The credibility plot for extreme explanations and all explanations in between

This editorial is about making statistics more useful by reframing them as the credibility of explanations for the results of trials. This approach is illustrated with a trial that reported postoperative organ dysfunction in 56/147 (38%) intervention participants and 75/145 (52%) control participants [1]. The rates in these samples are not the same, but so what? The `so what? ́ is how much credibility should we give to competing explanations for the relative rate in the intervention sample of 0.73 (73%)? An explanation that 38% of the intervention population develop postoperative organ dysfunction is obviously consistent with the sample rate of 56/147 (38%). But other explanations are also consistent with the sample rate, although less credible, for instance population rates of 27% or 51% or 34%. Similarly, an explanation that 52% of the control population develop organ dysfunction is consistent with the sample rate of 75/ 145 (52%), as aremany other less credible explanations. The credibility plot that different explanations account for an observed sample is called a probability density function. The credibility of different explanations changes as evidence increases; for instance, with each of 10 tosses of a coin for explanations that range from extremes of `all tosses are heads ́ to `all tosses are tails ́, or explanations for organ dysfunction that range from extremes of `no one gets organ dysfunction ́ to `everyone gets organ dysfunction ́, etc. Professor Richard McElreath, the director of the Max Planck Institute for Evolutionary Anthropology, uses this example in his excellent lecture series, `Statistical Rethinking 2022 ́, but instead of a coin toss he imagines sampling the unseen surface of a planet to determine the credibility of explanations that range from `all the surface is land ́ to `all the surface is water ́ [2, 3]. Each individual sample value – land or water – has its own credibility plot. The plot is the same for any dichotomous result; the toss of a coin, the diagnosis of postoperative organ dysfunction. The faint straight yellow line in Figure 1 is the credibility plot for a land sample. The credibility of explanations for this land sample decreases linearly from 2 for `all the surface is land ́ to 0 for the incredible explanation `all the surface is water ́. The credibility plot for a land sample is shaped like a backward slash (\). Similarly, the credibility plot for each water sample has the converse shape (/), with zero credibility for `all the surface is land ́ to a credibility of 2 for the explanation `all the surface is water ́. The credibility value of 2 for the explanation consistent with the sample is a logical necessity. The value of 2 is forced by: the plot is linear from this value to a value of zero for the opposite (impossible) explanation, for instance `all the surface is land ́ for a water sample; the convention that the plot area is 1; the horizontal axis length is 1. A straight line from a credibility of 2 to a credibility of 0 is the plot for any dichotomous result. The second value Professor McElreath samples from the planet surface is water. The sequence of values is now `land-water ́. We update the first `\ ́ function for land with a `/ ́ function for water. The credibility plots for each sample interact through their multiplication. The product of these two straight lines is the yellow convex curve in Figure 1: zero at the disproven extremes of `all land ́ and `all water ́, with highest density at `half land and half water ́. The subsequent