If people with mental disorders were more sleep-deprived and then consumed more sugar, wouldn't that also show up in the data?
> Although the study looked for this link, it was not seen in the data: men and women with mental disorders were not more likely to consume more sugar.
A causal A > B > C link would look similar (data-wise) to a A > C link.
Let group S be sugar eaters.
Let group A be all people w/o mental disorders.
Let group B be all people w/ mental disorders.
We have the main correlation between sugar and mental disorder:
p[A/S] <= p[B/S]
Using bayes rule:
p[A]p[S/A] <= p[B]p[S/B]
We also have "men and women with mental disorders were not more likely to consume more sugar":
p[S/A] >= p[S/B]
Comparing the inequalities gives us:
p[A] <= p[B]
So you're more likely to have a mental dissorder than not according to this study???
> Although previous studies have found an increased risk of depression with higher consumption of added sugars, none examined the role of ‘reverse causation’. If people with anxiety and/or depression tended to consume more sugary foods and drinks, this could be the real reason why a link between sugar intake and poorer mental health is observed. Although the study looked for this link, it was not seen in the data: men and women with mental disorders were not more likely to consume more sugar. As a result, the evidence that mental health is adversely affected by a high sugar intake is strengthened.