Statistical significance revisited

TL;DR

This paper revisits the concept of statistical significance, analyzing how many experiments are needed to achieve reliable results, especially considering the distribution of placebo effects, and highlights potential misconceptions in probability assumptions.

Contribution

It introduces a method to account for placebo efficacy distributions in calculating experiment repetitions for significance, extending beyond fixed-effect models.

Findings

Considering placebo distribution can double the required experiments for significance.

Neglecting distribution details may lead to underestimating necessary sample sizes.

Erroneous assumptions about probability distributions can cause misleading interpretations.

Abstract

Statistical significance measures the reliability of a result obtained from a random experiment. We investigate the number of repetitions needed for a statistical result to have a certain significance. In the first step, we consider binomially distributed variables in the example of medication testing with fixed placebo efficacy, asking how many experiments are needed in order to achieve a significance of 95 %. In the next step, we take the probability distribution of the placebo efficacy into account, which to the best of our knowledge has not been done so far. Depending on the specifics, we show that in order to obtain identical significance, it may be necessary to perform twice as many experiments than in a setting where the placebo distribution is neglected. We proceed by considering more general probability distributions and close with comments on some erroneous assumptions on probability distributions which lead, for instance, to a trivial explanation of the fat tail.