Full latent growth

Sometimes the actual inclusion of moderating variables and corresponding links in a model leads to problems; e.g., increases in collinearity levels, and the emergence of instances of Simpson’s paradox. The menu option “Explore full latent growth”, available in WarpPLS starting in version 6.0, allows you to completely avoid these problems, and estimate the effects of a latent variable or indicator on all of the links in a model (all at once), without actually including the variable in the model. Moreover, growth in coefficients associated with links among different latent variables and between a latent variable and its indicators, can be estimated; allowing for measurement invariance tests applied to loadings and/or weights.

Related YouTube video:

Explore Full Latent Growth in WarpPLS

http://youtu.be/x_2e8DVyRhE

Simpson’s paradox, moderation, and the emergence of quadratic relationships in path models

Among the many innovative features of WarpPLS are those that deal with identification of Simpson’s paradox and modeling of nonlinear relationships. A new article discussing various issues that are important for the understanding of the usefulness of these features is now available. Its reference, abstract, and link to full text are available below.

Kock, N., & Gaskins, L. (2016). Simpson’s paradox, moderation, and the emergence of quadratic relationships in path models: An information systems illustration. International Journal of Applied Nonlinear Science, 2(3), 200-234.

While Simpson’s paradox is well-known to statisticians, it seems to have been largely neglected in many applied fields of research, including the field of information systems. This is problematic because of the strange nature of the phenomenon, the wrong conclusions and decisions to which it may lead, and its likely frequency. We discuss Simpson’s paradox and interpret it from the perspective of path models with or without latent variables. We define it mathematically and argue that it arises from incorrect model specification. We also show how models can be correctly specified so that they are free from Simpson’s paradox. In the process of doing so, we show that Simpson’s paradox may be a marker of two types of co-existing relationships that have been attracting increasing interest from information systems researchers, namely moderation and quadratic relationships.

Among other things this article shows that: (a) Simpson’s paradox may be caused by model misspecification, and thus can in some cases be fixed by proper model specification; (b) a type of model misspecification that may cause Simpson’s paradox involves missing a moderation relationship that exists at the population level; (c) Simpson’s paradox may actually be a marker of nonlinear relationships of the quadratic type, which are induced by moderation; and (d) there is a duality involving moderation and quadratic relationships, which requires separate and targeted analyses for their proper understanding.

Enjoy!

Simpson’s paradox and unexpected results

The algorithms used in version 3.0 and later versions of WarpPLS have been revised so as to pick up instances of what is known as “Simpson’s paradox”. As a result, there may be changes in some coefficients and P values, when compared with previous versions.

Often the P value of the ARS fit index will go up, if instances of Simpson’s paradox are present in the model.

Simpson’s paradox is characterized by the path coefficient and correlation for a pair of variables having different signs. In this situation, the contribution of a predictor variable to the explained variance of the criterion variable in a latent variable block is negative.

In other words, if the predictor latent variable were to be removed from the block, the R-squared for the criterion latent variable would go up. A similar effect would be observed if the direction of the causality was reversed.

One widely held interpretation is that Simpson’s paradox could be an indication that the direction of a hypothesized relationship is reversed, or that the hypothesized relationship is nonsensical/improbable.

In the context of WarpPLS analyses, this is more likely to occur when nonlinear algorithms are used and/or full collineary VIFs are high, but may also occur under other conditions.