Beyond the two-trials rule: Type-I error control and sample size planning with the sceptical $p$-value

We study a statistical framework for replicability based on a recently proposed quantitative measure of replication success, the sceptical $p$-value. A recalibration is proposed to obtain exact overall Type-I error control if the effect is null in both studies and additional bounds on the partial and conditional Type-I error rate, which represent the case where only one study has a null effect. The approach avoids the double dichotomization for significance of the two-trials rule and has larger project power to detect existing effects over both studies in combination. It can also be used for power calculations and requires a smaller replication sample size than the two-trials rule for already convincing original studies. We illustrate the performance of the proposed methodology in an application to data from the Experimental Economics Replication Project.