HIGH-ORDER CONSTRUCTS FOR THE STRUCTURAL EQUATION MODEL

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1 HIGH-ORDER CONSTRUCTS FOR THE STRUCTURAL EQUATION MODEL Enrico Ciavolino (1) * Mariangela Nitti (2) (1) Dipartimento di Filosofia e Scienze Sociali, Università del Salento (2) Dipartimento di Scienze Pedagogiche, Psicologiche e Didattiche, Università del Salento Abstract: The aim of the paper is to present a structural equation model based on high order latent variables. The non parametric estimation method used is the partial least squares which enables the definition of complex structure in the data with few model assumptions. The paper contribution is the analysis of the latent dimensions with a third level of abstraction by considering two main approaches presented in literature: the repeated indicators and the two-step approach. Empirical evidences and simulations results are provided in order to show the methodology and check the reliability of the approaches at issue. Keywords: structural equation model, partial least squares, high-order construct, latent dimension of sense. 1. Introduction PLS enables researchers in many field of social sciences to investigate models at high level of abstraction. The dimensions of a higher-order construct could be conceptualized under an overall abstraction, and it is theoretically meaningful to use this abstraction for the representation of the dimensions, instead of merely interrelating them. In this paper, a third-order latent construct model estimated by PLS-SEM is presented. Two modelbuilding approaches (named repeated indicators and two-step approach) are compared through a simulation study for determining which method better represents the relationships among constructs levels. The work is articulated as follows: in section 2 PLS estimator and its extension to higherorder construct modeling are described; section 3 presents a case study in which Latent Dimensions of Sense (LDS) are modeled as a third-order latent variable;in section 4, the description of the montecarlo simulation is reported for the third-order model, in way to compare the two modelbuilding approaches and draw conclusion on their performances. 2. PLS-path modeling for higher-order constructs Due to its ability of estimating complex models, PLS Path Modeling can be used to investigate models with a high level of abstraction. As a matter of fact, many studies in social sciences involve the conceptualization of multi-dimensional constructs, that is, composed of different but strictly related conceptual dimensions. The basic PLS design was completed for the first time in 1966 by Herman Wold for the use in multivariate analysis, and subsequently extended for its application in the Structural Equation Modeling (SEM) in 1975 by Wold himself. An extensive review on PLS approach is given the Handbook of PLS (Esposito Vinzi, Chin, Henseler, Wang, 2010). The model-building procedure can be thought of as the analysis of two conceptually different models. A measurement (or outer) model specifies the relationship of the observed variables with their (hypothesized) underlying (latent) constructs; a structural (or inner) model then specifies the causal relationships among latent constructs, as posited by some theory. The two sub models' equations are the following: * Corresponding Author: enrico.ciavolino@unisalento.it ξ (m,1) =B (m,m) ξ (m,1) +τ (m,1)

2 x (p,1) =Λ (p,m) ξ (m,1) +δ (p,1) where the subscripts m and p are the number of, respectively, the latent (LV) and the manifest variables (MV) in the model, while the letters ξ, x, B, Λ, τ and δ indicates LV and MV vectors, the path coefficients linking the LV, the factor loading linking the MV to the LV, and the errors terms of the model. 2.1 The PLS algorithm The parameters estimation (Ciavolino et al., 2009) is based on a double approximation of the LVs ξ j (with j=1,...,m). The external estimation y j, obtained as the product of the block of MVs X j (considered as the matrix units for variables) and the outer weights w j (which represent the estimation of measurement coefficients, Λ). The internal estimation z j, obtained as the product of the external estimation of ξ j, y j, and the inner weights e j. According with the relationship among MVs and LVs hypothesized, outer weights are computed as: for Mode A (reflective relationship), and: w j = X j ' z j w j = (X j ' X j ) -1 X j ' z j for Mode B (formative relationship). The inner weights e j,i, in the centroid scheme, are defined as the sign of the correlation between the connected estimated y j and y i, with i j. The PLS algorithm starts by initializing outer weights to one for the first MV of each LV; then, the parameters estimation is performed, until the convergence, by iteratively computing: 1) external estimation, y j = X j z j ; 2) internal estimation, z j = Σ j i e j,i y i ; 3) outer weights estimation, with Mode A or B. The causal paths among LVs (the coefficients in the B matrix) are obtained through Ordinary Least Squares (OLS) method. 2.2 Modelling higher-order constructs Wold's original design of PLS path modeling does not consider higher-order LVs; each construct has to be necessarily related to a set of observed variables in order to be estimated. On this basis, Lohmöller (1989) proposed a procedure for the case of hierarchical constructs, the so-called Hierarchical Component Model or Repeated Indicators Approach, which is the most popular approach when estimating higher-order constructs through PLS. The procedure is very simple: a second-order factor is directly measured by observed variables for all the first-order factors. While this approach repeats the number of MVs used, the model can be estimated by the standard PLS algorithm (Reinartz et al., 2004). The manifests indicators are repeated in order represent the higher-order construct. A prerequisite for the repeated indicators approach is that all indicators of the first-order and the second-order factors should be reflective. This approach is the most favored by researchers when using PLS for modeling higher-order constructs. A disadvantage of this approach is a possibly biasing of the estimates by relating variables of the same type together through PLS estimation. Another way of building a higher-order model is the Two-Step Approach: the LV scores are initially estimated in a model without second-order constructs (Agarwal et al., 2000). The LV scores are subsequently used as indicators in a separate higher-order structural model analysis. It may offer advantages when estimating higher-order models with formative indicators (Diamantopoulos et al., 2001; Reinartz et al., 2004). The implementation is not one simultaneous PLS run. A clear disadvantage of such two-stage approach is that any construct which is investigated in stage two is not taken into account when estimating LV scores at stage one. The two approaches are illustrated in figure 1.

3 Figure 1. Model building: Repeated Indicators and Two-Step approaches In a recent study, Wilson et al. (2007) showed that the second-order constructs reliability does not depend on the approach adopted; anyway, the repeated indicators approach produce biased and less consistent estimates (in the case of small samples) with respect to the two-step approach. 3. Empirical evidence The study here presented concern the LDS underlying students' image of three different areas: Social Context (social behaviors, future perspectives, image on adults and young s, etc.), University (motivation, teachers and teaching, university functions, problems and development strategies, etc.), Psychology and Psychologist (field of interventions, functions, potential users, etc.). LDS is here conceived as a third-order latent construct affecting second-order dimensions (Social Context, University and Psychology), which in turn shape first-order latent variables underlying specific aspects of the second-order dimensions. A sample of 136 students of the first year of Psychology course at University of Salento were invited to fill in a 120 item multiple choice questionnaire, divided into 19 sections (first-order LVs). The questionnaire was designed to facilitate the expression of perceptions, opinion, judgment, about the three general area of experience, by referring to the repertoire of cultural meanings typical of the ISO method (Carli, Paniccia, 2002; Carli, Salvatore, 2001). In order to model the LDS, here called Image, the two approaches to higher-order constructs' modeling have been followed: 1) Repeated indicators approach: all of the MVs of the first level are repeated for the secondorder LVs (University, Social and Psychology) they belong to. The third-order LV Image is then linked with all the MVs used for measuring the second-order constructs. Here, the three levels are connected by means of path coefficients (β); 2) Two-step approach: the 19 first-order LVs are estimated and then used as MVs for measuring the second-order constructs which become MVs for determining the third-order LV Image. The link between the three levels is represented by weights (Λ). The two approaches, performed by a tailor-made algorithm developed with MATLAB 7.1, lead to the following results for weights and path coefficients linking the three levels: I Table 2. Parameter estimation with the two approaches 0,442 0,343-0,460 0,478-0,762-0,717 0,236-0,272 0,278 0,598 0,428 0,487-0,426 0,377 0,169 0,214 0,170 0,293 0,259 0,477 0,594 0,581-0,846 0,359-0,430 0,557 0,773 0,836 0,697 0,466 0,501 0,492-0,155-0,198 0,408 0,541 0,535 0,504 0,798 0,801 0,442 0,701 0,463 0,542 The parameters have been further investigated by using bootstrap procedure. The resulting means for the parameters of the two approaches are summarized in Table 3. The parameters are divided

4 according to the approach used, the construct level and its composition (e.g., Level II University contains the links between second-order LV University and its 9 first-order LVs). Table 3. Bootstrap results REPEATED INDICATORS: Beta I University Image Sample 0,53 0,44-0,36 0,44-0,12 0 0,25 0,01 0,17 0,70 0,49 0,44 mean Social Psychology 0,63 0,62 0,59-0,01 0,05-0,29 0,63 0,65 0,69 0,6 TWO-STEP: Weights I University Image Sample 0,36 0,34-0,27 0,29 0,08 0,1 0,12 0,16 0,13 0,67 0,25 0,37 mean Social Psychology 0,44 0,43 0,47 0,45-0,1-0,18 0,39 0,34 0,55 0,47 The differences in the strategy adopted for the higher-order LVs definition, as the type of link between levels (betas or weights) or the number of MVs per LV, are reflected in differences in the parameters' estimates. The main dissimilarities concern the second-order LV Social Context, in both the relations with first and third level. For instance, the link between Social Context with the 4 th LV of the first level is -0,01, for the repeated indicators approach, and 0,45 for the two step approach; furthermore, the relationship with the third-order LV Image is equal to, respectively, 0,49 and 0,25. In order to explore the conditions affecting the two approaches results and check their reliability, a Monte Carlo simulation has been defined. 4. Discussion This section describes the definition of a Monte Carlo experiment aimed at catching the differences between the two strategies of building a third-order LV model. Following the procedure used by Hulland et al., (2010) the data are generated, for each simulation design, as described below: 1. generation of the exogenous LVs with the specified sample size; 2. standardization of the generated LVs; 3. computation of the endogenous LVs as the weighted average (weights are the fixed path coefficient linking exogenous with endogenous LVs) of the exogenous standardized LVs (the model) plus an error component e drawn from a univariate normal distribution. Given a R 2 value, the error term standard deviation is chosen to satisfy the equation: R 2 = var (model)/[var(model) + var(e)] 4. standardization of the endogenous LVs; 5. computation of the observed variables from the standardized constructs, by fixing the value of the coefficients; 6. standardization of observed variables befor estimation. Factors varying across the simulation designs are sample size (n=50, 200, 500), number of replications (k=100, 500) and distribution probability of errors (symmetric and non-symmetric).

5 References Agarwal, R., Karahanna, E., (2000). Time Flies When You're Having Fun: Cognitive Absorption and Beliefs About Information Technology Usage. MIS Quarterly, 24 (4), Carli, R., Salvatore, S. (2001). L immagine della psicologia: Una ricerca sulla popolazione del Lazio [The image of psychology: Research on Latial population]. Roma: Kappa. Carli, R., Paniccia, R.M. (2002). L Analisi Emozionale del Testo: Uno strumento psicologico per leggere testi e discorsi [The Emotional analysis of the text: an instrument for reading texts and discourse]. Milano: FrancoAngeli. Ciavolino, E., Al-Nasser, A.D., (2009). Comparing Generalized Maximum Entropy and Partial Least Squares methods for Structural Equation Models, Journal of Nonparametric Statistics, 21 (8), Diamantopoulos, A., Winklhofer, H.M., (2001). Index construction with formative indicators: An alternative to scale development. Journal of Marketing Research, 38 (2), Esposito Vinzi, V., Chin, W.W., Henseler, J., Wang, H., (2010). Handbook of Partial Least Squares. Concepts, Methods and Applications. Springer Handbooks of Computational Statistics. New York. Hulland J., Ryan M.J., Rayner R.K., (2010). Modeling Customer Satisfaction: A Comparative Performance Evaluation of Covariance Structure Analysis versus Partial Least Squares, in Handbook of Partial Least Squares. Concepts, Methods and Applications. Springer Handbooks of Computational Statistics. New York. Lohmöller, J.B., (1989). Latent Variable Path Modeling with Partial Least Squares. Heidelberg: Physica-Verlag. Reinartz, W., Krafft, M., Hoyer, W.D., (2003). Measuring the Customer Relationship Management Construct and Linking it to Performance Outcomes. INSEAD Working Paper Series. 2003/02/MKT, Reinartz, W., Krafft, M., Hoyer, W.D., (2004). The Customer Relationship Management Process: Its Measurement and Impact on Performance. Journal of Marketing Research, 41(3), Wilson, B., Henseler, J. (2007). Modeling Reflective Higher-Order Constructs using Three Approaches with PLS Path Modeling: A Monte Carlo Comparison. In Thyne, M., Deans, K.R., Gnoth, J (eds.). Australian and New Zealand Marketing Academy Conference. Department of Marketing, School of Business, University of Otago. 3rd 5th December; Wold, H. (1966). Estimation of principal components and related models by iterative least squares. In Multivariate Analysis, P.R. Krishnajah, ed., Academic Press, NewYork, 1966, pp Wold, H., (1975). Path models with latent variables: The NIPALS approach, in Quantitative Sociology, H.M. Blalock, ed., Seminar Press, NewYork, 1975, pp

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