The New York Post reports vegetarians are twice as likely as meat eaters to be depressed. As is often the case, the article contains an implicit presumption that failing to eat meat causes depression. It is just as likely that depression causes vegetarianism.
In fact I’d argue it is even more likely that depressed people metaphorically grope around trying to find something to make themselves more cheerful. A number of them decide to eliminate meat from their diets and a viola a correlation is created. Of course it doesn’t work, except for whatever placebo effect is triggered.
I’ll bet, if studied, you’d find that depressed people are also more likely to use street drugs, practice yoga, try intermittent fasting, and whatever other quack nostrums are being offered.