# Partial Least Squares Essay

Partial least squares (PLS) is a structural equations modeling (SEM) technique and, therefore, allows simultaneous testing of the measurement and structural models. In PLS, constructs can be modeled as reflective or formative. Formative constructs refer to situations in which the observed variables cause the latent variable and each of the corresponding indicators represents different dimensions of the construct. They are similar to indexes. In contrast, reflective constructs represent situations in which the observed variables are caused by the latent variable and indicators are unidimensional and correlated with each other. They are similar to scales. The following three equations illustrate a formal specification of the PLS model, as given by Jan-Bernd Lohmöller (1989):

η = β0 + Bη + υ (structural relation)

y = π0 + ∏η + ε (measurement relation for reflective constructs)

η = ω0 + Ω’y + δ (measurement relation for formative constructs)

Where:

η = latent variables (LVs)

β0 = location parameters for LVs (structural relation) B = path coefficients

υ = inner residuals (structural)

y = manifest variables (MVs)

π0 = location parameters for MVs

ε = outer residuals (measurement for reflective constructs)

ω0 = location parameters for LVs (measurement)

Ω = weight coefficients

δ = validity residuals

PLS minimizes the differences between observed and calculated data values. It assesses how well the variance of the dependent variables is accounted for by their corresponding independent variables. In contrast to covariance-based structural equation modeling, PLS can be used for exploratory as well as confirmatory analysis. PLS can also work with small samples and is less sensitive to deviations from normality. In fact, some observed variables (or indicators) can be categorical. However, the model should be recursive (all arrows in one direction). PLS is also useful in situations where the researcher works with a complete population instead of a representative sample.

PLS does not have well-accepted overall model fit measures. Instead, the model should be evaluated by examining the size and significance of (1) loadings from reflective constructs to their indicators, (2) weights to formative constructs from their indicators, (3) standardized regression coefficients between constructs, and (4) coefficients of multiple determination (R-squares) for endogenous constructs (dependent variables).

Finally, there are several conditions in which PLS is preferred over covariance-based SEM. Some examples are: (1) when a strong theory about the phenomenon does not exist, (2) when the model will include some categorical manifest variables, (3) when there is some degree of unreliability in manifest variables, and (4) when data come from nonnormal or unknown distributions.

Bibliography:

1. Chin,W.W., and P. R. Newsted. “Structural Equation Modeling Analysis with Small Samples Using Partial Least Squares.” In Statistical Strategies for Small Sample Research, edited by Rick H. Hoyle, 307–341.Thousand Oaks, Calif.: Sage Publications, 1999.
2. Falk, R. Frank, and Nancy B. Miller. A Primer for Soft Modeling. Akron, Ohio: University of Akron Press, 1992.
3. Gil-Garcia, J. Ramon. “Using Partial Least Squares in Digital Government Research.” In Handbook of Research on Public Information Technology, edited by G. David Garson and Mehdi Khosrow-Pour, 239–253. Hershey, Pa: IGI Global, 2008.
4. Lohmöller, Jan-Bernd. Latent Variable Path Modeling with Partial Least Squares. Heidelberg, Ger.: Physica-Verlag, 1989.

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