The posterior probability of a null hypothesis given a statistically significant result
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Some researchers informally assume that, when they carry out a null hypothesis significance test, a statistically significant result lowers the probability of the null hypothesis being true. Although technically wrong (the null hypothesis does not have a probability associated with it), it is possible under certain assumptions to compute the posterior probability of the null hypothesis being true. We show that this intuitively appealing belief, that the probability of the null being true falls after a significant effect, is in general incorrect and only holds when statistical power is high and when, as suggested by Benjamin et al., 2018, a type I error level is defined that is lower than the conventional one (e.g., α = 0.005). We provide a Shiny app (https://danielschad.shinyapps.io/probnull/) that allows the reader to visualize the different possible scenarios.
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