The Case of the Insignificant Clue

INSPECTOR LESTRADE: My research shows that the average Type Z personality is more likely to be a convicted of murder than the average Type Q personality. Obviously, this has important implications for the present case. We have two suspects--one has Type Z personality (Mr. Zimski) and one has Type Q (Mr. Quirk). Clearly, the Type Z suspect is our murderer because being a Type Z causes people to murder.

HOLMES:  Inspector, you don't know that being a Type Z causes one to commit murders. Correlational evidence is very good for describing what has happened, but as evidence of why (causality), it is circumstantial at best. You don't know if these people would have scored a certain way on  your test--what you call a Type Z personality--before they committed murder. Thus, although you think that being a Type Z causes one to be a murderer, it could be be that committing a murder causes one to score as a Type Z on your scale. For example, I can easily imagine how the effects of being convicted of murderer-- the guilt, the shame, the life sentence, to name a few--could affect a person's responses to certain questions.

INSPECTOR LESTRADE: Are you finished?

HOLMES: No, there's also the possibility that some third factor accounts for the statistical relationship you've found between murder and Type Z. It may be  that Type Zs are more likely to be men than women, that they are more likely to come from disadvantaged backgrounds, or that they are more likely to be unemployed. Any of these factors might cause someone to both score a certain way on your test and to be more likely to commit murder. For example, men are more likely to commit murder and may respond differently to some questions on your scale than women. Similarly, its quite possible that two effects of alcohol abuse may be to answer certain questions in a particular way and to increase the chances of committing murder.

INSPECTOR LESTRADE: So, you disagree with me that Zimski committed the murder because he's a Type Z because I haven't proved that being a Type Z causes murder. Fair enough. But you will agree that since correlational evidence is useful for describing what happens and predicting what will happen, my personality test results are a vital piece of the puzzle--even if I can't use them to establish why Type Zs tend to be murderers.

HOLMES: I will agree that the link between your personality types and murder is highly significant.

INSPECTOR LESTRADE: So, I'll arrest Zimski as I planned. As usual, I don't really need the great Sherlock Holmes, as stimulating as our discussions may be. Good day.

WATSON: Lestrade has gotten sharper, eh, Holmes. I guess you can teach a dog new tricks. A little knowledge doesn't hurt anyone.

HOLMES: On the contrary, Lestrade's behavior in this case is proof that a little knowledge can be dangerous.

WATSON: But you yourself congratulated him.

HOLMES: I did no such thing, Watson. I merely admitted that the link between murder and his personality types was statistically significant.

WATSON: Isn't that the same thing as agreeing that his research was important and relevant?

HOLMES: Definitely not. I said nothing about the importance or relevance of his research.

WATSON: But you did say there was a strong link between murder and his personality types.

HOLMES: Watson,  you disappoint me. Know you nothing of statistics? A significant correlation is not necessarily a large one--it merely means that there probably is some relationship between the variables. To know the strength of a relationship, we need to square the correlation coefficient to get the coefficient of determination. When we calculate the coefficient of correlation for Lestrade's test, we find that the relationship between personality type and murder is miniscule. Specifically, 51% of all murderers are Type Z whereas 49% are Type Q. This difference is small and of no practical value.

WATSON: So, if the correlation was large and significant, we would know that the Type Z suspect was the murderer. For example, if Lestrade had given the suspects your Moriarity test and one had scored higher than the other, we would know that the high scorer would be the liar because the correlation between scoring high on the Moriarty test and honesty is -.8, meaning that the coefficient of determination is .64 (-.8 * -.8).

HOLMES: In that case, we would be much more likely to suspect that the high scorer on the Moriarty test was the one who was lying--but we could not be sure. The only way we could be sure is if the coefficient of determination was 1.00. In other words, we would need a statistically significant correlation of -1 to be sure that the high scorer on the Moriarty scale was the one who was lying the most.

WATSON: So, how will we know who the murderer is?

HOLMES: Look for the motive, Watson. Find out why the murder occurred. Since nothing was taken and since the poor woman was repeatedly stabbed, I think we can say that this was a crime of passion, which suggests it was Quirk.

WATSON: Is there any other evidence pointing to Quirk?

HOLMES: Yes, the dog's strange behavior.

WATSON: What do you mean, the dog's strange behavior? The dog did nothing.

HOLMES: Precisely. Since the dog was on the grounds and was not drugged, the only reason it would not have barked at someone entering and leaving the grounds after midnight was that it knew the "intruder." The dog didn't know Zimski, but it did know Quirk.

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