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(The linked one is a short news summary.)
Their code is public: https://github.com/AnuSub/Stata-and-R-codes/
Some of the criticisms here can be answered by the paper and the code. To handle the "green jellybean" problem of multiple testing, they apply the Bonferonni correction. (Actually, the code uses the Simes variant of Bonferonni, so I think the article wording is not precise but it's not a major difference.)
I believe the code has a flaw in that it does not apply a multiple testing correction to the confidence intervals in the diagrams. (See lines 3199-3217). This makes the diagram inaccurate but does not alter the number of symptoms that would meet the bar of corrected-p < 0.05.
The deeper challenge is of course that it is a retrospective study with synthetic controls, so it will not be a gold standard. I think the best criticism is the one they note themselves:
"Conversely, with the evolving awareness of long COVID, it is possible that patients with a history of COVID-19 may have been more likely than those without to access primary care and alert clinicians of their symptoms, which could potentially lead to an inflation of the observed effect sizes. This is potentially supported by the increased aHRs observed for symptoms such as cough, sneezing, fever and allergies among patients who were infected during the second surge of the pandemic, compared to those infected during the first surge"
How likely are you to go to the doctor if you symptom is reduced libido and you have no recent major medical issues? But if you have diverse symptoms AND recently recovered from COVID, you may be much more likely to seek a medical opinion.
My comments above are methodological and not at all meant to deny that people do suffer long after acute COVID. Some of the ratios (like anosmia being 6x more likely in COVID group) are pretty large and that to me would continue to merit further investigation.
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