AN INTRODUCTION TO PARTIAL LEAST SQUARES PATH …

1 STA201 Analyse multivari e approfondie AN INTRODUCTION TO. PARTIAL LEAST SQUARES . PATH modeling . Giorgio Russolillo CNAM, Paris An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling Component-Based vs Factor-Based Sructural Equation Models An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 2. Covariance Structure Analysis and K. J reskog Karl J reskog is Professor at Uppsala University, Sweden In the late 50s, he started working with Herman Wold. He discussed a thesis on Factor Analysis. In the second half of the 60s, he started collaborating with Duncan and A. Goldberger. This collaboration represents a meeting between Factor Analysis (and the concept of latent variable) and Path Analysis ( the idea behind causal models).

2 In 1970, at a conference organized by Duncan and Goldberger, J reskog presented the Covariance Structure Analysis (CSA) for estimating a linear structural equation system, later known as LISREL. An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 3. Soft modeling and H. Wold Herman Wold (December 25, 1908 February 16, 1992). Econometrician and Statistician In 1975, H. Wold extended the basic principles of an iterative algorithm aimed to the estimation of the PCs (NIPALS) to a more general procedure for the estimation of relations among several blocks of variables linked by a network of relations specified by a path diagram.

3 The PLS Path modeling avoids restrictive hypothesis, multivariate normality and large samples, underlying maximum likelihood techniques. It was proposed to estimate Structural Equation Models (SEM) parameters, as a Soft modeling alternative to J reskog's Covariance Structure Analysis An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 4. Two families of methods The aim is to reproduce the sample covariance matrix Covariance-based of the manifest variables by means of the model Factor-based parameters: Methods the implied covariance matrix of the manifest variables is a function of the model parameters it is a confirmatory approach aiming at validating a model (theory building).

4 SEM The aim is to provide latent variable scores (proxy, composites, factor scores) that are the most correlated to each other as possible (according to path diagram structure) and the most representative Variance-based of their own block of manifest variables. Composite-based it focuses on latent variable scores computation Component-based it focuses on explaining variances Methods it is more an exploratory approach than a confirmatory one (operational model strategy). An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 5. Structural Equation Models: two approaches In component-based SEM the latent variables are defined as components or weighted sums of the manifest variables they are fixed variables (linear composites, scores).

5 In factor- based SEMs the latent variables are equivalent to common factors they are theoretical (and random) variables This leads to different parameters to estimate for latent variables, : factor means and variances in covariance-based methods weights and scores in component based approaches An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 6. Reflective and Formative indicators Latent Emergent Construct Construct x1j x2j x3j x4j x1j x2j x3j x4j Reflective (or Effects) Indicators Formative (or Causal) Indicators Consumer's attitudes, feelings Social Status, Perceptions Constructs give rise to observed variables Constructs are combinations of (unique cause unidimensional) observed Aim at accounting for observed variables(multidimensional).

6 Variances or covariances Not designed to account for observed These indicators should covary: changes variables in one indicator imply changes in the These indicators need not covary: others. changes in one indicator do not imply Internal consistency is measured changes in the others. (es. Cronbach's alpha) Measures of internal consistency do not apply. An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 7. PLS Path modeling : inner, outer and global model An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 8. Drawing conventions or Latent Variables (LV).

7 X or x Manifest Variables (VM). or Unidirectional Path (cause-effect). or Bidirectional Path (correlation). Feedback relation or reciprocal causation or or Errors An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 9. Notations P manifest variables (MVs )observed on n units xpq generic MV. Q latent variables (LVs). q generic LV. Q blocks composed by each LV and the corresponding MVs Q. in each q-th block pq manifest variables xpq , with p q =P. q=1. Greek characters are used to refer to Latent Variables . Latin characters refer to Manifest Variables An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G.

8 Russolillo slide 10. A Path model with latent variables Path 11 x11 Coefficients x13 13. 21 x21 1 1. x23 23. 31 x31. 3 x33 33. 12 x12 x43 43. 12 2 External x53 53 Weights 2. 22 x22 22 Inner or Structural model Outer or Measurement model An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 11. PLS Path Model Equations: inner model The structural model describes the 3. relations among the latent variables 1 13. 3. 23. 2. For each endogenous LV in the model it can be written as: J. q* = jq* j + q*. j=1. where: - jq* is the path-coefficient linking the j-th LV to the q*-th endogenous LV.

9 - J is the number of the explanatory LVs impacting on q*. An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 12. PLS Path Model Equations: outer model The measurement model describes the relations among the manifest variables Latent Construct and the corresponding latent variable. q 1q 2q 3q 4q x1q x2q x3q x4q For each MV in the model it can be written as: x pq = pq q + pq where: - lpq is a loading term linking the q-th LV to the p-th MV. An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 13. Weight relation Linear composite In component-based approach a weight relation defines each latent variable score as a weighted aggregate of its own MVs: q =Xqwq An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G.

10 Russolillo slide 14. PLS-PM Algorithm An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G. Russolillo slide 15. PLS-PM approach in 4 steps 1) Computation of the outer weights Outer weights wq are obtained by means of an iterative algorithm based on alternating LV. estimations in the structural and in the measurement models 2) Computation of the LV scores (composites). Latent variable scores are obtained as weighted aggregates of their own MVs: q X q w q 3) Estimation of the path coefficients Path coefficients are estimated as regression coefficients according to the structural model 4) Estimation of the loadings Loadings are estimated as regression coefficients according to the measurement model An INTRODUCTION to PARTIAL LEAST SQUARES Path modeling G.