Your criteria for ruling out other factors is very vague and the example does not prove your point.
To be honest, there wasn't a point I was trying to prove, only a family of variables to eliminate for the sake of scientific study. People who kill to survive are certainly bad people, but they are at least responding to a biologically viable stimulus: i.e., don't die. Never mind if they will or will not survive based on the crime they commit, as long as they
perceive a threat to their own survival and a need to commit that crime. Despite their willingness to commit crimes we can (arguably) say that their brains are reacting rationally to some sort of valid stimulus:
I am poor. I need money. I will steal money from someone and kill them if I have to to get it would be an understandable (though not forgivable) line of reasoning. What is more difficult to understand is why someone would kill simply for the sake of killing, with little or no payoff.
We can just as easily assume that the armed robber killed the clerk for fun and that the money is just a prize. In giving the criminal a motive we've created and accepted a non-testable hypothesis.
If our hypothetical armed robber's primary motivation for killing the clerk was fun and not money, then there would be no reason to exclude him from this study as his (or her) ability to reason is probably impaired. Also, the idea is not necessarily to assign motive, but to glean it from study of these types of individuals. Motives that have economic rationale, IMO, would not be suitable for the study because their brains are working fine (although they are still jerks); crimes without rationale and that suggest some sort of synaptic misfiring would be perfect.
Also, it might not be that difficult to establish correlation. For the very worst offenders, serial killers, serial rapists, etc., getting good clean data is probable I should think, considering that many of these types are very methodical to the point of pathological OCD in everything that they do (especially crime). They are liable to eat the same things, at the same times, with very little variation. Self-reporting should not be an issue either, since there are no awards to win, and telling the data collector the "right" combinations of foods gets him out of prison no sooner. I would think someone asking a serial killer what they ate would seem extremely mundane. And even if they wanted to lie about what they ate, what criteria would they use to base their fabrication on? To be safe, a data collector could just omit what he really wants to know about the criminal (i.e., dietary habits).
I agree in advance that I am generalizing the serial killer/rapist mental state. Obviously I am aware that the possibility exists of just such a criminal without OCD tendencies. But I would hazard to say that most fit the general mold. After all, many of them are caught on the basis of not being able to vary their routines enough, therefore ultimately giving themselves away. But given an extreme propensity for obsessive tendencies as well as acting to such violent extremes, often without any rational reason, these types of people would be perfect for studying possible links between poor nutrition and anti-social behavior, IMO. Hell, submitting to such a study is the least they can do for society anyway ...
So I guess my hypothesis is:
Poor nutrition causes varying degrees of anti-social behavior; furthermore, good nutrition may play a role in mitigating anti-social behavior.And I liked that article!