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Obviously, both refer to the same thing. But, which is more correct/more common in biomedical papers? By alpha, I mean alpha level (Also see one). (And, should it be $p < 0.05$ or $p = 0.05$ if it's not alpha?)

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Definitely α; α is the threshold you set, p is the probability you measure, you compare p to α. If α = 0.05, then you consider p < 0.05 significant. See e.g. https://en.m.wikipedia.org/wiki/P-value

I wouldn't necessarily let a majority of biomedical papers sway you one way or another with statistics, because the stats in so many of those papers are sub-ideal in that area. Biologists and physicians do not get enough statistical training relative to its importance, especially as you go back decade by decade.

I don't think confusion with other distributional parameters is plausible if you are describing a threshold for statistical significance - there is only one meaning in that context.

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Definitely p = 0.05, this is because α is a parameter, e.g. in the gamma distribution and therefore avoids any confusion. The gamma distribution models over-dispersion for example mutation rate heterogeneity across a gene.

α = 0.05 is a bit low for α in e.g. a gamma distribution but it is certainly possible, usually its >0.1.

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    $\begingroup$ I think we are talking about a different alpha. I mean this: google.com/search?q=alpha+level $\endgroup$
    – BigMistake
    Aug 4, 2023 at 13:10
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    $\begingroup$ Indeed but it avoids any confusion. $\endgroup$
    – M__
    Aug 4, 2023 at 13:28

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