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Bonferroni Test Definition
It is a type of multiple comparison testing used in statistical analysis when an experimenter performs enough tests. After performing the tests he or she will eventually end up with a result that shows statistical significance even if there is none. If a particular test results in correct results 99% of the time, running 100 tests could lead to a false result somewhere in the mix and the Bonferroni's test attempts to prevent data from incorrectly appearing to be statistically significant by lowering the alpha value. The Bonferroni test which is also known as the "Bonferroni correction" or "Bonferroni adjustment" suggests that the "p" value for each test must be equal to alpha divided by the number of tests.
Bonferroni method is valid for equal as well as unequal sample sizes and it is being restricted to only linear combination of treatment level which means pairwise comparisons as well as contrasts which are special cases of linear combinations.
Since the Bonferroni is a multiple-comparison correction which is used when several dependent or independent statistical testing are being performed simultaneously when an alpha value is given where the may be appropriate for each individual comparison. It is not used for the set of all comparisons and in order to avoid a lot of spurious positives, the alpha value has to be decreased to account for the number of comparisons that is being performed.
The simplest and most conservative approach is the Bonferroni correction which assigns the alpha value for the entire set of comparisons which is equal to by taking the alpha value for each comparison which is equal to. Explicitly, given tests for hypotheses where under the assumption that all hypotheses are false, and if the individual test critical values are . Then the experiment-wide critical value is and in the equation form it is represented as follows:
for, then
which is from the Bonferroni's inequalities.
There are also other types of multiple comparison tests such as Scheffe’s test as well as the Tukey-Kramer method test and a major criticism of the Bonferroni tests is that it is too conservative and so it may fail to catch some significant findings.
