Hi,

a few points

a) I personally would not interpret the fit of your CFA as "acceptable". The term acceptable is something subjective and it may be that you accept the model, but objectively, the model shows a significant (beyond chance) deviation from the data. I would invest in it to get to a clear measurement structure.

The large df for the CFA suggests that you had a large number of observed variable and a small number of latent variable. This promotes misfit as such a model makes such a huge number of assumptions (in terms of local independence) that it is a-prori unrealistic. Better select the key indicators that reflect those latents you imagine. This at leaste increases a chance to fit the model

b) If you mediation/SEM model has 0 df than this means that 1) you have made your model with manifest composites? Otherwise the SEM would have at least the same number of df than the CFA/measurement models. Furhter: Did you specify a full mediation or partial mediation. The 0 df suggests the latter.

It would be helpful if you upload a full path diagram (including th emeasurement models).

I would not necessarily agree with Byrne's statement (although I find the quote quite funny, as the recommendations in her book regarding fit evaluation are otherwise rather lax), that a saturated model is useless. The only thing is that it has no testable implications and, thus, many alternative structures with equal fit. But in principle that is true (in degrees) for every model. Models vary from extremely restrictive/bad fitting (i.e., null model) to saturated / perfectly fitting. Within this range of these models fit increases but also number of alternatives/equivalent models. Therefore, good practice would be that--besides the model fit--researchers should discuss in their paper, how strong the support of their model is (how many permutations exist) and more importantly, which partitions of the model are unequally supported. With some background knowlegde on equivalent models, this is not sooo difficult.

This paper by Judea Pearl and Rex Kline (unfortunately not yet publshed...) as well as Chen and Pearl

are a good basis:

https://pdfs.semanticscholar.org/6ed0/1cfc00d19ed7e6899999b8500bd29b50ccae.pdf

ftp.cs.ucla.edu/pub/stat_ser/r432.pdf

Best,

Holger

Holger Steinmetz Thank you so much for your detailed reply. The attachments are the model diagrams.

I will have a read at the references you mentioned and I studied first. Sorry for my limited knowledge and understanding..In terms of "manifest composites", does it mean the way how I compute the latent variables? I made them from computing the mean value of constructs for each unobserved variable.

Thank you a lot for your help and reply!

Hi,

as I expected. Your full SEM has many indicators and this will in most cases will cause the misfit. This model however,

a) has huge df stemming from the measurement model but

b) zero df from the structural model (as all latent variables are connected).

You could try to bring in other external variable whith very specific effects (only on one or two latent variables (e.g. "instrumental variables"). This not only leads to testable implications but also (probably) reduces alternatives (e.g., turning arrows).

Start by reading the two references :)

Best,

Holger

Holger Steinmetz Thank you! I appreciate your advice.

Holger Steinmetz Thank you for your recommanded references especially about the bow-free graph to assures identification. And I tried the way of adding instrumental variables. And would instrumental variables be variables such as age and year? But I'm confused with it now...In SEM, will instrumental variables be control variables? It seemed that there are some overlap..Thank you a lot for answering based on my limited understanding!

Hi,

a) instrumental variables are no control variables, to the contrary: If a focal effect of X on Y is confounded by the joint influence of some hidden variable(s), simply controlling for an instrument will amplify the bias. This is called z-bias:

Ding, P., VanderWeele, T., & Robins, J. (2017). Instrumental variables as bias amplifiers with general outcome and confounding. Biometrika, 104(2), 291-302. doi:10.1093/biomet/asx009

Kim, Y. (2019). The causal structure of suppressor variables. Journal of Educational and Behavioral Statistics, 1076998619825679. doi:10.3102/1076998619825679

b) an instrument W is a variable that correlates strongly with X but is conditionally independent from Y given X. And the correlation between W and X must not stem from X affecting W (it can, however, be the result of a unobserved confounder). In SEM parlance, this means that W has an effect on X but not on Y

c) Age and Year in your model are simply further predictors of Y (i.e., WE), the key issues is however the effect on X (TS). Furthermore, in your model, you simply let Age/Year covary with TS. You could without doubts estimate an effect (instead of a covariance) on TS as it can be excluded that TS causes Age nor Year nor that any other hidden variable causes TS and Age/Year (i.e., Age/Year must be causes and cannot be dependent variables). But even in the present form, you now have testable implications in your new model, namely that Age/Year are conditionally independent from PM and IU once you control TS. If your model does not fit, THESE assumptions are violated (both or one of them).

d) While Age/Year are no instrumental variables of the TS-WE effect (because of their effect on WE), they could (as fair as I can guess at the moment) be indeed be regarded as instruments for PM and IU. But this depends on the strength of the effect of Age/Year on TS.

e) Having said all of this, I doubt that Age / Year have effects on TS or WE and if they have: could you really claim that it has no effects on PM/IU? Of course, you can try it.

Please not that all of this requires you to dig a bit more into the principles of graph theory, d-separation ("conditional independence") and instrumental variable estimation. This is nothing you can do on a sunday with some "throwing in some variables". I already provided some references. Please read them. This is surely not easy but will profit you enormously.

What I can further recommend is the browser software daggity.net. You can draw path diagrams and play a little with possibilities. Two important outcomes of this software is that it displays a) the testable implications and b) whether variables can act as instruments (I used it for the above guesses).

Best,

Holger

Holger Steinmetz Thank you so much for your consistent help. I will go on reading and studying.