Data
Data for the study was sourced from the 2011 Access, Bottlenecks, Cost and Equity (ABCE) project conducted by the Institute for Health Metrics and Evaluation (IHME) in collaboration with Ghana Ministry of Health (MOH), Ghana Heath Service (GHS), Ghana UNICEF office and UNICEF. The data collects information from facilities at all levels of the health sector across several countries. Information collected include facility finances and input, facility management, equipment and capacity as well as facility outputs. A total of 87 health centers were included in the final sample. The inclusion of facilities was determined by data availability.
The stochastic frontier analysis (SFA) model
Following Danquah et al., the starting point of the SFA model is to specify a relationship between a set of health facility inputs that produce an output [
14]. This can be specified as
$$ {y}_i= f\left({l}_i,{k}_i,{q}_i\right) $$
(1)
Where y is the output measure (in this case, number of out-patient visit), l is labour (total number of clinical and non-clinical staff), k is capital (proxied with number of hospital beds and rooms) and q is a set of other potential determinants of output (such as age of health facility).
Assuming that some heath facilities do not efficiently employ inputs to produce output, we can capture the actual observable output in the following stochastic frontier production function
$$ {y}_i= f\left({l}_i,{k}_i,{q}_i;\beta \right){TE}_i{e}^{v_i} $$
(2)
Where β is an estimable parameter, TE captures technical efficiency and measures the deviation of a health facility from the stochastic frontier.
v
i
is a random error term. Equation
(2) can be re-specified in a log-linear form as follows
$$ \mathit{\ln}{y}_i=\mathit{\ln} f\left({l}_i,{k}_i,{q}_i;\beta \right)+\mathit{\ln}{TE}_i+\mathit{\ln}{e}^{v_i} $$
(3)
Assume
TE
i
= exp(−
u
i
) we can reformulate Eq.
(3) as
$$ \mathit{\ln}{y}_i=\mathit{\ln} f\left({l}_i,{k}_i,{q}_i;\beta \right)+{e}_i-{u}_i,\kern1.75em {u}_i>0 $$
(4)
As noted earlier, the error term is composed of a sum of normally distributed disturbance (
v
i
) which accounts for measurement and specification error and a one-sided disturbance (
u
i
) which measures inefficiency. Both
v
i
and
u
i
are assumed to be independent and identically distributed across observations. An exponential assumption [
u
i
~
ε(
δ
u
)] proposed by Meensen and VanBroeck [
15], was made about the distribution of the inefficiency term [
16].
The estimation of the stochastic frontier (SF) requires a functional form of the production function. Two specifications are popular in the literature; the Cobb-Douglas (CD) and Tanslog functional forms. We specify the CD production function in this study as
$$ {y}_i=\beta {l}_i^{\alpha}{k}_i^{\pi}{q}_i^{\theta},\kern1em \alpha +\pi +\theta =1 $$
(5)
All inputs are as defined in Eq.
(1) above. α, π and θ are parameters to be estimated. Equation
(5) can be linearized to generate a linear production function as follows;
$$ \mathit{\ln}{y}_i=\mathit{\ln}{\beta}_0+\alpha ln{l}_i+\pi {k}_i+\theta {q}_i+{\varepsilon}_i $$
(6)
Where ε
i
is an error term that can be decomposed into a normal white noise error term and an inefficiency term.
The Translog form of the above production function can be specified as
$$ \mathit{\ln}{y}_i=\mathit{\ln}{\beta}_0+\alpha ln{l}_i+\pi {k}_i+\theta {q}_i+0.5\left[{\alpha}_1{ l n}^2{l}_i+{\pi}_1{ l n}^2{k}_i+{\theta}_1{ l n}^2{q}_i\right]+0.5\left[{\alpha}_2\mathit{\ln}{l_i}^{\ast}\mathit{\ln}{k}_i+{\alpha}_3\mathit{\ln}{l_i}^{\ast}\mathit{\ln}{q}_i+{\alpha}_4\mathit{\ln}{q_i}^{\ast}\mathit{\ln}{k}_i\right]+{\varepsilon}_i $$
(7)
We estimated both the CD and Translog production functions. To decide on which specification is appropriate, we used the likelihood ratio test. The test was conducted on the null hypothesis that the multiplicative terms in the Translog function are simultaneously equal to zero. Results of the likelihood ratio test (LR Chi2 = 7.26;
p-value = 0.7011) suggests that the CD function is favorable relative to the Translog function. The lack of statistical significance (high
p-value) indicates that we fail to reject the null hypothesis. This implies that the CD functional form is sufficient for our analysis. We, however, reported results from the Translog specification as a Additional file
1.
The choice of SFA over the data envelopment analysis (DEA) was motivated by its flexibility to allow control variables aside the direct inputs. Further, while the DEA is the most used in the estimation of health system efficiency among the two models, it is weak in the sense that it is extremely sensitive to the presence of outliers, which define the frontier. Its nonparametric nature also implies that it is unable to address random variations in the data which are then captured as inefficiency. While the SFA addresses these weaknesses, it is also limited in the imposition of some functional form on the production function which, in some cases, become difficult to estimate [
17‐
20]. As discussed earlier, we minimize this weakness in the SFA by statistically deciding which functional form is appropriate for this study.
The choice of inputs and outputs was mainly based on availability of data. A search of the literature suggested many variables considered to be standard measures of facility inputs and outputs. Following these, the current study used the following variables in measuring inputs and outputs at the primary health facilities.
Output variable: Number of outpatient visits was used as the output variable of interest. This is because most health centers in Ghana only provide outpatient services. Inpatient services are mostly not available at this level of health care.
Input variable: The main input variables used in the efficiency estimation include number of personnel, hospital beds and expenditure on other capital items and administration. Other control variables used include rural/urban location, public/private facility type, age of facility, display of fee list and number of rooms available.
Computing efficiency gain
The potential gains from efficiency was computed by finding the proportion of a facility’s revenues (
R
i
) that could be saved if efficiency was improved. This is presented in Eq.
(2) as the proportion of facility revenue that is lost to inefficiency.
$$ {rev}_i=\left({eff}_{max}-{eff}_i\right)\times {R}_i $$
(8)
where rev
i
represents revenue of the ith facility that could be gained if inefficiencies were corrected, eff
max
is maximum efficiency level (1.00 in this case) and eff
i
is actual efficiency score of the ith facility, predicted from the SFA specification above.
The potential savings in total facility revenue (rev
i
) also shows the potential fiscal space for health available for the ith facility if efficiency were improved.
Nopo matching decomposition
Disparities in efficiency between public and private facilities was estimated using the Nopo matching decomposition approach. The Nopo procedure even, though similar to the famous Oaxaca-Blinder approach, is considered to be better for two main reasons. One is that it is fully parametric and in the case of the current study it requires a linear regression model for efficiency. Secondly, it does not restrict comparison to facilities with comparable characteristics, i.e. it ignores the common support problem.
The Nopo decomposition approach uses an algorithm to match a public facility with a similar private facility at the primary level. This implies that the facility type becomes the treatment variable in this decomposition analysis. Four steps were followed to complete the procedure.
1 These are outlined below;
1.
Select one public facility without replacement from the sample
2.
Now select all private facilities that have similar characteristics as the public facility selected in step 1
3.
Construct a synthetic facility from all the facilities from step 2 whose characteristics are equal to average of all of them and match it to the facility in step 1.
4.
Put the observations of both facilities (the synthetic private and the public facility) in their respective new samples of matched facilities
5.
Repeat steps 1 through 4 until it exhausts the original public facility sample
Following this characterization, the disparities in efficiency between the matched public and private facilities was then computed. This total gap (Δ) was further disaggregated into four components as described below
Δ0 = This is the part of the efficiency gap that cannot be explained by differences in facility characteristics. This is also considered to be the residual part of the decomposition.
Δpu = This component explains the disparities between public facilities that are matched and those that are unmatched
Δpr = Similarly, this component shows differences between matched private facilities and unmatched private facilities
Δx = This component shows that part of the efficiency gap between private and public facilities that can be explained endogenously (differences in covariates).