[Yuhan HU]

It seems possible that your main x variable is not a very strong predictor of your outcome (note that statistical significance is not a measure of strength). Mediation is not assessed by looking at whether the coefficient for x loses its significance in the presence of the mediator. If you are using the Barron and Kenny framework, then you need to multiply the two path coefficients (x ->m)*(m -> y) by each other using nlcom. See example 42g in the Stata sem manual. See here for some example code.

Thanks a lot Erik! I just confused by two fully mediation effects, but I have found literature arguing that we should lose focus on the significance of coefficient for x.
Rucker, D. D., Preacher, K. J., Tormala, Z. L., & Petty, R. E. (2011). Mediation analysis in social psychology: Current practices and new recommendations. Social and personality psychology compass, 5(6), 359-371.
Thanks again for your help!

[Erik Ruzek]

That reference deals with the question of whether mediation can be present when the c' effect is non-significant. It uses the Barron and Kenny approach (a path*b path) to assess mediation. That is, the paths are multiplied and statistical significance is assessed using either bootstrapping or another approach (e.g., Bayesian estimation). See this paragraph on page 363:

Tormala et al. started with a simple mediation model. In this analysis, number of positive thoughts requested (i.e., X) significantly affected both M1 (a1 = -1.25, p < .01) and attitudes (c = -1.18, p = .03). Participants reported less confidence in their positive thoughts and less positive attitudes when asked to generate 10 (difficult) versus 2 (easy) positive thoughts. When both X and M1 were included as predictors of attitudes, M1 remained significant (b1 = .56, p < .01), whereas X did not (c' = -.47, p = .36). A bias-corrected bootstrap 95% CI indicated that the indirect effect through M1 was significant, a1 X b1 = -.71, 95% CI: [-1.48, -.14].

Look at Preacher's work with Zyphur and Zhang in 2011 for more detailed explanation of how to assess mediation.

[Yuhan HU]

That reference deals with the question of whether mediation can be present when the c' effect is non-significant. It uses the Barron and Kenny approach (a path*b path) to assess mediation. That is, the paths are multiplied and statistical significance is assessed using either bootstrapping or another approach (e.g., Bayesian estimation). See this paragraph on page 363:

Look at Preacher's work with Zyphur and Zhang in 2011 for more detailed explanation of how to assess mediation.

Thanks a lot, Erik!