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Considering that the p-value of a chi statistic(df=1) is uniform, the expected genomic inflation factor is λ=1. I show its calculation in R with the next chunk of code:

chi = qchisq(df$pval,df=1,lower.tail=FALSE)
factor = median(chi)/qchisq(p=0.5,df=1)

My question comes here. Imagine that I compute λ on several (e.g. >30 independent GWAS) studies. My null hypothesis is that all of them lack true-polygenic signals (i.e. uniform p-value distribution). What test can I do to reject H0? What is the distribution that I can assume for the inflation factors?

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    $\begingroup$ Why is p fixed at 0.5? Anyway it is unlikely to matter here: the answer could only be worked out from simulation studies in my opinion and a 0.05% threshold assigned against the resulting distribution. $\endgroup$
    – M__ ♦
    Commented Jan 5, 2023 at 15:03
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    $\begingroup$ You are right! I can simply do the simulation. On the other matter, I am not a statistician but I guess is related to the fact that 0.5 is the expected (p) value. $\endgroup$
    – Gero
    Commented Jan 5, 2023 at 16:06
  • $\begingroup$ Given the distribution is bell-shaped (looks normal) then *p*=0.5 makes sense, that can be justified. The key is getting the right simulation to test your statistic to model what your statistic is testing. If that test works well, it doesn't really matter how you got there. Obviously simulations need replications and that sort of thing. $\endgroup$
    – M__ ♦
    Commented Jan 5, 2023 at 17:06

1 Answer 1

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The distribution as I suspected seems very much like a normal distribution. However if someone knows to explain the reason (limit central theorem?) I will give select their answer.

    # Do simulation    
    factor_calc = function(pvals){
      chi = qchisq(pvals,df=1,lower.tail=FALSE)
      return(median(chi)/qchisq(p=0.5,df=1))}
    pvals_studies = sapply(1:10000, function(i){runif(10000)})
    factor_distribution = apply(pvals_studies, 2, function(pvals){factor_calc(pvals)})

    # Plot resulting distribution
    hist(factor_distribution)

enter image description here

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