Bonferroni of Bergamo

Bonferroni corrected

Uncorrected (random results)

I enjoyed a fine afternoon in the old Citta Alta of Bergamo in northern Italy – a city in the sky that the Venetians, at the height of their power as the “most serene republic,” walled off as their western-most outpost in the 17 century.

In statistical circles this town is most notable for being the birthplace of Carlo Emilio Bonferroni.  You may have heard of the “Bonferroni Correction” – a method that addresses the problem of multiple comparisons.

For example, when I worked for General Mills the head of quality control in Minneapolis would mix up a barrel of flour and split it into 10 samples, carefully sealed in air-tight containers, for each of the mills to test in triplicate for moisture.  At this time I had just learned how to do the t-test for comparing two means.  Fortunately for the various QC supervisors, no one asked me to analyze the results, because I would have simply taken the highest moisture value and compared it to the lowest one.  Given that there are 45 possible pair-wise comparisons (10*9/2), this biased selection (high versus low) is likely to produce a result that tests significant at the 0.05 level (1 out of 20).

This is a sadistical statistical scheme for a Machiavellian manager because of the intimidating false positives (Type I error).  In the simulation pictured, using the random number generator in Design-Expert® software (based on a nominal value of 100), you can see how, with the significance threshold set at 0.05 for the least-significant-difference (LSD) bars (derived from t-testing), the supervisors of Mills 4 and 7 appear to be definitely discrepant.  (Click on the graphic to expand the view.) Shame on them!  Chances are that the next month’s inter-laboratory collaborative testing would cause others to be blamed for random variation.

In the second graph I used a 0.005 significance level – 1/10th as much per the Bonferroni Correction.  That produces a more sensible picture — all the LSD bars overlap, so no one can be fingered for being out of line.

By the way, the overall F-test on this data set produces a p value of 0.63 – not significant.

Since Bonferroni’s death a half-century ago in 1960, much more sophisticated procedures have been developed to correct for multiple comparisons.  Nevertheless, by any measure of comparative value, Bergamo can consider this native son as one of those who significantly stood above most others in terms of his contributions to the world.


  1. #1 by Eric Kvaalen on June 16, 2010 - 11:33 am

    The so-called Bonferroni test is based on the rather trivial Boule inequality, whereas Bonferroni gave a much more sophisticated set of inequalities. So it’s actually a misnomer which, in my opinion, doesn’t do him honor.

You must be logged in to post a comment.