reference Manual. Geneva: World Health Organization (WHO); 2013. Available at: http://www.who.int/healthinfo/systems/SARA_Reference_Manual_Full.pdf (Accessed November 2013).Google Scholar
 Horrace WC, Schmidit P. Confidence statements for efficiency estimates from stochastic frontier models. J Prod Anal. 1996;7:257–82.View ArticleGoogle Scholar
 Cliff AD, Ord JK. Spatial Processes: Models and applications. London: Pion; 1981.Google Scholar
 Hadri K. Estimation of a doubly hetroscedastic stochastic frontier cost function. J Bus Econ Stat. 1999;17:359–63.Google Scholar
 Ord K. Estimation Methods for Models of Spatial Interaction. J Am Stat Assoc. 1975;70(349):120–6.View ArticleGoogle Scholar
 Cook DG, Pocock SJ, Pocock SJ. Multiple Regression in Geographical Mortality Studies, with Allowance for Spatially Correlated Errors. Biometrics. 1983;39(2):361–71.View ArticlePubMedGoogle Scholar
 LeSage J. Maximum likelihood estimation of spatial regression models. Toledo: University of Toledo; 2004.Google Scholar
 Marida K. Maximum Likelihood Estimation for Spatial Models, Proceedings of the Symposium on SPATIAL STATISTICS: Past Present and Future, held at Syracuse University. New: York; 1989.Google Scholar
 Drukker DM, Prucha IR, Raciborski R. Maximum likelihood and generalised twostage least squares estimators for a spatialautoregressive model with spatialautoregressive disturbances”. STATA J. 2013;13(2):221–41.Google Scholar
 Elisa Fusco and Francesco Vidoli. 2013. “Spatial stochastic frontier models: controlling spatial global and local heterogeneity”, International Review of Applied Economics, DOI: 10.1080/02692171.2013.804493.Google Scholar
 Belotti F, Daidone S, Ilardi G, Atella V. Stochastic frontier analysis using STATA”. STATA J. 2013;13(4):719–58.Google Scholar
 Dmitry P. Distingushing between spatial hetogrnuity and inefficiency: Spatial stochastic frontier analaysis of European Airports”. Transp Telecommunication. 2013;14(1):29–38. doi: 10.1080/02692171.2013.804493.Google Scholar
 Areal, F.J, Balcombe, K. and R Tiffin. 2010. “Integrating spatial dependence into stochastic frontier analysis”, Munich Personal RePEc Archive (MPRA) Paper No. 24961, posted 14. September 2010 11:38 UTC, http://mpra.ub.unimuenchen.de/24961/
 Battese GE, Coelli TJ. A stochastic frontier production function incorporating a model for technical inefficiency effects, Working Papers in Econometrics and Applied Statistics No. 69. University of New England, Armidale: Department of Econometrics; 1993.Google Scholar
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