COMPARACIÓN DE LA PROPORCIÓN DE PROBLEMAS ENTRE CENTROS DE SALUD

As was discussed in the previous section, the two stage model is the most common approach for dealing with environmental factors in DEA literature, which use regression in the second stage. In this chapter we propose a two stage analysis in order to deal with such environmental factors in DEA; a DEA is used to measure hospital efficiency while, SEM, which is a statistical technique for testing and estimating causal relations, is used to determine the direct and indirect effect of the environmental variables on efficiencies. Hence, SEM is used in the second stage rather than standard regression as the nature of the summary data for this study. In particular, most of our environmental factors result from the patient level, such as age, gender and GCS.

Despite the fact that these factors are summarised in order to be in a hospital level, there is a possibility of a casual relationship between these environmental factors and between these factors and efficiencies. For example, gender or age of patient (environmental factor) could affect Glasgow Coma Score GCS for patients (environmental factor), which consequently affects the recovery of the patient or the efficiency of the hospital. Thus, SEM enables a possibility to estimate and test the direct effect of gender on efficiency, as well as the indirect effect of gender on efficiencies through GCS, in order to obtain the total effect of patient

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gender on efficiencies by combining the direct and indirect effects. Therefore, SEM is proposed through this research, which is the first study to combine SEM with DEA in order to treat uncontrollable factors.

In addition to the previous reason for choosing this method in the current research, SEM has some advantages over the regression. Initially, it is a very flexible and comprehensive approach, which permits latent variables as well as multi-dependent variables. Secondly, it has the ability to deal with complex data, including missing data, non-normal data and time series with auto-correlated error. Moreover, variables in SEM could be independent and dependent, whereas variables in standard regression are either independent or dependent. Unlike multivariate regression, SEM has the ability to solve the equations of the model construct relationships simultaneously. Finally, a graphical presentation provides a convenient approach and powerful picture to explain a very complex relationship in SEM. For illustrative purposes, this methodology has been used to investigate the effect of environmental factors on the performance of 256 BTI hospitals. DEA scores provide important information for the performance of hospitals, while SEM exposed additional and valuable details that have not been identified from previous studies.

5.3.1 Introduction of Structural Equation Models

One specific statistical multivariate technique, which is very proficient, is through Structural Equation Modeling (SEM), as it functions through various methods of analysis. Hence, the researcher becomes capable of measuring the effects that are both direct and indirect by creating a performance of test models that exist with multiple dependent variables, whilst implementing different equations of regression at the same time. SEM is considered as a graphical model that is formed through econometrics, even though, due its historical development in the area of genetics, it has advanced with an introduction into sociology, which was referred to as path analysis. In fact, SEM contrasts from the single-linear regression models used for fitting the relationship between two groups of variables. In other words, SEM examines and confirms the causal relationships between the exogenous and

endogenous variables, and they are termed as causal models for correlational data, (Fox, 1984). In SEM, it is possible for a variable to be a predictor (such as environmental variables) in a specific equation, whereas it would be a response in another equation. Additionally, variables can influence one-another, either directly or through another variable (indirect

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effect) (See Figure 1). Invariably, endogenous variables are defined as variables, whose values are predicted by other variables (for example, Y1, Y2, and Y3 in Figure 1). Therefore,

the remaining variables are called exogenous variables. The SEM shown in Figure 1 can be written by a linear model of the form:

(5.1)

The vectors Y, X and ε consist of endogenous variables, exogenous variables and disturbance terms, respectively. The parameter matrix B represents the structural coefficients relating to the endogenous variables, whilst Γ relates to the exogenous variables with endogenous variables.

Figure 5.1: Example of path diagram for SEM

5.3.2 Direct, Indirect and Total Effect

In SEM, there are three types of effects: direct, indirect and total effects. The total effect measures the effect of X by external intervention on Y. The direct effect is defined as the effect of X on Y without any intervention (mediation) of any other variable, such as the direct arrow from X2 to Y2. On the other hand, the indirect effect involves one or more intervening

variables which mediate the effect, such as the effect of X3 on Y3 through Y2. In Statistics,

the indirect effect is defined as the difference between direct and total effect.

X

Y

Y

X

X

X

Y

X

97 5.3.3 DEA with SEM Methodology

In the current study, the two stages methodology is used to deal with environmental factors. In the first stage, the DEA model is applied with only controllable variables. In the second stage, SEM is conducted with efficiency scores (obtained from first stage) and environmental factors. Hence, in the second stage of this study, the researcher aims to study the simultaneous relationships among a set of environmental predictors, as well as these environmental factors with the efficiency score response obtained at the DEA first stage, in order to determine the sources of inefficiencies.

Structural equation models (SEM) will enable the possibility to examine those relationships using Multi-equation regression. Thus, SEM investigates the direct effect of the environmental (independent variables) on the efficiency scores (dependent variable), as well as the indirect effect of the environmental variables on efficiency scores through other environmental variables (dependent and independent variables). Even though there are multi dependent variables in SEM model (efficiency scores and the environmental mediators), the main interesting dependent variable is efficiency scores, which is limited variable between 0 and 1. The other dependent variable in our SEM model is continuous, which fits liner regression. Therefore, the study has focused on how model efficiency scores are variables in SEM.

It has been exhibited that in order to carry out the second DEA analysis, there are two main approaches for the interpretation of such an efficiency score variable in the second stage, as discussed in Macdonald (2009). The first and most common approach is to consider this efficiency score as an observed variable of DMUs efficiency. This is to show that efficiency scores are considered as descriptive measures of the efficiency score of the unit sample. Consequently, the frontier can be treated as an (within sample) observed frontier. Hence, in stage two, the efficiency scores can be viewed as other dependent variables in regression methodology, and therefore, standard inference of parameter estimation for the second stage is valid.

A second approach for interpretation is that the efficiency score is an estimated variable of 'true' efficiency scores relative to a 'true' construct. Given this interpretation, standard estimation of second stage is inconsistent and inference is invalid because of the uncertainty due to sampling variation, as well as the dependency of DEA scores on each other, which

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violates the assumption of within sample independence in regression analysis. Therefore, the second stage of DEA analysis should take these issues into account in order to get consistent estimations, such as methodology proposed by Simar and Wilson (2007), as well as Banker and Natarajan (2008) methodology. In the current study, the first interpretation framework is applied, and hence, the important point relates to choosing a suitable model for the DEA scores, which is a continuous limited dependent variable.

The most common and natural approach to investigate the relationship between DEA scores and environmental variables is the tobit regression, which is convenient with a censored or a corner solution dependent variables, of which DEA scores consider as the second type. A corner solution variable is "continuous and limited from above or below or both and takes on the value of one or both of the boundaries with a positive probability" (Hoff, 2007: p. 426). An alternative approach for modelling DEA scores against environmental variables is linear specification model estimated by ML or OLS. This linear specification model has been supported by both papers of Hoff (2007) and Macdonald (2009) who both concluded in their simulation studies that linear regression is sufficient and a consistent estimator in second stage DEA modelling, which has the advantage of the simplicity and familiarity compared with others. In addition, Banker and Natarajan (2008a) provide proof that linear regression estimated by (OLS) or (ML) in the two stage yields consistent estimators. Therefore, in this study, Tobit and linear specifications that use ML are both applied for modelling DEA scores as the dependent variable in SEM analysis.