The PLS approach to structural equation modelling

In this study, the structural equation modelling (SEM) method has been applied for the purpose of statistical analysis. SEM allows the simultaneous modelling of relationships among multiple independent and dependent constructs and differen-tiates between dependent and independent variables such as the endogenous and exogenous latent variables. The exogenous latent variables are not explained by the postulated model (acting as the independent variables) whereas the endoge-nous variables are explained by the relationships contained in the model (Haen-lein and Kaplan 2004). Thus, SEM is particularly suitable for measuring and es-timating a theoretical model with linear relations between variables, which may be either observable or directly unobservable and may only be measured imper-fectly. SEM enables an explicit modelling for the measurement error for the ob-servable variables and avoids potential bias, thus allowing constructing unobserv-able variunobserv-ables which can be measured by indicators.

Two approaches have been suggested in literature for estimating the parameters for SEM; covariance-based and the variance- or component-based approach. The covariance-based approach minimizes the difference between the sample covari-ances and those predicted by the theoretical model, whereas the variance-based approach minimizes the variance of the dependent variables explained by the in-dependent variables. As Chin (1998:295) mentions, ‘to many social science re-searchers, the covariance-based procedure is tautologically synonymous with the term SEM’. However, an alternative method for estimating parameters of SEM in the variance-based approach is partial least squares (PLS). The variance-based PLS approach has been applied in this study.

The trade-off between covariance- and variance-based SEM is guided by the re-search objectives, the epistemic view of data to theory and the properties of data at hand. This study utilizes PLS variance-based SEM due to the small sample size (the number of observations is slightly higher than the number of variables) and the epistemic view of data to theory. First, none of the independent and dependent variables are absolutely measureable. Further, rather than aiming for producing the covariance matrix as close as possible to the theoretical model, the aim has been set as analyzing the degree or level of speed and the success of export ex-pansion. In such a situation the variance-based PLS approach seems suitable.

In the PLS approach, the theoretical model is shown in the form of a path diagram which is a graphical representation of how the various elements of the model re-late to one another. Similar to a complete covariance-based structural model, a PLS model is also composed of two parts: the outer (measurement) and inner (structural) models. The outer model describes how each latent variable is meas-ured by corresponding observed indicator variables. This part of the model pro-vides information on the validities and reliabilities of the observed indicators. The inner model describes the relations between the latent variables (independent and dependent) themselves and indicates the degree of variance (Diamantopoulos and Siguaw 2000:4).

The structural and measurement models of the theoretical framework are shown in Figure 14. The third component of the PLS model however, is the weight rela-tions, which are used to estimate case values for the latent variables.

Figure 14. The structural and measurement models of the theoretical frame-work.

Contextual factors

Marketknowledge competence

Marketing planning and implementation

capability

Alliance management

capability

New product dev capability

Experiential Knowledge

Customer Knowledge

Alliance learning capability

Export expansio n Speed

Succes s

Outer measurement model Inner measurement model

Sample size requirements for PLS: The PLS technique uses simple or multiple regression depending on the mode of each block of indicators and the inner weighting schemes block for the sample size consideration. Only the part of the model that requires the largest multiple regression (arrow schemes linking to various other blocks) is important. Furthermore, the number of variables associ-ated with the latent variables in this part must be considered for sample size re-quirement. In another case, the block with the largest measurement equations or the dependent latent variable with the largest number of independent latent vari-ables impacting on it could also serve as a guide for the sample size selection (Chin 1998:311). Using the regression analysis-related rule of thumb, a minimum of 5 or a maximum of 10 cases per indicator should suffice. In the theoretical framework of the study, the outer measurement model contains the largest num-ber of measurement equations. The total numnum-ber of indicators for the outer meas-urement block of the framework is 50 in the case of this study. Thus, a sample size of 250 would suffice for 100 % of the sample size requirement. A total of 100 usable samples constitute 40 % of the required sample size of 250 in this study. However, the PLS method allows flexibility in achieving good results for a smaller sample size as low as 50 (Chin and Newsted 1999).

5.2.1 Description of PLS path modelling algorithm

The PLS modelling lacks a global goodness-of-fit measure. For that reason a rig-orous method of testing the reliability and the validity for the outer and inner models is considered important in this technique. Chin (1998) suggests a system-atic two-step process to assess partial mode structures. In the first step, the sessment of the outer models reveals the measurement reliability and validity as-sociated with their particular formative or reflective modes of the model. If the latent variable scores indicate sufficient reliability and validity, the assessment of the inner path models is followed as a second step.

The basic algorithm of PLS modelling calculates the estimated values for each latent variable in the data set by estimating the unobservable variables as exact linear combinations of their indicators. Similar to a principal component analysis, weights are assigned to each case value in a way that the result captures most of the variance of the independent latent variables, which may predict the dependent variable. Then, weights approximation is determined for the inner model by cal-culating a weighted average of indicators for each unobservable variable. After-wards, a simple ordinary linear squares regression is followed up. The process of inside-outside approximation is repeated until convergence of the case values is achieved. In sum, first a weight relation for the outer and inner models is

esti-mated, and then this is followed by an ordinary linear squares regression to de-termine the parameters of the structural relations (Haenlein and Kaplan 2004;

Fornell and Bookstein 1982).

The model evaluation for PLS involves prediction-oriented significance testing.

For estimating the stability of estimates, the individual item loadings assessment and the composite reliability measurement, jackknifing and bootstrapping meas-ures are employed. For model prediction purposes, the Stone-Geisser test for pre-dictive relevance and Fornell and Larcker’s average variance extracted (AVE) measures are used. Below a short description of each of these measures is given.

In order to assess the measurement models, the individual item indicator loading is considered first. To assess the internal consistency for a given block of indica-tors the composite reliability ρ^c is measured by utilizing the standardized regres-sion weights for indicators and latent variables. The acceptable value of the indi-vidual items loadings with their latent variable has been suggested as 0.5 at the early stages of scale development and as 0.7 or greater (Chin 1998:325) for de-veloped theory. The acceptable value of 0.5 suggests that a latent variable may explain on average half of the variance of its indicators.

To assess the convergent validity of the reflective block the average variance ex-tracted (AVE) is applied. The AVE is the amount of variance that is captured by the construct in relation to the amount of variance due to the measurement error.

Next, to assess the discriminant validity of the measures, the Fornell-Larcker cri-terion of the square root of AVE for each latent variable and the cross loading is applied. The AVE signifies that a latent variable shares more variance with its own indicators than with any other latent variable. Thus, the AVE of each latent variable should be greater than the highest squared correlation with any other la-tent variable. However, it has been acknowledged in literature that clear guide-lines as to how much larger the squared correlation should be are not provided.

Chin (1998) suggests taking the square root of the AVE for each latent variable to assess the discriminant validity of the reflective latent variables.

Another measure to assess the discriminant validity at the indicator level is by the cross loadings for the reflective mode. Here, the loadings of each indicator should be greater than all of its cross loadings in other latent variables. Thus, the correla-tions between latent variable component scores and other indicators besides its own block are calculated. If an indicator loads higher on another latent variable, it shows that the indicator shares more variance with that latent variable. In such situations, it is recommended that the indicator should be removed.

5.2.2 Assessment of the structural model

The inner structural model specifies the relationships among the exogenous and endogenous latent constructs. In the assessment of the inner model, the coefficient of determination R²of each latent endogenous variable is looked at first. R² of 0.67, 0.33 and 0.19 have been mentioned as substantial, moderate and weak re-spectively (Chin 1998). Moderate value applies when in an inner path model, a latent endogenous variable is explained by few (one or two) latent exogenous variables. On the other hand, when a latent endogenous variable relies on several latent exogenous variables, R²should indicate a substantial value. For further evaluation, the total effect or sum of all the direct and indirect effects of a particu-lar latent variable on another are recommended as the next step. Chin (1998:316) also recommends exploring the change in R² by calculating the effect size f ² to see the impact of a particular independent latent variable on a dependent latent variable. f²of .02, .15 and .35 represent small, medium and large effects respec-tively at the structural level.

Next, another criterion for the assessment of the predictive relevance of the struc-tural model, the Stone-Geisser criterion, Q²is applied. Depending on the form of prediction, different forms of Q² can be obtained. If the prediction of the data points is made by the underlying latent variable score, a cross-validated commu-nality is obtained. On the other hand, when the latent variables from the block in question are used for prediction, a cross-validated redundancy Q²is obtained.

Q2represents a measure of validating the predictive relevance of the model for parameter estimates. Q²greater than 0 indicates predictive relevance, whereas Q2less than 0 represents a lack of predictive relevance of the model for parameter estimates. Similar to f²the values of Q²are .02, .15 and .35 for small, medium and large effects.

To analyze the acceptance or rejection of hypotheses in partial least squares, bootstrapping is the standard recommended procedure to generate standard errors and t-values. In this study, the size for the number of samples for a bootstrapping run was set to 200 and the degrees of freedom as N-1. For a 90 % significance level, t-statistics values above 1.96 % indicated a significant threshold. To assess the acceptance or rejection of an individual hypothesis the probability of the t-values for each structural relationship are compared to the standard probability values. Further, the values and signs on the individual beta path coefficients of the inner, structural model are assessed for hypothesis significance of the theoreti-cally assumed relationships between latent variables. Insignificant path

coeffi-cients or an algebraic sign contrary to the a-priori hypotheses from the theoretical model are considered as rejected. Chin (1998) recommends that standardized path coefficients should be 0.2 in order to be significant.