Previous studies have not conclusively demonstrated behavioral effects of artificial food colorings … This study, which was designed to maximize the likelihood of detecting a dietary effect, found none.
Rephrased to say precisely what the study found:
This study tested and rejected the hypothesis that artificial food coloring causes hyperactivity in all children.
Interesting. Those two statements seem quite different; more than just a rephrasing.
Probabilistically, it sounds like the study found P(hyper|dye) = P(hyper|~dye), that is they rejected P(hyper|dye) > P(hyper|~dye), and concluded P(hyper|dye) = P(hyper|~dye) (no connection) correctly.
I think your logical interpretation of their result throws out most of the information. Yes they concluded that it is not true that all children that ate dye were hyperactive, but they also found that the proportion of dye-eaters who were hyperactive was not different from the base rate, which is a much stronger statement, which does imply their conclusion, but can’t be captured by the logical formulation you gave.
Probabilistically, it sounds like the study found P(hyper|dye) = P(hyper|~dye), that is they rejected P(hyper|dye) > P(hyper|~dye), and concluded P(hyper|dye) = P(hyper|~dye) (no connection) correctly.
You are making the same mistake by ignoring the quantification. The test used to reject P(hyper|dye) > P(hyper|~dye) uses a cutoff that is set from the sample size using the assumption that all the children have the identical response. They didn’t find P(hyper|dye) = P(hyper|~dye), they rejected the hypothesis that for all children, P(hyper|dye) > P(hyper|~dye), and then inappropriately concluded that for all children, !P(hyper|dye) > P(hyper|~dye).
Interesting. Those two statements seem quite different; more than just a rephrasing.
Probabilistically, it sounds like the study found
P(hyper|dye) = P(hyper|~dye), that is they rejectedP(hyper|dye) > P(hyper|~dye), and concludedP(hyper|dye) = P(hyper|~dye)(no connection) correctly.I think your logical interpretation of their result throws out most of the information. Yes they concluded that it is not true that all children that ate dye were hyperactive, but they also found that the proportion of dye-eaters who were hyperactive was not different from the base rate, which is a much stronger statement, which does imply their conclusion, but can’t be captured by the logical formulation you gave.
You are making the same mistake by ignoring the quantification. The test used to reject P(hyper|dye) > P(hyper|~dye) uses a cutoff that is set from the sample size using the assumption that all the children have the identical response. They didn’t find P(hyper|dye) = P(hyper|~dye), they rejected the hypothesis that for all children, P(hyper|dye) > P(hyper|~dye), and then inappropriately concluded that for all children, !P(hyper|dye) > P(hyper|~dye).