How does model complexity in SEM affect regression coefficients (+ significance) of links between latent variables

Question

I have been running a SEM in the lavaan package, testing how different specifications affect model fit etc.

My question is regarding the following issue:

I compared two models, one with 7 latent variables and one with 10 latent variables. The second also has a lot more paths between latent variables, so is more complex than the first.

While model fit indicators are very good for both, I noticed that in the more complex model, the coefficients (+ their significance levels) for paths between latent variables were different, even for the same pair of latent variables (those listed under the headline "regressions" in the lavaan output). So much so, that in the less complex model a path/relationship was significant, which was not significant in the more complex model. I have also seen that coefficients change from being positive to negative, depending on complexity of the model.

As I am still quite new to SEM I was wondering if anyone could explain the reasons for this - why do we see these discrepancies in coefficients and significance levels?

(And, related to this question, consider that I found the following statement in a SEM study: "While we had specific hypotheses for each of the constructs involved, we tested all direct relationships between variables and retained only those in the model that turned out to be significant." Isn't that highly problematic, just testing every potential path between all latent variables available, if model complexity affects the significance of paths/relationships? What is the meaning of significance and coefficients at all, if they can change when we add additional variables into the model?)

I would be extremely grateful for an answer or comment!

Answer 1

Changing your model can change your parameters, just as changing your model in a multiple regression model can change your parameters.

Complexity of the model doesn't necessarily have any effect. If your model is correctly specified, adding additional complexity (also correctly specified) will not necessarily change your parameters - but it can.

Consider a simple example. Three variables, X, Y and Z, all correlate 0.7.

A model which correlates Y and Z will have a correlation of 0.7.

If you regress Y and Z on X, the correlation between Y and Z becomes a correlation of the residuals - and it's now a partial correlation, so it will be different.

(I would ask another question, if you have another question - answering two different questions in one answer gets confusing.)