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This is sort of like the opposite of the gamblers fallacy, although it's not the "inverse gamblers fallacy". For instance, if I observe that in some condition, the expression of Gene A is elevated with p-value 1e-3, and the expression of Gene B is elevated with p-value 1e-4, I might say that this event has p-value = 1e-7. Except that implicitly assumes these two genes are uncorrelated (which is not usually true in practice even under the null hypothesis of neither gene being associated with the condition).

Is there a name for this kind of error?

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    $\begingroup$ This error (which I don't really have a specific name for, but let's call it the "independence fallacy*) often goes hand in hand with the prosecutor's fallacy (conflating P(innocent|evidence) with P(evidence|innocent), also called the conditional probability fallacy) by first lining up a string of conditions that are treated as independent (making P(E|I) low). Possibly the most famous instance of your independence fallacy combined with the prosecutor's fallacy is People v. Collins $\endgroup$ – Glen_b Feb 1 '16 at 0:09
  • $\begingroup$ @Glen_b The prosecutor's fallacy sounds like it's mixing up likelihood and and posterior probability, not quite the same. This recently came up for me in the context of discussing the Sally Clark trial, where one "expert" witness makes both mistakes. "Independence Fallacy" sounds good. I was thinking "amateur statisticans fallacy" because it seems like a mistake people make when they know just a little bit of statistics. $\endgroup$ – Jacob Feb 1 '16 at 7:58
  • $\begingroup$ My characterization of it as P(E|I) vs P(I|E) comes from the wikipedia page. Not that this is any reason to think it's correct of course; the wikipedia page can easily be wrong. $\endgroup$ – Glen_b Feb 1 '16 at 8:01
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"Pseudo-replication" is one such name for the result of the error, if not the act of believing the premise. In a repeated measurements design, for instance, ignoring the correlation structure arising from the groups would make it look like you had more information than you actually did. (I remember an example where biology students divided tissue samples into smaller pieces to get a higher sample size and achieve significance.) But the naive point estimates would still be okay. In fact, sometimes they are used intentionally with the variance estimates corrected by a robust formula.

Another name might be "working assumption," though you wouldn't be guilty of belief either. Back when I worked with gene expression data, I remember the "independence working assumption" being used for constructing likelihoods for the vectors of gene expression measurements. As there were thousands of measured genes and the inter-dependencies were not understood, this independence assumption was tolerated.

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