There is one huge reason not to take it seriously, if you accept chemistry, biology, or toxicology: The senior author is from the National Institute of Environmental Health Sciences, which for the last decade has competed with the International Agency for Research on Cancer (IARC) and Ramazinni Institute (Picchiatello) for how best to erode confidence in science and health among the public, by using statistics to undermine scientists.
And that is all it is, statistics based on notoriously flaky recall of how much hair product people used. They say "study participants" but actually did no studies. Instead, they looked at a group of sisters and their self-reported claims about hair products, added up how many got breast cancer, declared statistical significance, and went to publication.
Though their statistical claims are just correlation, they know media will be off and running with a way to implicate any company by linking them to cancer. Their paper is exploratory but it won't matter, they use all of the right buzzwords like endocrine disrupting chemicals, carcinogens, and tell us women of color are most at risk. They know journalists won't care that this was not just correlation, it was weak correlation.
*models adjusted for age, race, education, ever oral contraceptive use (OC), parity, age at first birth, smoking status, BMI, age at menarche
†For hazard ratio estimates associated with personal use in the 12 months before enrollment and application to others in the 12 months before enrollment, “No use” is the referent category for “Any use”. For frequency of use, “No use” is also the referent category for “<4 times per year” and “Every 5 weeks, 1+month” and the p for trend is provided.
Even if it were not a weak association, it wouldn't mean a causal relationship so women of color can straighten their hair if they want until scientists show otherwise. On actual strong associations related to endocrine disruption, NIEHS remains silent. The biggest endocrine disruptor out there, birth control pills, bind 20,000X as well to estrogen receptors as any chemical in these hair products, and has recently been linked to smaller hypothalamus volume in the brains of users, but NIEHS has yet to jump on the 'stop giving women birth control because of endocrine disruption' bandwagon. The reaction to that would be pretty funny, but it would put into perspective how NIEHS epidemiologists torture data until it confesses to the crime of the month.
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