Specification of direct and total effects have been mistreated for years that will mislead to incorrect conclusion about statistical inferences about indirect effect. We prove that for detection of presence of mediation, it requires only to test the association between the predictor and mediator giving the reason that how Baron and Kenny (1986)’s three steps of tests has low power. We also provide theoretical proofs for the observation of Hayes (2009) for absence of total effect but there is indirect effect and the observation of Palmatier et al. (2009) for total effect containing no indirect effect. With regression function being formulated in terms of distributional parameters of variables of response, predictor and mediator, we allow to quantify the information of mediation existed in their joint distribution to be removed to specify unambiguous direct effect and total effects. One important discovery is that the mediation causes not only affect the regression slope parameters but also the regression intercept. So, instead of limited use of slope type effects, we introduce regression setup direct, indirect and total effects expanding to the scope of effect prediction. Statistical inferences for these effect regressions are introduced and evaluated.