A Comparison of Approaches for the Analysis of Interaction Effects Between Latent Variables Using Partial Least Squares Path Modeling

Abstract

In social and business sciences, the importance of the analysis of interaction effects between manifest as well as latent variables steadily increases. Researchers using partial least squares (PLS) to analyze interaction effects between latent variables need an overview of the available approaches as well as their suitability. This article presents 4 PLS-based approaches: a product indicator approach (Chin, Marcolin, & Newsted, 2003), a 2-stage approach (Chin et al., 2003; Henseler & Fassott, in press), a hybrid approach (Wold, 1982), and an orthogonalizing approach (Little, Bovaird, & Widaman, 2006), and contrasts them using data related to a technology acceptance model. By means of a more extensive Monte Carlo experiment, the different approaches are compared in terms of their point estimate accuracy, their statistical power, and their prediction accuracy. Based on the results of the experiment, the use of the orthogonalizing approach is recommendable under most circumstances. Only if the orthogonalizing approach does not find a significant interaction effect, the 2-stage approach should be additionally used for significance test, because it has a higher statistical power. For prediction accuracy, the orthogonalizing and the product indicator approach provide a significantly and substantially more accurate prediction than the other two approaches. Among these two, the orthogonalizing approach should be used in case of small sample size and few indicators per construct. If the sample size or the number of indicators per construct is medium to large, the product indicator approach should be used.

Notes

¹This phenomenon can best be observed at an interaction model with one indicator per latent variable. Usually, the path coefficient β₃ as estimated by the product indicator approach will not equal the regression parameter β₃ of the product term in a multiple regression between the indicators unless indicators are standardized.

²More precisely, the outer estimate.

³We considered the following software distributions (author in parenthesis): LVPLS (Lohmöller, 1984) including later graphical extensions, PLS-Graph (Chin, 1993–2003), SmartPLS (Ringle et al., 2005), and SPAD-PLS (DECISIA, 2003).

⁴We thank an anonymous reviewer for this observation.

⁵SmartPLS, as Java-based software, principally allows for plug-ins. However, this functionality had not yet been provided by the programmers when this article was written.

⁶“The data were obtained from a single organization that had recently installed electronic mail. A total of 60 questions relating to a recent introduction of electronic mail were presented. Of the 575 questionnaires distributed, 250 usable responses were analyzed representing 43.5 percent of those surveyed. On average, the respondents had been using electronic mail for 9 months, sent 2.53 messages per day (s.d. = 2.36) and received 4.79 messages per day (s.d. = 3.49). Respondents were on average 39 years old (s.d. = 9.28) and had worked for the company an average of 11 years (s.d. = 6.9). Sixty percent of the respondents were male. The respondents came from various levels in the organization, 13 percent were managers, 12 percent were engineers, 38 percent were technicians, and the remaining 37 percent were clerical workers” (Chin et al., 2003, online appendix, p. 9).

⁷Chin, Marcolin, and Newsted (2003) also investigated the case of 20 observations. However, taking into account that in the orthogonalizing approach, the indicators of the interaction term are regressed on all indicators of the exogenous variables, several conditions would have led to singularities. For instance, having 10 indicators per construct, a regression with 20 independent variables would have to be estimated by means of 20 observations.

⁸We thank an anonymous reviewer for this hint. It must be stated, though, that until now, existing implementations of the bootstrap in PLS software do not allow inclusion of this recalculation.

⁹Although we encountered significant sphericity, with Bartlett's test indicating a χ²(3) of 1135.473, we decided against a multivariate analysis of variance (MANOVA), because MANOVA does not allow for random factors, and—albeit significant—correlations between the relative biases of the three path coefficients were well below 0.1, thus not substantial.

¹⁰If prediction is the only purpose of the moderated path analysis, any adjustment of the interaction term's standard deviation is arbitrary, and can thus be ignored.

¹¹Note that Chin et al. (2003, p. 211) mistakenly labeled R ² _excl instead of R ² _incl in the denominator of this formula, thereby provoking an underestimation of f ².