Do we have a mathematical formula to compute the p-value of an observation from the Dirichlet distribution in exact sense at https://en.wikipedia.org/wiki/Exact_test?
I think MGLM package in R can help. Read the documentation and some examples of the package.
Yes, the p value is always < 0.05 100% of the time in an exact test. You use the probability distribution to determine the actual p value. P is the probability that the test statistic is equal to or less than the value actually observed based on your sample under H0
Jamilu Yunusa Falgore and Andrew Paul McKenzie Pegman can we think of a mathematical formula in a closed-form?
The PDF formula has the support, the 'x' vector of concentration parameters (in this case all categories are from 0 to 1). The observed values is within the support (aka any combination of 'x' values between 0 and 1 that sums to 1). A p-value is the probability of an x-vector being equal to or more extreme than the observed x-values. The p-value is found by essentially just integrating the PDF function from x-values of 0 (minimum) to x-values of the observed (left tail) or from observed to 1 (right tail). However this p-value idea this gets kinda weird in the multivariate setting depending on how you define 'more extreme' because they must sum of 1, so by definition it you make one parameter more extreme (smaller) you make another parameter less extreme (bigger) because of the constraint, hence why no CDF function is provided. Anyways, look at the ternary plot and visually define the region that is 'more extreme' then send of the integral of the PDF to integrate over that region of the parameters. So its literally just the PDF formula integrated from x1=[0, observed], x2=[0, observed], x3=1-(x1+x2). Note how x3 is constrained.
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