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Wow. That is the worst critique of anything I've ever seen.
The subjects in the test were given a fifty question questionnaire and only 13 questions are used, and this jackass is complaining about that? What the fuck? That is how you get honest answers in social science! If all the questions are related to the topic, the people answering can get a sense of what the study is about and begin to second guess the answers that the study "wants". The participants try to "win" by figuring out the answers that the scientists want to hear and then giving those answers.
You get more accurate answers by throwing in ringer questions that are not scored. If the overall test is filled with irrelevancies, the participants can't figure out what the overall goal of the test is (because they don't know which ones are real and which ones are fake), and the overall answers are more accurate to their real feelings. This is a documented effect, and it's how all top level social science is done.
If someone is going to complain about that in an effort to debunk a study, they are not worth listening to. William M. Briggs is a social science illiterate, or he's deliberately lying about the study because he doesn't like the implications. Since he's an adjunct statistics professor at Cornell and couldn't possibly be unaware of how one goes about getting results from survey questions, I believe the only possible conclusion is that he is a liar. Kindly never link to that asshole again.
It's linked to from your link. The original author says he was aware of this technique; but he did cite the use of this technique as a criticism of the study without mentioning that it is an accepted methodology. That tells me that his analysis is both biased and deliberately misleading.
I tend to question the use of IQ tests as a measure of intelligence. The criticism you cite raises some legitimate questions. But it raises them as if they are specific to this one study, they aren't.
I think some of his criticisms are valid. But, based on the fact that he is being deliberately misleading, I wouldn't take his article seriously.