Nonrecursive vs. recursive models in WarpPLS
Ned Kock
As we proceed with our discussions here, I think it is useful to clarify the difference between nonrecursive relationships, denoted by reciprocal arrows, and those denoted by bidirectional arrows.
Nonrecursive models are those in which some direct relationships among latent variables are denoted by two arrows, one arrow going in one direction and the other in the opposite direction.
For example, a model showing the following direct relationships would be nonrecursive: A > B, A > C, B > C, C > B, B > D and C > D.
In the model above, the nonrecursive relationship that characterizes the model as nonrecursive is indicated by the two direct links: B > C and C > B.
By the way, the fact that such models are called “nonrecursive” illustrates the confusing jargon often used in multivariate statistics, which I sometimes refer to as “statospeech”. I’d argue that, based on the notion of recursion in mathematics, one would tend to call models with reciprocal arrows as “recursive”. Instead, they are called “nonrecursive”.
Nonrecursive relationships are different from those indicated by bidirectional arrows in SEM. The latter simply indicate the expectation of a correlation. The former actually implies that some serious extra calculations are expected, to estimate the path coefficients in both directions.
Estimation of nonrecursive relationships with WarpPLS (and, to the best of my knowledge, PLS-based SEM software in general) is not done automatically. Users can employ a procedure frequently called “two-stage least squares” (2SLS), which is easy to implement with WarpPLS. The link below is for one of the many resources online discussing this procedure.
https://www3.nd.edu/~rwilliam/stats2/l93.pdf
If you need a reference, Kline’s (1998) widely-cited SEM book discusses this two-stage procedure, and even recommends it in the context of covariance-based SEM. Please see the WarpPLS User Manual for the reference (http://warppls.com).
The procedure is relatively simple, essentially requiring users to build two models. In PLS-based SEM, it requires the use of PLS regression (implemented by WarpPLS). If employed with PLS modes A or B (used by most publicly available PLS-based SEM software) it will lead to unreliable results, because of the influence of the inner model structure on the estimation of latent variable scores.
More advanced users may correctly point out that the 2SLS will lead to trivial results (i.e., the same path coefficients in both directions) in very simple nonrecursive models, such as one with only two latent variables and the following links: A > B and B > A.
This is correct only for linear analyses. A nonlinear 2SLS analysis of a simple nonrecursive model with two latent variables typically will lead to nontrivial results. That is, it will typically lead to a different path coefficient for each direction.
And yes, in case you are wondering, this property of nonlinear analyses can be used for causality assessment, which is likely to be a very fertile area for future research.
If you are generally interested in the intricacies of causality assessment using various techniques, including techniques based on Simpson’s paradox (which WarpPLS is sensitive to; see User Manual and the WarpPLS blog), the following is required reading.
Pearl, J. (2009). Causality: Models, reasoning, and inference. Cambridge, England: Cambridge University Press.
Future versions of WarpPLS, and its User Manual, will have more on causality assessment. There will even be several causality assessment coefficients reported. Stay tuned!
Ned