Bootstrapping DEA

The bootstrap is a method of drawing by replacement from a data sample, which replicates the data generating process of the model and generates estimates that are used for statistical calculation. DEA has certain inherent inefficiency created by noise, formed by the distance from the efficient boundary. Moreover, bootstrapping helps to overcome these efficiencies for bias and to develop the correct confidence intervals, whilst accepting that the data has random noise. In Bootstrapping, the probability of distribution of the inefficiencies in DEA follows the true, but the unknown distribution of data. By taking a sample from the DEA inefficiencies, the researcher is actually taking out data from the population. By taking

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repeated samples, it is possible to build an empirical sample distribution for all the DEA efficiencies. This sample is then used to develop the confidence intervals for DEA efficiencies (Efron, 1987).

3.2.3.1 The Concept of Bootstrapping

Bootstrapping is used in a number of instances, such as hypothesis testing when it is not possible to form a statistical inference. By using re-sampling with bootstrapping, the assumed randomness of the data is redistributed, and this randomness is seen when variables from the model show deviations from their estimated value calculations. When the variance is higher in the residual data, then it means that the confidence intervals of the bootstrap model will be wider. Accuracy of the bootstrap model is derived from the bias of the process and the variance in the residuals, and these depend on the sample size. Residual variance creates differences in the bootstrapping distribution. What is more, the centre point of the bootstrap distribution curve must be equal to the computed value, and this variance is known as the

bootstrap bias, caused by the random sampling method. With smaller samples, observations

are erratic and the bias increases. In some cases, the bootstrap estimator can also fall to bias, and it will show variance from the true values (Simar & Wilson, 2007).

The steps in using the bootstrapping method are indicated through a series of stages (Simar & Wilson, 2000). Firstly, use DEA and calculate the efficiency scores for the data. The next step is to obtain through replacement from the empirical distribution of the scores from the first step. Indeed, if the distribution is smoothened, it provides better results. The original efficient input levels must be divided by the new or pseudo efficiency score, obtained from the empirical distribution and this step provides the bootstrap results for the new inputs. Subsequently, the following step is to calculate the bootstrapped efficiency scores by applying DEA for the newly obtained inputs with the same outputs. Overall, the previous steps can be repeated along with the bootstrapped scores to test the hypothesis and obtain the statistical inference of the results.

According to Simar and Wilson (2008), in order to construct a set of homogenous bootstraping efficiency estimates for the original DEA efficiency scores { } for an observed point (xn,yn), there are eight steps to be

implemented as follows:

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1. To calculate the DEA efficiency scores by using the original data set. Then, for simplicity, these efficiency scores are parameterised by in order to

avoid creating estimated lower bounds for confidence intervals that are negative. The corresponding parameterised bootstrap efficiency estimates is

.

2. To choose a smoothing parameter, the bandwidth h that is discussed in Silverman (1986) to calculate this bandwidth parameter. In the current study,

.

3. To generate ,..., by drawing with replacement a random sample of efficiency from the constructed set of 2n reflected efficiencies out of the n

computed in step 1; ={ }. Drawing from the data set of instead of the efficiency computed in step 1 is to permit for the possibility that DEA efficiency has an upper bound of 1.

4. To adjust the sample of efficiencies drawn in step 3 by drawing , independently from the kernel function K (.) and find the values for + for each n = 1, ..., N.

5. To calculate the values for ,

,

where:

is the value of the variance seen in the probability density function in the kernel function. Subsequently, the value of is calculated as:

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6. The bootstrap sample is created as: = {( },

where: = .

7. To complete the set of the bootstrap DEA efficiency estimate (xn, yn)

for the original sample observations with the reference set of .

8. The steps 3-7 are repeated B times, which is at least 2000 times to derive the bootstrap set estimate of { (x, y) | b = 1, ...,B}.

The bootstrap bias is estimated for the original DEA estimator as follows:

(3.9)

B is the number of instances that the process was carried out, , which provides the bootstrap DEA scores, and is the DEA score. For this equation, the biased corrector

estimator is the unknown true efficiency of :

(3.10)

Efron and Tibshirani (1993); Simar and Wilson (2008) argue that this bias correction can introduce extra noise. Therefore, the sample variance of the bootstrap value must be recalculated as:

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It may be required to avoid the bias from the above equation, unless:

(3.12)

According to Daraio & Simar (2007) and Simar and Wilson (2008), in comparison to the original DEA values, the estimates for bias corrected values (bootstrap DEA values) must be preferred in consideration when the bias is more advanced than the standard deviation ( ).

3.2.3.2 Studies using DEA and Bootstrapping Approaches

A number of researchers have used DEA with bootstrapping methods to analyse the performance and efficiency of hospitals and the healthcare sector organisations. Staat (2006) has researched the performance and efficiency of German hospitals by using the DEA- bootstrapping procedure. The process was applied to two data sets of hospitals, and all hospitals had comparable quality and range of services. Furthermore, this helped to overcome the earlier issues of DEA efficiency analysis with regression analysis.

Bernet et al. (2008) examined data from two geopolitical regions of Ukraine to compare polyclinics in Ukraine in order to analyse whether the inflexibility of Soviet system of planned economies developed lower economic efficiency in eastern regions, and the DEA with bootstrapping methods was used in the evaluation. Assaf and Matawie (2010) used the DEA bootstrapping approach to analyse the efficiency of health care foodservice operations in the USA. The process helped to derive the bias from estimates and the confidence intervals of DEA efficiency score, as well as to resolve the co-relation problem of DEA efficiency scores is the second stage anlysis. Halkos and Tzeremes (2011) examined the Greek public healthcare delivery efficiency by using data envelopment analysis and the bootstrap method. The efficiency levels of the hospitals were analysed by using convex and non-convex models with bootstrap techniques, and overall the analysis helped to find the misallocation of healthcare resources among the Greek regions.

In other similar studies, Kounetas and Papathanassopoulos (2013) used different input–output combinations to identify factors that influence the Greek hospital performance. Invariably,

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they used the DEA bootstrapping method to evaluate the productive efficiency of different hospitals in the data set.

The bootstrapping DEA method is an advanced methodology to overcome the disadvantage associated with the standard DEA, which is due to the deterministic nature. However, there are just a few health care applications for such approaches, as mentioned previously in Chapter One. Therefore, the present study applies the above methodology for the empirical analysis of HTI care in Chapter Six.