Adding variances of independent probability distributions to estimate probabilities of replication

Abstract

If the prior probability distributions of all possible hypothetical true means and all possible observed means of a continuous variable are conditional on the universal set of all numbers (i.e., before the nature of a study is known and a Bayesian prior distribution can be estimated), that prior probability distribution will be uniform. It would follow that a Gaussian probability distribution and a Gaussian likelihood distribution based on the same data set are identical. Replication involves doing two independent studies and is thus modelled by adding the variance of the probability distribution based on the observed data to the variance of the expected probability distribution of the replicating study based on its proposed sample size. This allows an estimate to be made of the probability that the P value will be less than or equal to 0.05 two sided (or any other specified value) in any replicating study. The same model can be used to estimate sample sizes when planning the required power of the initial study. If this requires doubling the variance, then this will require double the sample size estimated using current power calculations, suggesting that studies using current methods are underpowered. These considerations might be used to explain the replication crisis.