It would be great if one answers with literature that justifies their argument. :)

The Sobel test is generally frowned upon these days as it has low statistical power.

Assuming a multiple regression context. The mediation effect is a * b where a is the coefficient predicting m (mediator) from x and b is the coefficient from predicting y by m + x. If both a and b effects are statistically significant you have evidence of mediation (an indirect effect). This joint test approach is very simple to conduct and is generally safer than other approaches (with good Type I error and power).

If you have control variables logically I believe should be in both models as they make the tests of the a and b paths more sensitive (i.e., by reducing error) even if they aren't correlated with x or m.

thank you Thom Baguley ! Indeed, my main methodology follows the test approach you explain. Using this approach, the indirect effect was shown to be not significant. I intended to do the Sobel-Goodman Test as a robustness check, hence, as a second approach to corroborate the findings. However, the indirect is significant in this test once I include my controls, whilst when I exclude them it is insignificant. From here onwards, my question arose as to whether include my controls or not... since I am simply looking for robust results of my main approach.

Azamussan Syed thank you for your reply. Do you perhaps have an article that support this argument?

Azamussan Syed Thank you for you detailed answer and for looking into this matter for me. It helps a lot!

As you say, it seems like Hayes & MacKinnon do not dive deep into control variables. However, I completely agree with what you say and will continue my analysis based on this discussion. I will for sure go through Gaskin's StatWiki. Thank you so much for you extreme help and providing me with the sources!

I believe that it must have that the indirect effects are significant but also that it reduces the direct effect to be not significant to fulfill the model of total mediation, if the direct effect stays significant it corresponds to partial mediation and it is a questionable result if it uses only PROCESS and not by means of method of structural equations that allow to propose global theoretical models beyond the coeficient of regression, but considering the indexes of adjustment SEM that determines the proposal of the mediating model. In addition, the importance of the control variable is that it can be evaluated one by one to determine which has the greatest suppressive effect that can have importance in the relationship of the variables of its study.

Article Introduction to mediation analysis with structural equation modeling

Article MacKinnon DP, Krull JL, Lockwood CM. Equivalence of the medi...

Sydney Straver, Mediation models should be estimated using SEM. In doing so, you also include the control variables in the model as suggested by Thom Baguley. Moreover, instead of Sobel's test, bootstrap test of indirect effects is recommended. Here is the link to a paper about a user-written Stata package that facilitates mediation analysis within the framework of SEM:

Article medsem: a Stata package for statistical mediation analysis

I'd disagree slightly. There is no general reason to use SEM for a mediation model without latent variables. There are specific reasons why SEM approaches might be helpful by making the analysis a single step or by providing fit statistics or in fitting a wider range of models. However, a simple mediation analysis fit as a path model using multiple regression should be equivalent to the SEM formulation. There are also intermediate approaches like piecewise SEM models that can be more attractive than full SEM (if you don't have latent variables).

Generally most regression models including mediation models can be specified in equivalent ways using different modeling approaches.

I agree with Thom Baguley, that a simple mediation model estimated using a series of regressions does technically correspond to SEM estimation (apart from the fact that the former uses OLS and the latter ML). However, if your software (R, Stata, SPSS, SAS etc.) facilitates SEM, then the series-of-regressions procedure becomes redundant as SEM is a more flexible and a versatile technique. Having said that, I would still recommend the series-of-regressions procedure as a pedagogical approach to understand what mediation analysis is about. Thank you Thom Baguley for informing about piecewise SEM, which I will certainly look into.

Article On the Interpretation and Use of Mediation: Multiple Perspec...

For piecewise SEM this is a good online book (covering SEM and piecewise SEM).

https://jslefche.github.io/sem_book/index.html