Hello All,
I am doing a study that includes two constructs, using PLS-SEM. Based on previous literature, those two constructs can be included in the same category. Considering the discriminant validity, I found that HTMT between them is lower than 0.25. (And as a contrast, HTMT between different dimensions of another second-order variable is about 0.9.)
Should I integrate them in model as a second-order variable with two dimensions? or as two different variables?
Hello Ziwei,
I cannot say something about estimation issues within PLS but you stepped in a trap of thinking that a factor (may it be a primary factor or secondary factor) represents some aggregate of the indicators (latent or manifest) --or "categories" as you label it.
A factor is a common cause of two or several other variables. These variables can be observed (when the factor is a primary latent variable) or latent (when the factor is a second-order factor). This factor is a unidimensional, singular, specific-as-possible dimension, not an aggregate.
As in any other case of causal modelling, the common factor model is ONE of several possible causal structures, with the other option being, for instance, that both primary variables causally influence each other.
Hence, to make a factor a valid scientific term, the correlations between both (or the set of primaries) should be as high as possible to make the common-cause perspective a valid options. This correlation will transfer to the estimated factor loading which represents--assumed that the factor represents something real and has the presumed causal influence--the causal effects of the factor on the primaries.
I cannot all much about any HTMT value. But as in any factor model, the question of whether the factor represents something real is vital. Especially in a second-order factor model, it has been criticized that a second-order factor may be pure noise
Lee, N., & Cadogan, J. W. (2013). Problems with formative and higher-order reflective variables. Journal of Business Research, 66(2), 242-247. doi:10.1016/j.jbusres.2012.08.004
If the (second-order) factor is valid than the best way to go is to embed the factor model into a model that has other external variables. For instance, if your "construct" (which the second-order factor model represents) is assumed to affect Y1 and Y2 than placing the second-order factor in a model with effect on Y1 and Y2 would test the restriction whether covariances between X1 and X2 (the primaries of the second-order factor) and Y1/Y2 are spurious and exist only because of the joint effects of the second-order factor on all 4 variables. A direct effect of an X on a Y would be a possible problem (albeit not necessarily) and would be the basis for further diagnostics, model enlargements, and replication.
HTH
--Holger
Check this paper:
Kepner G. R. (2014). Dimensional analysis yields the general second-order differential equation underlying many natural phenomena: the mathematical properties of a phenomenon's data plot then specify a unique differential equation for it. Theoretical biology & medical modelling, 11, 38. https://doi.org/10.1186/1742-4682-11-38
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