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With q1 = 1−p1 and q2 = 1−p2 we obtain: <img src="http://journals.plos.org/plosone/article/asset?id=info?doi/10.1371/journal.pone.0003178.e024.PNG" class= inline-graphic"/> The summation is necessary because i successes can be obtained from any combination of j p1 and i−j p2 trials, where j can assume any value from 0 to i.
As such, the pairwise comparison ratios can be used to obtain the components of a binomial process in which w i successes have been observed in (w i + w j ) trials subject to an unobserved preference parameter, p i.
Assuming random independent sampling of families, the probability to sample f or less families out of F possible is P + = ∑ i = 0 f bin p +, i, F where bin(p+, i, F) is the binomial probability to get i successes out of F trials with the probability of success in a single trial p+.
To quantify the evidence against the presence of the reference sequence r in s, we perform a Binomial test at each position r i, to generate a one-sided P value P i to assess the probability of observing a i - b i successes in a i trials, with a probability of success of 1- e i.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com