> The requirement of "experiment independence" is a very commonly
> misunderstood aspect of the Bonferroni method. In fact, Bonferroni
> developed the method to deal with the fact that there are instances in
> which you want to compare two means/effects and you know there is some
> degree of interdependence, but you have no way to ascertain the nature or
> magnitude of that interdependence. That is, Bonferroni developed the most
> conservative correction for multiple comparisons to deal with the worst
> case senarios of very high levels of interdependence. Thus, exactly the
> opposite of the misunderstood requirement you inquired about.
>
> "This calculus does not require the hypothesis that the assured lives
> should be independent, as is usual in treatments of this problem."
>
> Bonferroni CE. 1935. Il calcolo delle assicurazioni su gruppi di teste. In
> Studi in onore del professore Salvatore Ortu Carboni, ed. SO Carboni, pp.
> 13-60. Rome: Tip. del Senato.
>
Dear Irwin,
I have some reservations on your assessment on the Bonferroni. Let me explain
why:
1) "The Bonferroni inequality has very little to do with the Bonferroni
correction used in multiple comparisons".
The reference you point at [1], that is an application, and the more theoretical
one of 1936 [2] contain the so called Bonferroni inequalities but they have
nothing to do with the famous-infamous Bonferroni's correction. The latter is
indeed based just on Boole's inequality (Seneta, 1993).
This is a wonderful example of "Stigler's law of eponymy" [4] that asserts that,
in science, discoveries usually bring the name of the wrong person.
Therefore the abstract you are citing refers to the wrong subject.
2)You are correct in saying that Bonferroni's correction does not require
indepence, but if depence exists than the correction is too conservative. In the
limit, a correlation=1 would not require any type of correction.
3)I believe that what Marco was referring to is technically equivalent but a
little more subtle from the inferential point of view, something we can refer to
"inferential independence". This indeed is very consistent with the
Neymann-Pearson paradigm of testing, to which multiple corrections belong, that
requires the set of multiple hypotheses to be unrelated a-priori. By unrelated
it is meant that not only no "measureable correlation" exists, but also that
inferences should be independent.
Marco seems to doubt that this is the case in our field. All I can say is that I
completely agree with him.
Best regards
Federico
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References:
[1] Bonferroni CE (1935). Il calcolo delle assicurazioni su gruppi di teste. In
Studi in onore del professore Salvatore Ortu Carboni, ed. SO Carboni, pp.13-60.
Rome: Tipografia del Senato.
[2] Bonferroni CE(1936) "Teoria statistica delle classi e calcolo delle
probabilita'," Pubblicazioni del Reale Istituto Superiore di Scienze Economiche
e Commerciali di Firenze 8: 3-62.
[3] Seneta E (1993) "Probability inequalities and Dunnett's test," In F M Hoppe,
Multiple Comparisons, Selections and Applications in Biometry, New York: Marcel
Dekker pp. 29-45.
[4] Stigler SM (1980) "Stigler's law of eponymy," Transactions NY Acad. Sci.
Series 2, 39 (1980), 147-157.
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