(last updated October 18, 1997)

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Partial Least Squares (PLS) can be a powerful method of analysis because of the minimal demands on measurement scales, sample size, and residual distributions. Although PLS can be used for theory confirmation, it can also be used to suggest where relationships might or might not exist and to suggest propositions for later testing.

Compared to the better known factor-based covariance fitting approach for latent structural modeling (exemplified by software such as LISREL, EQS, COSAN, and EZPATH), the component-based PLS avoids two serious problems: inadmissible solutions and factor indeterminacy (Fornell and Bookstein, 1982). The philosophical distinction between these approaches is whether to use structural equation modeling for theory testing and development or for predictive applications (Anderson and Gerbing, 1988). In situations where prior theory is strong and further testing and development is the goal, covariance based full-information estimation methods (i.e., Maximum Likelihood or Generalized Least Squares) are more appropriate. Yet, due to the indeterminacy of factor score estimations, there exists a loss of predictive accuracy. This, of course, is not of concern in theory testing where structural relationships (i.e., parameter estimation) among concepts is of prime concern.

For application and prediction, a PLS approach is often more suitable. Under this approach, it is assumed that all the measured variance is useful variance to be explained. Since the approach estimates the latent variables as exact linear combinations of the observed measures, it avoids the indeterminacy problem and provides an exact definition of component scores. Using the iterative estimation technique (Wold, 1981), the PLS approach provides a general model which encompasses, among other techniques, canonical correlation, redundancy analysis, multiple regression, multivariate analysis of variance, and principal components. Because the iterative algorithm generally consists of a series of ordinary least squares analyses, identification is not a problem for recursive models nor does it presume any distributional form for measured variables.

Sample size can be smaller, with a strong rule of thumb suggesting that it be equal to the larger of the following: (1) ten times the scale with the largest number of formative (i.e., causal) indicators (note that scales for constructs designated with reflective indicators can be ignored), or (2) ten times the largest number of structural paths directed at a particular construct in the structural model. A weak rule of thumb, similar to the heuristic for multiple regression (Tabachnik and Fidell,1989, p. 129), would be to use a multiplier of five instead of ten for the preceding formulae. An extreme example is given by Wold (1989) who analyzed 27 variables using two latent constructs with a data set consisting of ten cases.

Second order factors can be approximated using various procedures. One of the easiest to implement is the approach of repeated indicators known as the hierarchical component model suggested by Wold (cf. Lohmöller, 1989, pp. 130-133). In essence, a second order factor is directly measured by observed variables for all the first order factors. While this approach repeats the number of manifest variables used, the model can be estimated by the standard PLS algorithm. This procedure works best with equal numbers of indicators for each construct.

Finally, PLS is considered better suited for explaining complex relationships (Fornell, Lorange, and Roos, 1990; Fornell and Bookstein, 1982). As stated by Wold (1985, p. 589), "PLS comes to the fore in larger models, when the importance shifts from individual variables and parameters to packages of variables and aggregate parameters." Wold states later (p. 590), "In large, complex models with latent variables PLS is virtually without competition."

Nevertheless, being a limited information method, PLS parameter estimates are less than optimal regarding bias and consistency. The estimates will be asymptotically correct under the joint conditions of consistency (large sample size) and consistency at large (the number of indicators per latent variable becomes large). Furthermore, standard errors need to be estimated via resampling procedures such as jackknifing or bootstrapping (cf. Efron and Gong, 1983). Rather than being viewed as competitive models, the covariance fitting procedures (i.e., ML and GLS) and the variance-based PLS approach have been argued as complementary in nature. According to Jöreskog and Wold (1982, p 270):

"ML is theory-oriented, and emphasizes the transition from exploratory to confirmatory analysis. PLS is primarily intended for causal-predictive analysis in situations of high complexity but low theoretical information."

The significance of paths can also be determined by using the jackknife statistics resulting from a blindfolding resampling procedure (Lohmöller, 1984, pp. 5-09 through 5-12). The blindfolding procedure omits a part of the data matrix for the construct being examined and then estimates the model parameters. This is done a number of times based on the blindfold omission distance. Results obtained from this resampling procedure include the jackknifed estimated means and standard deviations. Also, it should be noted that by using the PLS algorithm under a reflective mode for all constructs, we eliminate any concerns of collinearity within blocks of variables used to represent underlying constructs.

REFERENCES

Anderson, J.C. and Gerbing, D.W. (1988). "Structural Equation Modeling
in Practice: A Review and Recommended Two-Step Approach," *Psychological
Bulletin*, 103(3), 411-423.

Efron, B. and Gong, G. (1983). "A Leisurely Look at the Bootstrap, the
Jackknife, and Cross-Validation" *The American Statistician*, 37(1),
36-48.

Fornell, C., and Bookstein, F. (1982). "Two Structural Equation Models:
LISREL and PLS Applied to Consumer Exit-Voice Theory," *Journal of Marketing
Research*, 19, 440-452.

Fornell, C., Lorange, P., and Roos, J. (1990). "The Cooperative Venture
Formation Process: A Latent Variable Structural Modeling Approach," *Management
Science*, 36(10), 1246-1255.

Jöreskog, K.G. and Wold, H. (1982). "The ML and PLS Techniques
For Modeling with Latent Variables: Historical and Comparative Aspects,"
in H. Wold and K. Jöreskog (Eds.), *Systems Under Indirect Observation:
Causality, Structure, Prediction (Vol. I)*, Amsterdam: North-Holland,
263-270.

Lohmöller, J.-B. (1984). *LVPLS Program Manual: Latent Variables
Path Analysis with Partial Least-Squares Estimation*, Köln: Zentralarchiv
für empirische Sozialforschung.

Tabachnick, B.G. and Fidell, L.S. (1989). *Using Multivariate Statistics*,
Second Edition, New York: Harper and Row.

Wold, H. (1981). "The Fix-Point Approach to Interdependent Systems:
Review and Current Outlook," in H. Wold (Ed.), *The Fix-Point Approach
to Interdependent Systems*, Amsterdam: North-Holland, 1-35.

Wold, H. (1985). "Partial Least Squares," in S. Kotz and N. L. Johnson
(Eds.), *Encyclopedia of Statistical Sciences (Vol. 6)*, New York:
Wiley, 581-591.

Wold, H. (1989). "Introduction to the Second Generation of Multivariate
Analysis," in H. Wold (Ed.), *Theoretical Empiricism*. New York: Paragon
House, vii-xl.

The DOS version of PLS-PC 1.8 developed by Lohmöller is distributed for free by Dr. Jack McArdle Click here to download.