Reproducibility is the essence of a scientific research. Focusing on two-sample problems we discuss in this paper the reproducibility of statistical test results based on p-values. First, demonstrating large variability of p-values it is shown that p-values lack the reproducibility, in particular, if sample sizes are not enough. Second, a sample size formula is developed to assure the reproducibility probability of p-value at given level by assuming normal distributions with known variance. Finally, the sample size formula for the reproducibility in general framework is shown equivalent to the sample size formula that has been developed in the Neyman-Pearson type testing statistical hypothesis by employing the level of significance and size of power.
When a causal effect between treatment and outcome variables is observed, effects on the outcome are of interest to investigate the mechanisms among the outcome and treatment. Indirect effect is defined as the causal effect of the treatment on the outcome via the mediator. Direct effect is defined as the causal effect of the treatment on the outcome that is not through the mediator. In this paper, we discuss the estimation of direct and indirect effects based on the framework of potential response models focusing on the 4-way decomposition. Direct and indirect effect estimations are illustrated with two examples where the outcome, mediator, covariate variables are continuous and categorical data. Moreover, we discuss the estimation of clausal effects and the effect decomposition in the settings that include confounder of mediator and outcome affected by treatment, multiple mediators, or time-varying treatment in the presence of time-dependent confounder.