# Scale Validity Assessment And Testing Of Model Accounting Essay

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The results of this analysis appear in Table 5.16 and evaluation of the correlation matrix through the KMO and Bartletts test resulted in high KMO statistics and a significant probability level (p<0.001) for the Bartlett's test of all constructs. This indicates that sufficient correlations were found within the correlation matrix for factor analysis to proceed. In addition, item-to-item correlations within each construct were inspected and all coefficients fell within the acceptable range for factor analysis of 0.40 to 0.80. Cronbach's alpha for each construct was computed, indicating good reliabilities, of the measures ranging from 0.845 to 0.951.

## 5.2.6 Discriminant Validity

Having computed the composite measures, an assessment of discriminant validity as recommended by Gaski (1984) was initiated. Gaski (1984) argues that if the correlation between two composite constructs is not higher than their respective reliability estimates, then discriminant validity exists. This being the case, construct correlations were examined and compared to the reliabilities calculated through Cronbach's alpha in the initial data analysis. The results indicated that all reliability estimates were greater than correlations between the constructs, therefore, verifying discriminant validity. In addition, the data were tested for common method variance in accordance with Harmon's one factor test (Malhotra et al. 2006, Podsakoff et al. 2003, Podsakoff and Organ 1986). The results of this test indicated that the majority of variance was not accounted for by one general factor; therefore a substantial amount of common method variance was not evident.

## Community Expectations

KMO statistics

0.852

0.943

0.856

0.918

Bartlett's test statistics significance

0.000

0.000

0.000

0.000

Cronbach's alpha

0.845

0.951

0.847

0.921

Number of Items

30

33

25

27

Variance Explained

61.39%

65.24%

63.48%

62.44%

No of Factors

8

4

7

4

## 5.3 Scale Validity Assessment and Testing of Model

Structural Equation Modeling (SEM) [1] with maximum likelihood estimate was used to test the measurement and structural model of stakeholder's expectations. The factor analytic theory on which SEM operates, assumes linear relationship of internal attributes (latent variables) on surface attributes (observed variables). SEM conducts two basic types of factor analyses: exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). As any theory, the theory of factor analytic needs an evaluation of fit between real world and itself. That is how well the theory accounts for the observed phenomena in our study. The issue of goodness of fit (i.e. our model represents the reality of each stakeholder expectations) of the theory to the observed data will have to be considered as a routine part of any application of factor analysis. Byrne (2006) suggested that "If the goodness of fit is adequate, the model argues for the plausibility of postulated relations among variables; if it is inadequate, the tenability of such relations is rejected".

Unlike other multivariate procedures, structural equation modeling takes a confirmatory approach rather than exploratory approach to the data analysis. It asks for a priori specification of the pattern of inter-variable relations. SEM provides explicit estimation and correction of all the measurement error parameters and does not ignore the error in explanatory variables. It takes into account both the observed and unobserved (latent) variables in data analysis. Therefore, SEM easily determines both the direct and indirect effect in multivariate analysis.

In structural equation modeling first the conceptual model derived from the theory is transformed to model specifications that always form a path diagram. The conceptual model must be identified in order to determine whether unique values can be found for the parameters to be estimated in the theoretical model (Kaplan, 2000, p.8). The purpose of model identification is to find whether or not there is a unique set of parameters consistent with the data. Model identification process measures the fit of the transposition of the variance-covariance matrix of observed variables (the sample data) into the structural parameters of the research model. If a unique solution for the values of the structural parameters can be found, the model is considered to be identified. Consequently the parameters are considered to be estimable and the model therefore is testable empirically. A model cannot be identified, if it indicates that the parameters are subject to arbitrariness thereby implying that different parameters value define the same model. In such cases, the estimates of the model parameters are not consistent and cannot be evaluated empirically. SEM models should meet the criteria of over identification. In an over identified model the degree of freedom is positive and it allows for the rejection of model and hence it retains scientific interest of testing. The number of parameters to be estimated is less than the number of variance and covariance of the observed variables (data points). This model may or may not give unique solution for all parameters and it can be accepted/rejected hence it retain scientific interest.

Once a model has been identified, it can be used to obtain estimates of the free parameters from a set of observed data. Some authors recommend a two-stage approach in which the measurement model is estimated first, followed by the structural model and others recommended that the full model be estimated all at once. This is then followed by the assessment of the goodness of fit of the model. A structural model is said to fit the observed data to the extent that the covariance matrix it implies is equivalent to the observed covariance matrix (Hoyle, 1995). If the model fit does not meet the standard, the researcher could try to modify the model in terms of sound theory. There are two ways to modify a model. One is to free parameters and the other is to fix parameters. If the model fits the requirements, then the consequences are explained. Amos 7 software was used to test the proposed model. For the assessment of the fit of the proposed model, guidelines of literature were followed. The fit of overall model and the fit of internal structure were evaluated (Bagozzi and Yi, 1988; Bollen, 1989; Brown and Cudeck, 1993; Byrne, 1998; Chou and Bentler, 1995; Joreskog and Sorbom, 1993; Hair et al., 1998).

## 5.3.1.1 Absolute Fit Measures

Chi-square test simultaneously tests the extent to which specification (null hypothesis, H0) of the factor loadings, factor variances/covariance, and error variances for the model under study are true. The probability value associated with chi-square represents the likelihood of obtaining a chi-square value that exceeds the chi-square value when Ho is true. Thus, higher the probability associated with chi-square, the closer is the fit between the hypothesized model (under Ho) and the perfect fit (Bollen, 1989). The value of chi-square should not be significant that is its p-value should be more than 0.1. However as the chi-square statistic is sample size sensitive several authors have recommended considering the chi-square to degree of freedom ratio of less than 3, to assess the fit.

Researchers have addressed the chi-square limitations by developing goodness of fit indexes such as GFI, AGFI, RMR, and RMRSEA as that take a more pragmatic approach to the evaluation process. The goodness of fit index (GFI) [2] is a measure of the relative amount of variance and covariance in sample data with the theoretical model.

Adjusted Goodness of Fit Index (AGFI) takes into consideration the degree of freedom it compare the hypothesized with no model at all (Hu and Bentler, 1995). The adjusted goodness of fit index (AGFI) values should be larger than 0.90. The Root Mean Square Residual (RMR) represents the average residual value derived from the fitting of the variance-covariance matrix for the hypothesized model Î£ (Î¸) to the variance covariance of the sample data (S). The standardized RMR represents the average value across all standardized residuals and ranges from zero to 1.00. In a well fitting model standardized RMR value will be less than 0.05.

Root Mean Square Error of Approximation (RMRSEA) is one of the most important and robust informative criteria in covariance structure modeling. It takes into account the error of approximation in the population and tests the fit of unknown optimally chosen parameter values with the population covariance matrix (Brown and Cudeck, 1993). This index is sensitive to the number of parameters to be estimated. Root mean square error of approximation (RMSEA) values less than 0.05 indicates good fit. RMSEA values ranging from 0.05 to 0.08 indicates fair fit. The values ranging from 0.08 to 0.10 are indicative of mediocre fit. RMSEA values larger than 0.10 are indicative of a poor fit.

The Expected Cross-Validation Index (ECVI) measures the discrepancy between the fitted covariance matrix in the analyzed sample, and the expected covariance matrix that would be obtained in another sample of equivalent size. The model having the smallest ECVI values exhibits the greatest potential for replication. The Expected cross validation index (ECVI) values for the theoretical model less than that of independent model and saturated model are indicative of the acceptance of the model.

## 5.3.1.2 Incremental Fit Measures

Normed Fit Index (NFI), comparative fit index (CFI) are incremental indices of fit and are based on the comparison of the hypothesized model against some standard and their values range from zero to 1, where 1 shows the exact fit. However NFI underestimates the model in case of a small sample size. Both the measures show the complete covariation in the data. According to Bentler (1992) a value of more than 0.90 is representative of well fitting model. Later Hu and Bentler (1999) gave a revised cut off value of more than 0.95 for a well fitting model. However Bentler (1990) suggested the use of CFI as the index of choice. The relative fit index (RFI), the incremental index of fit (IFI) and the Tucker-Lewis index (TLI) should be higher than 0.90 for a better fit of the model. According to Hu and Bentler (1999) the TLI should be more than 0.95 in case of a large sample.

## 5.3.1.3 Parsimonious fit measures

The Parsimonious Normed Fit Index (PNFI) values and parsimonious goodness of fit index (PGFI) values should be larger than 0.5. The parsimony goodness of fit index (PGFI) takes into account the complexity in terms of number of estimated parameters of the hypothesized model in the assessment of overall model fit. It provides a more realistic evaluation of the goodness of fit of the model (Mulaik et al., 1989). Akaike Information Criterion (AIC) for the theoretical model less than that of independent model and saturated model are indicative of an acceptance of the model. The Hoelter (1983) Critical N (CN) values of large than 200, are indicative of an acceptance of the model. The Normed chi-square values ranging from 1.0 to 5.0 are indicative of an acceptance of the model.

## 5.3.1.4 Measures of fit of Internal Structure

For the measurement model, the test of parameters estimates of observed variables should be significant. If they are significant, this means that they can effectively reflect latent variable. Average variance explanation of the construct should be larger than 0.50 in cases of more complex and abstract construct and larger than 0.60 for fairly complex construct and 0.70 for less complex construct. The reliability of the construct should be larger than 0.60. The test of the structural model included direction, magnitude, and R2 of parameter. Parameters estimates should be significant. The direction must be corrective and R2 must have enough magnitude of explanation.

## 5.3.2 Application of Confirmatory Factor Analysis (CFA)

Since the stakeholder expectations model, its item measures and constructs were formulated on the basis of the literature and qualitative research insights, a confirmatory factor analysis was performed to test it. Structural equation modeling approach was used in two steps (James et al., 1982; Swafford et al., 2006). The first step involved analysis of measurement models to assess psychometric properties. The second step was to define the structural model by specifying direct and indirect relations among the latent variables. The literature supports the fact that separating the measurement and structural model testing, ensures the model identification. If measurement models are identified independently, the structural model will also be identified. As per the guidelines of Bagozzi (1994), and Bagozzi and Yi (1989), the following measurement prosperities were considered important for assessing the measures developed in this research: content validity, internal consistency of operationalization (unidimensionality and reliability), convergent validity, discriminant validity, and predictive validity.

## 5.3.2.1 Content Validity

A measurement instrument is said to be content valid if there is a general agreement among the subjects and researchers that the instrument has measurement items that cover all important aspect of the variables being measured. Content validity depends on how well the researchers create measurement items to cover the domain of the variable being measured (Nunnally, 1978). The evaluation of content validity is a rational judgmental process not open to numerical evaluation. Usual method of ensuring content validity is an extensive review of literature for the choice of the items and getting inputs from the practitioners and academic researchers on the appropriateness and completeness of items that measure underlying theoretical construct.

## 5.3.2.2 Unidimensionality

Unidimensionality indicates that all of the items are measuring a single theoretical construct. It refers to the existence of a single concept underlying a group of measures and is important to assess before structural model testing is done (Gerbing and Anderson, 1988). Unidimensionality was tested in the initial stage of scale development using exploratory factor analysis based on the Eigen values and based on the Scree plots (Rencher, 1995). A rule thumb is that Eigen values greater than 1.0 for the first factor/dimension. And Eigen values of the second dimension is less than or equal to Eigen value of the first dimension minus 1, supports the constructs exhibiting unidimensionality. Eigen values and the percentage of explained variances for each construct are discussed in the previous section. Researchers argue that exploratory factor analysis does not always establish the unidimensionality of the construct and several researchers recommend the application of confirmatory factor analysis to test the unidimensionality by using the structural equation modeling approach (Li Suhong et al., 2005; Joreskog and Wold, 1982). Confirmatory measurement models or confirmatory factor analysis using the maximum likelihood estimate is recommended as the best method in achieving unidimensionality of the measurement for theory testing and development (Joreskog and Wold, 1982). Structural equation modeling provides multiple fit criteria to test for unidimensionality and reduce any measuring biases inherent in different measures. GFI and RMR are the two goodness of fit index used most often to test the unidimensionality in confirmatory factor analysis. GFI values of more than or equal to 0.90 and RMR values of equal to or less than 0.05 suggest no evidence of a lack of unidimensionality.

## 5.3.2.3 Convergent Validity

Convergent validity was tested in both the EFA and CFA stages of evaluation (Atuahene-Gima and Evangelista, 2000; Moorman, 1995). To assess convergent validity of constructs we look at each item in the scale as a different approach to measure the construct and determine if they are convergent. Convergent validity represents how well the item measures relate to each other with respect to a common concept, and is exhibited by having significant factor loadings of measures on hypothesized constructs (Anderson and Gerbing, 1998). In EFA a construct is considered to have convergent validity if its Eigen value exceeds 1.00 and also all the factor loadings must exceeds the minimum value of 0.30 (Hair et al., 1995). In CFA, convergent validity can be assessed by testing whether or not each individual item's coefficient is greater than twice its standard error (Anderson and Gerbing, 1998). According to Bollen (1989), larger the values of t-test or relationship, the stronger are the evidence that the individual items represent the underlying factors. Further, the proportion of variance (R2) in the observed variable, accounted for by the theoretical constructs influencing them was used to estimate the reliability of an indicator. R2 value exceeding 0.30 were considered acceptable (Carr and Pearson, 1999). To achieve higher statistical power in testing, Joreskog and Sorbom's (1993) recommendation was followed and each measurement model was analyzed independently. Since the items were rigorously developed following experts rating and opinions, most of the items were significantly loaded on their respective factors. All factor loadings for the constructs shown in the measurement models are significant, except for the factor "influence" under perceived control, which was retained to maintain content validity. Therefore, convergent validity of the scales was established. To assess the convergent validity of constructs Bentler-Bonett Coefficient delta (Bentler and Bonnet, 1980), was determined. It is the ratio of the difference between the chi-square value of the null measurement model and the chi-square value of the specified measurement model to the chi-square value of the null model. A value of 0.90 and above demonstrates strong convergent validity (Hartwick and Barki, 1994; Segar and Grover, 1993). The fact that all the constructs have Bentler-Bonett coefficient delta values of 0.91 or above, demonstrates the presence of strong convergent validity.

## 5.4 Measurement Models

Since data sources for all four constructs were different, four first order initial measurement models were constructed. Each construct was tested for unidimensionality by using the Maximum Likelihood (ML) estimation method in CFA. The initial models of fitness were assessed and subjected to re-specification.

In the next stage, second order confirmatory factor analysis was performed based on the re-specified model. Re-specified nested models were reported by producing over-identified models (Figure 5.3, 5.4, 5.5 and 5.6). To produce over-identified models regression path in each measurement component was fixed at 1. The criteria used to evaluate the items were: each item's error variance estimate; evidence of items needing to cross-load on more than one component factor as indicated by large modification indices; the extent to which items give rise to significant residual covariance; parsimony purpose; regression coefficient of each item; reliability of the item and the reliability of the whole construct. In addition, the logic and consistency of data with the theoretical framework was considered when evaluating each item. The CFI other relevant model fitness indices for all the constructs i.e., stakeholder expectations presented are shown in Table 5.17. All the CFI values were above 0.90, implying that there is strong evidence of unidimensionality for the scales.

57.935

57.281

89.618

81.114

39

30

60

49

1.486

1.909

1.494

1.655

0.046

0.062

0.047

0.056

0.092

0.049

0.147

0.053

0.958

0.954

0.947

0.936

0.952

0.954

0.916

0.86

0.933

0.931

0.89

0.812

0.984

0.977

0.97

0.938

## Figure 5.6: Scales for Shareholders' Expectations

Figure 5.3 confirms the CSR in action framework of Taneja et al. (2011) that local community expect good corporate communication, continuous efforts on CSR activities, sustainable business practices by keeping in mind triple bottom line approach in mind from the companies.

Employees of companies expect fulfillment of their social needs with transparent recruitment selection system; clear job descriptions and healthy participative work place environment supported by organizational system for their development (Figure 5.4). Fulfillment of expectations of the employees will lead higher job satisfaction, lower employees' turnover ratio which has been directly linked to rising employee recruitment and training costs, low levels of employee morale, and customers' perceptions of service quality and lower profitability (Dermody et al., 2004). Allen et al. (2003) concluded that employees' perceptions of supportive HR practices such as participation in the decision-making process, growth and development opportunities, and fairness of rewards and recognition consistently positively related to increasing employees' productivity and efficiency.

Customers expect perfect CRM practices; quality assured products as specified, at fair price with factual and informative promotional activities (Figure 5.5). Therefore, to enhance customer satisfaction other than price satisfaction (Matzler, 2007), product quality, customer relationship management practices are equally important (Matzler and Sauerwein, 2002). Customer satisfaction will not only give managerial strength but also enhance profitability of the company (Kelsey and Bond, 2001).

Market value of a company is a function of the market's (existing and prospective shareholders) perception of a business's ability to generate returns today and in the future. Put simply,

## Shareholder Value = Current Performance + Expectations for Future Performance

(Kaplan and Norton, 1992)

Hence, shareholders expect good returns today on their investments (Anand et al, 2005) as well as the ability to generate profits into the future. They further expect the working of the companies should be ethical and responsible not only to them but also to all other stakeholders for their sustainable development and growth in future (See Figure 5.6).

Chapter 6 - Quantifying and Evaluating Corporate Economic and Social Performance

Based on the answer to third research question [3] i.e. for constructing valid and reliable scales for measuring social performance of the companies, the focus of this chapter is to answer the remaining two research questions:

How to integrate economic performance of the organizations with social performance of the company to determine overall performance of the company/

How to evaluate the overall performance of an organization in comparison to other set of organizations?

To answer these questions, a three stage approach was used. In the first stage, overall efficiency scores (i.e. both economic and social performance) of all firms under study were calculated by using output oriented approach of Data Envelopment Analysis. Operational performance of the companies was taken as input and financial and social performance was taken as output measures to calculate overall technical, managerial and scale efficiencies of the companies with Variable Return to Scale (VRS) Model (Banker et al., 1984). Operating expense ratio was an indicator for operational performance. Stakeholders' perceptions about respective companies on the developed scale (chapter 5) have been considered as social performance indicator. Return on Investment of each sample firm has been considered as an indicator for financial performance.

In the second stage, from the respective benchmarks of each firm, slacks were calculated to evaluate the performance of each sample company. These calculated slacks are basically identified performance gaps and highlight the future course of actions in terms of future target performance for each indicator for each decision making unit under review.

In the third and final stage, a sensitivity analysis has been employed to test and validate the obtained results by using various combinations of firms. In the subsequent sections, a brief description of Data Envelopment analysis (DEA) has been given along with reasons of selection of DEA over other methods. Then in the next section, the input output model used for the study has been defined and answers to research questions have been illustrated by using performance indicators of sample firm. In the last section, testing and validation of economic and social performance model has been illustrated with help of sensitivity analysis carried on obtained results.

## 6.1 Why Data Envelopment Analysis?

To assess the overall performance of firms [4] , empirical studies on performance measurement had used simple linear aggregations, weighted or non-weighted approaches (Freeman 2003, Kuosmanen and Kortelainen 2007; Chen and Delmas, 2011). These types of approaches seem appropriate only on those cases when the weights to the performance indicators are exogenously given (Bird et al. 2007, Hillman and Keim 2001, Mitchell et al. 1997). But this choice of weights becomes herculean task for the corporate managers who work in diverse competitive environment and facing a variety of stakeholder pressures (Clarkson 1995, Delmas and Toffel 2008; Chen and Delmas, 2011). Furthermore, assessment of CSP and CEP (CFP+COP) together would contain both negative and positive indicators to represent strengths and concerns regarding business practices. Chen and Delmas, (2011) carried out a sensitivity analysis by using various aggregation methodologies and demonstrated that the scores resulting from these methodologies differ in terms of their median and variance and are sensitive to changes in aggregation weights (Chatterji and Levine, 2006; Rowley and Berman, 2000; Delquié, 1997). Mattingly and Berman (2006) classified of 12 KLD (Kinder Lydenburg Domini) rating, corporate social actions variables in to four latent factors on the basis of co-variation present in KLD data with the help of exploratory analytical method.

Considering the multiple dimensions of the corporate overall performance (CSP+CEP) construct and limitations of the existing performance aggregation methodologies, Jones (1995), Chen and Delmas (2011) and Belu & Manescu (2013) suggested data envelopment analysis (DEA), a mathematical programming method for evaluating the relative efficiencies of firms (Charnes et al. 1978, Cook and Zhu 2006) that does not require a priori weights to aggregate different CSP and CEP issues.

Bendheim et al. (1998) has also used DEA as a methodology to develop a comprehensive and empirically testable [5] understanding of multi-fiduciary stakeholder relations (Goodpaster, 1991).

DEA calculates an efficient frontier that represents the best performing entities in a peer group. The DEA (CEP+CSP) score represents the distance of a firm to the efficient frontier. This score further represent the extent to which a firm can reduce its current concerns with its given strengths relative to those of the best performers. DEA has several advantages and edges over other methodologies in addressing the challenges of assessing overall performance. Some of these advantages are

DEA produces a ratio index that incorporates both good and bad performance indicators i.e. it does not assume normal distribution of data (Charnes et al. 1994, p. 8).

DEA does not require an a priori weight specification for different performance indicators. Further, DEA score represents the distance to the efficient frontier which helps to compare firms' overall performance both within and across industries (Chen and Delmas, 2011).

DEA does not require explicit specification of the functional forms relating inputs to outputs unlike traditional regression approaches. Further, in DEA more than one function (e.g., more than one production function) is admitted, and the DEA solution can be interpreted as providing a local approximation to whatever function is applicable in the neighborhood of the coordinate values formed, from the outputs and inputs of the DMUo being evaluated (Bowlin, 1998).

DEA is oriented toward individual decision making units which are regarded as responsible for utilizing inputs to produce the outputs of interest. It therefore, utilizes n optimizations, one for each DMU, in place of the single optimization that is usually associated with the regressions used in traditional efficiency analyses. Therefore, the DEA solution is unique for each DMU under evaluation.

The main deficiency of all of the regression approaches is their inability to identify sources and estimate the inefficiency amounts associated with these sources. Hence, no clue as to corrective action is provided even when the dependent variable shows that inefficiencies are present. DEA provides both the sources (input and output) and amounts of any inefficiency (Chaffai, 1997).

Data Envelopment Analysis (DEA) is preferred over Stochastic Frontier Analysis (SFA) for measuring efficiencies of entities due to model specification issues. SFA is considered appropriate for single output studies DEA is designed to handle multi outputs cases (Oh and Lööf, 2008).

## 6.2 What is Data Envelopment Analysis (DEA)?

Data Envelopment analysis [6] (DEA), has been widely used as an efficiency measurement tool in a variety of fields (Mostafa, 2007). Though, the DEA estimator was firstly introduced by Farrell in 1957 but it has become a popular tool for efficiency measurement after seminal work of Charnes, Cooper, and Rhodes in 1978. DEA is basically used for the purpose of evaluating the efficiency of entities (e.g., programs, service organizations etc.) which are responsible for utilizing resources to obtain outputs of interest (Bowlin, 1998). DEA is primarily a benchmarking technique, originally developed to evaluate nonprofit and public sector organizations, has also been proven to locate ways to improve manufacturing and service organizations, not visible with other techniques (Sufian et al., 2007).

The major advantage of DEA technique using is that it does not require a priori restrictive functional form of production function as well as cost function. However, this method, a nonparametric one, does not take into consideration the random effects in measuring efficiencies [7] . The DEA approach to measuring efficiency is analogous to measuring engineering efficiency, i.e. in terms of the ratio of output to input, of any business enterprise (Sherman and Zhu, 2005 and Chakrabarti and Chawla, 2005). The DEA approach involves the use of linear programming methods to construct a nonparametric piecewise frontier over the data, so as to be able to calculate efficiencies relative to this surface. In other words, the purpose of DEA is to construct a non-parametric envelopment frontier over the data points such that all observed points lie on or are below the production frontier (Sufian et al., 2007). It is basically a fractional programming model that can include multiple outputs and inputs without recourse to a priori weights (as in index number approaches) and without requiring explicit specification of functional relations between inputs and outputs (as in regression approaches). With the help of DEA technique one can compute a scalar measure of efficiency and determines efficient levels of inputs and outputs utilization by the subjects under evaluation [8] .

In simple terms, DEA Model is based on extension of simple usual measure of efficiency ratio of output and input i.e.:

deatt1

However, this simple efficiency measurement concept is only useful in case where there is only one input and one output for an organization. Practically, organizations exist with multiple inputs and outputs related to different resources, activities and environmental factors. To cater to this need, Farrell (1957) and Farrell and Fieldhouse (1962) had developed a concept of 'relative efficiency measurement' by which the problem of measuring efficiency from multiple inputs and outputs can be addressed. In this concept, a weighted average of efficient units, which act as a comparison for an inefficient unit, is calculated while focusing on the construction of a hypothetical efficient unit (Emrouznejad, 2007). In simple words the relative efficiency is measured as

Charnes et al. (1978) re-wrote this ratio by introducing the usual mathematical notation as follows:

deat3

Here, it is important to note that efficiency is usually constrained to the range [0, 1]) as this is based on simple assumption that productivity can never be more than 100% and less than 0% (Emrouznejad, 2007). Further, relative efficiency measure concept is based on the fundamental assumption that it requires a common set of weights to be applied across all decision making units (for whom efficiency is required to be calculated) (Emrouznejad, 2007).

The next cause of concern arises for the researchers was how such agreed common set of weights can be obtained when decision making units might value inputs and outputs differently and therefore adopt different weights (Bowlin, 1998). Charnes, Cooper and Rhodes (1978) recognized this difficulty and proposed that each unit should be allowed to adopt a set of weights which shows it in the most favorable light in comparison to the other units (Emrouznejad, 2007). In light of the solution efficiency of a target unit j0 can be obtained as a solution to the following simple linear programming problem:

Maximize the efficiency of Decision making Unit (DMU) j0 under considerations,

Subject to the efficiency of all DMUs being < =1.

The solution variables of the above problem are the weights and the solution produce the weights most favorable to unit j0 and a measure of efficiency. The initial algebraic form of this model was given by Charnes, Cooper, and Rhodes (CCR) [9] (1978) and further developed by Banker, Charnes, Cooper (1984) [10] have been illustrated and discussed below. For the purpose of this study both CCR and BCC models have been used. The details of CCR model has been discussed in following sub-sections followed by discussions on enhanced BCC model.

## 6.2.1 Constant Return to Scale Model of DEA {Charnes, Cooper, Rhodes (CCR)}

Charnes, Cooper, and Rhodes (CCR) (1978, 1979, and 1981) had given the following form of ratio for DEA:

This model was designed to evaluate the relative performance of some decision making unit (DMU), designated as DMUo, based on observed performance of j = 1, 2, â€¦â€¦â€¦n DMUs. A DMU is to be regarded as an entity/organization responsible for converting inputs into outputs (Bowlin, 1998; Emrouznejad, 2007). In this study, it is an individual sample company like Indian Oil Corporation Ltd etc.

The yr j; xi j > 0 in the model are constants which represent observed amounts of the rth output and the ith input of the jth decision making unit which we refer to as DMUj in a collection of j= 1, 2,â€¦â€¦.., n entities which utilize these i= 1, 2,â€¦â€¦â€¦, m inputs and produce these r=1, 2,â€¦â€¦â€¦.s outputs (Bowlin, 1998; Emrouznejad, 2007. One of the j =i,â€¦â€¦â€¦n DMUs is singled out for evaluation, accorded the designation DMUo, and placed in the functional to be maximized in equation (1) while also leaving it in the constraints. It then follows that DMUo's maximum efficiency score will be h*< 1 by virtue of the constraints, where Îµ>0 in equation (1) represents a non-Archimedean constant which is smaller than any positive valued real number (Bowlin, 1998; Emrouznejad, 2007).

The numerator in equation (1) represents a set of desired outputs and the denominator represents a collection of resources used to obtain these outputs. The result of this ratio will be a scalar value. The value h*o obtained from this ratio satisfies 0 < h*o <1 condition (Bowlin, 1998; Emrouznejad, 2007). The interpretation of this ratio is if an efficiency rating is equal to one i.e. h*o= 1 it represents full efficiency and incase ratio h*o< 1 means inefficiency is present in that particular DMU. The star (*) indicates an optimal value obtained from solving the model (Bowlin, 1998; Emrouznejad, 2007).

It is important to note here that no weights have been specified a priori in order to obtain the scalar measure of performance. The optimal values u*r, v*i may be interpreted as weights when solutions are available from equation (1). But, they are determined in the solution of the model and not a priori. To emphasize differences from more customary (a priori) weighting approaches, the u*r, v*i values secured by solving the above problem are called virtual multipliers and interpreted in DEA so that they yield a virtual output, Yo=Î£u*r yro (summed over r = 1â€¦â€¦â€¦. s), and a virtual input, Xo= Î£v*i xio (summed over i=1â€¦â€¦â€¦â€¦â€¦..m), which can allow us to compute the efficiency ratio ho= Yo/Xo (Bowlin, 1998; Emrouznejad, 2007).

It can be observed from equation (1), that h*o is the highest rating that the data allow for a DMU (Emrouznejad, 2007). No other choice of u*r and v*i can yield a higher h*o and satisfy the constraints. These constraints make this a relative evaluation with:

for some j as a condition of optimality (Bowlin, 1998; Emrouznejad, 2007).

Similarly efficiency evaluations has been obtained for each of the j= 1,â€¦â€¦.,n DMUs listed in the constraints of equation (1) by according them the same treatment, i.e., positioning them in the functional as DMUo, one by one, while also leaving them in the constraints (Bowlin, 1998; Emrouznejad, 2007).

These efficiency ratings are more than just index numbers which indicate a ranking of DMUs based on their efficiency. The value of h*o has operational significance in that 1-h*o provides an estimate of the inefficiency for each DMUo being evaluated. Carried further, as shown below, this characterization makes it possible to identify the sources and amounts of inefficiency in each input and output for every one of the DMUs being evaluated.

Further, it should be noted that the orientation of DEA is generally toward relative efficiency as determined by the above optimization applied to the data. For any DMUo being evaluated, the optimization implies that the evaluation will be effected by reference to the subset of j =1,â€¦â€¦â€¦..,n DMUs for which:

(2)

Here, the stars (*) indicate that these ur and vi values are optimal, and hence, make ho maximal for DMUo; the DMU being evaluated. In addition, kÎµK indicates the subset of DMUs that have attained the value of unity (an efficient DMU), which is the maximum value allowed by the constraints. It is to be noted that these kÎµK DMUs have attained the value of unity with the same u*r and v*i that are best for DMUo (Bowlin, 1998; Emrouznejad, 2007).

Using U*; V* to represent vectors with components u*i ; v*r which are optimal for DMUo in equation (1), we observe that such a choice will not give h*o = 1 unless DMUo is also in the set kÎµK. If h*o < 1, then DMUo will be characterized as inefficient relative to the set of DMUs in equation (2) which attain 100% efficiency with these same U*; V* values. In other words, DMUo is rated relative to an efficient subset of DMUs with; in general, different efficient subsets being utilized for the wanted efficiency ratings as different DMUs are brought into the functional of equation (1) for evaluation.

## 6.2.2 Linear Programming Form Model

The data in equation (1) can be computationally intractable if addressed directly. The theory of fractional programming, as first given in Charnes and Cooper (1962), makes it possible to replace equation (1) with an equivalent linear programming problem. The transformation [11] will result equation (1) in to following form (Charnes et al. 1978):

The first set of j = 1,â€¦â€¦â€¦..,n constraints in (3) come from the less-than-or-equal-to unity requirements in equation (1) while ur,vi > Îµ > 0, r, i, come from the non-Archimedean conditions in equation (1). Also, Î£vi xio = 1 guarantees that it is possible to move from equation (3) to equation (1), as well as from equation (1) to equation (3) (Bowlin, 1998). The theory of fractional programming [12] insures that

(4)

Here the stars indicate optimal values in equation (1) and equation (3), respectively.

Since equation (3) (Bowlin, 1998) is a linear programming problem, it has a dual which can be represented as:

and Î¸ is unrestricted in sign.

The name data envelopment analysis technique has been derived equation (5) only. As any admissible choice of Î»j provides an upper limit for the outputs and a lower limit for the inputs of DMUo and against these limits Î¸ is tightened with, Î»*j, si-*, sr+* > 0 representing optimizing choices associated with minimize Î¸ = Î¸*. The collection of such solutions then provides an upper bound which envelops all of the observations, and hence, leads to the name Data Envelopment Analysis (Bowlin, 1998). The companies on upper bound in this study will be termed as efficient companies and companies which will under the upper bound will be termed as inefficient companies.

Recalling that xio; yro are represented in the constraints as well as the functional in (1), it is clear that equation (5) always has at least the solution Î¸ =1, Î»o = 1 and all other Î»j, si-*, sr+*= 0 when DMUo is the DMU under evaluation. It follows that an optimum will be attained with 0<Î¸*< 1. Because equation (5) has a finite optimum, the duality theory of linear programming gives (Bowlin, 1998):

Note that Î¸*=1 does not imply h*o =1 unless also sr+*, si-* = 0 for all r and i. That is, all slack must also be zero in equation (6). Conversely, sr+*, si-*=0 for all r and i does not imply h*o = 1 unless Î¸*=1 also. In other words, it is necessary to have both Î¸* =1 and zero slack for efficiency or, conversely, h*o =1 implies Î¸* = 1 and all slack equal to zero in an optimum solution to equation (6) in order for DMUo to be characterized as fully (100%) efficient via DEA (Bowlin, 1998). Therefore, h*o = 1 if and only if DMUo is efficient (Bowlin, 1998).

## 6.2.3 Variable Return to Scale Model {Banker, Charnes, and Cooper (BCC)}

The scale of the operations of the sample companies in this study is varied. CCR model bases the evaluation on constant returns to scale i.e. it does not consider the scales of operation. Banker, Charnes, and Cooper (BCC) model (1984) has given a further extension of CCR model to treat of returns to scale. To recognize the impact of scale of operations, CEP and CSP performance results were also analyzed by using the BCC version. Following is the equivalent of equation (5) (Bowlin, 1998) for the BCC formulation.

The difference between the CCR model (equation (5)) and the BCC model (equation (7)) is that the Î»js are now restricted to summing to one. This has the effect of removing the constraint in the CCR model that DMUs must be scale efficient. Consequently, the BCC model allows variable returns to scale and measures only technical efficiency for each DMU. That is, for a DMU to be considered as CCR efficient, it must be both scale and technical efficient. For a DMU to be considered BCC efficient, it only need be technically efficient.

The separate evaluation of returns to scale in the BCC model is more evident in the dual to equation (7) (Bowlin, 1998) which we write as follows:

In this model, the u*o (the * indicates an optimal value determined via model (8)) indicates the return to scale possibilities. A u*o <0 implies local increasing returns to scale. If u*o = 0, this implies local constant returns to scale. Finally, a u*o > 0 implies local decreasing returns to scale. It is to be noted that the CCR model previously discussed simultaneously evaluates both technical and scale efficiency in the aggregate. The BCC model, however, separates the two types of inefficiencies in order to evaluate only technical inefficiencies (i.e. due to managerial inefficiency) in the envelopment model (equation 7) and scale inefficiencies (not able to get harness the benefits of size i.e. scales of operations) in the dual to equation (7) (equation (8). In other words, the technical efficiency calculated through BCC version separate the inefficient organizations for poor management of resources and improper utilization of scale of operations.

## 6.3 Specification of Input, Output and Analysis Model

Specification of input and output variables is a key consideration in using DEA (Bowlin, 1998). Choosing correct inputs and outputs is important for the effective interpretation, use, and acceptance of the results of the DEA Analysis (Mahadevan and Kim, 2000; Sherman and Rupert, 2006). DEA has mainly been employed by the researcher in an input/output framework, where inputs are entries into a production process (e.g. raw materials, labor, machinery, capital resources, energy etc.) and outputs usually are physical quantities of goods produced in the manufacturing industry and number of clients serviced in the service sector (Belu, 2009). In simple words, in DEA the resources are referred as "inputs" and the outcomes of a process is referred as "outputs", and a Decision Making Unit i.e. a production process transforms inputs into outputs (Thanassoulis, 2001). For the purpose of calculating CSP + CEP performance index selected eight companies are taken as the DMUs. Hence, in total there were 8 DMU under consideration. Manandhar and Tang (2002) suggested that in DEA, the definition of term efficiency should not be just confined to traditional sense of operating efficiency rather it should be generalized to represent relative evaluation of performance (Mostafa, 2009).

Bendheim et al. (1998) had taken CSP scores on community relations, employees' relations, environment performance, customers (product) relations, and 10-year total return to shareholders as output measures. For input measure they had taken a "dummy" input (i.e. assigning a score of 1 to all firms) to study an isoquant situation to evaluate firms exclusively with respect to their output shortfalls.

On the similar lines, Belu (2009) argued that for socially responsible companies 'production' process is basically the conversion of economic results into socially and environmentally sustainable achievements. Therefore, he considered the economic and financial results of a particular company as inputs and the sustainability achievements of the companies (given by the marks awarded by the specialized screening companies, for each dimension of interest) as output.

Both of the studies are considering CFP is input and CSP is output function in production process which is not supported by either literature (Section 3.1, Chapter 3) or practice as the companies do not set it as their main objective to maximize CSP. Rather companies want to maximize both CSP and CFP with their operations (Eells and Walton, 1974; Kaplan and Norton, 1992; Rowley and Berman, 2000; Taneja et al. 2011). The study has proposed to consider operational performance as input and social performance and financial performance as output variables (outcome of operational performance) (See Table 6.1).

## DEA Variable

Input

Operational Performance

Material Cost Ratio

Operating Expenses Ratio

Employee Cost Ratio

Other Operating Expense Ratio

Output

Social Performance

Customer Perception

Stakeholders' Perception

Shareholder Perception

Community Perception

Employees Perception

Financial Performance

Return on Investment