How to test for a partially mediated model? I have a dataset with three variables: Outcome, Exposure, and Mediator.
My hypothesis is that the variables are related as in the following DAG:

In particular I want to test that "Mediator" in effect partially mediate the relationship between Outcome and Exposure.
That is, Exposure causes Mediator and Mediator causes Outcome. But also, Exposure causes Outcome directly too.
How can I test whether Mediator is a partially mediate the relationship between Exposure and Outcome?
 A: Following my comment, here are answers to the three question.
How to test if there is a mediated effect? There is many ways to test indirect, the recommended methods is the bootstrap methods. You can find some useful informations in this paper. Mplus, R, SPSS (via process), SAS, MATLAB, etc. can all carry the analysis.
If the mediated effect is significant? Using the confidence interval computed via bootstrap you check if the indirect is significant. Is the interval holds 0, then, the null hypothesis of no indirect effect if not rejected. Otherwise, you can reject the null hypothesis.
Is the indirect effect substantial (of practical importance)? This is probably the hardest problem about indirect effects. There is no definitive effect size to directly compare to, like you would with cohen's $d$ or correlation coefficients $r$, or any others. Historically, some has used the Baron and Kenny method distinguishing fully vs partly mediated (see paper for details), but I would discourage it use. Some would use the proportion of mediated effect. But this effect size has more challenges than an actual answer in practices. I would recommend to use your own judgments as well as following the recommandation in your specific field.
