# Can we simplify a 4-layer conceptual model with control variables?

I have a quick question on a conceptual model in an empirical research. Say that I have two antecedents A and B of C. I can use some theory to hypothesize that A and B affect C, and based on results from previous studies, I know for sure that C affects D. Finally, I use another theory to hypothesize that D affects E.

Now, my question is: if my only concern is to test the potential relationships between ($$A$$ or $$B$$) and $$E$$, can I simplify the conceptual model so that it could have: $$A/B\rightarrow D\rightarrow E$$? In other words, can I force C to be a control variable? Or can I simply ignore this mediating factor C, and accept that the final estimation between $$A/B\rightarrow E$$ might not be accurate as a consequence (aka. wrong result in terms of confirming/refuting the hypothesis on the indirect effect between $$A/B\rightarrow E$$ via $$D$$)?

The answer is yes, psychologists do that all the time. However, testing the relationship will be very challenging if the mapping $$A\rightarrow{E}$$ or $$B\rightarrow{E}$$ is not monotone. Imagine the case where $$E\propto{f}(\sin(A))$$.

If the relationship is linear, then Pearson's Product Moment Coefficient will test what you are looking at. Note that it is bidirectional $$A\rightarrow{E}$$ and $$E\rightarrow{A}$$. The idea of a "left-hand side" and a "right-hand side" so common in economic thought will just vanish. There is no endogeneity issue, though because it doesn't matter which way the relationship runs.

If your relationship isn't linear but it is monotonic, then you could use either Spearman's $$\rho$$ or Kendall's $$\tau$$. They have different properties, so you would have to do some research on which to use. They are also not directional.

If you believe that $$A/B$$ cause $$E$$, then there is a unidirectional measure of monotonic association, Somers' D. It can differentiate between it is cloudy, therefore it is raining, from it is raining, therefore it is cloudy.

Correlation only shows association but your question was "can I test it?" Yes, it can be tested.

The alternative would be to use a regression model, but you are nesting relationships. Depending on what the real world problem is, it may not be a trivial thing to accomplish.

Also, correlation makes weaker claims about your knowledge of the real structure in nature than a regression would make.

There is one easy way that your chain could break down. Imagine that $$A,B,Z\rightarrow{C}$$ and almost all of the value and variability of $$C$$ is due to $$Z$$. Then even though $$A$$ and $$B$$ influence $$C$$, the impact may be lost by the time it gets down the chain of variables to $$E$$.

If you do test the correlation of the two variables to the final variable, you will need to do a correction for multiple comparisons in your testing such as the Holm-Bonferroni correction.

• Thank you very much for providing great details in your answer. It really helps strengthen my understanding. I have a follow-up question though: since what I aim to achieve is to estimate a potential causal effect of $A/B\rightarrow E$ (and I use some theories such as Social Capital Theory/Transaction Cost Economics/Flow Theory/etc. to derive the path model described in the OP), would it be fine, from a technical perspective, to apply Partial Least Square - SEM (PLS-SEM) method here?
– ghjk
Aug 31 at 4:09
• I would suggest turning your comment into a question. Your question depends on both factor modeling and structural equation modeling. There are objections to the method in the literature. It would require more detail to discuss. It is important to remember that when you are orthogonalizing data, you are transforming it in a manner that may be opaque. It isn't an area of literature I have spent much time with because what I work on isn't helped by using principal components or factor analysis. Aug 31 at 5:01