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How do they attribute a cause of cancer to something like drinking small amounts of alcohol?

How do you know that a person having 1 drink per day is the reason they got cancer?

They don't. What they can do is observe cancer rates in populations that have 1 drink per day and compare that to rates of cancer in other populations. There are a bunch of statistical tools you can use to try to identify and correct for other confounding variables.

One of the cited links references such a study. They found no reduction in risk for those who ceased drinking who were previously only drinking one drink per day, while there was a reduction in risk for those who ceased drinking who had been drinking three or more drinks per day.

https://www.nejm.org/doi/10.1056/NEJMsr2306723

https://news.ycombinator.com/item?id=30009529

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