Cortical Functional Mapping by High-Resolution EEG - Quantitative EEG Analysis Methods and Clinical Applications - page 364

assumptions did not hold. In this respect the good news is that the robustness of the
Student's test from the assumption of homoscedasticity and normality are generally
very high, and hence we can use the test even though there is no precise information
related to the Gaussian and the homoscedastic nature of the data [16]. However, the
appropriate statistic to deal with the heteroscedastic case is the Sattertwaite or
Cochran and Cox estimation of the standard error to insert in the formulation pre-
sented above. Such calculations can be found in a standard statistic textbook [16],
but they lead to results very similar to those obtained with the standard Student's
approach.
As a final statistical issue, it is well known that many univariate statistical tests
(like those presented in this application, one for each cortical dipole modeled) can
easily generate the appearance of false positive results (known as type I errors or
alpha inflation ). The large number of univariate tests performed results in statisti-
cally significant differences being found between two analyzed samples when no real
differences exist. The usual conservative approach in this case is to use a Bonferroni
correction, a procedure that simply defines as statistically significant at 5%, for
instance, all of the statistical results that are still significant when their probability is
divided by the number of univariate tests performed. Let N be the number of
univariate tests to be performed and t 0.05 be the statistical threshold for a single
univariate t -test to perform for our contrast at a 5% level of statistical significance.
We can state that the results will be statistically significant at the 5% level
Bonferroni corrected ( t 0.05Bonf *) all of those cortical dipoles that present a t -value
associated with a probability p 0.05 higher than

p * = p N (13.21)

005 . 005 .

Usually, the Bonferroni correction is a very conservative measure of statistical
differences between two groups during the execution of multiple univariate tests.
We can verify this with a practical example: If the number of cortical dipoles to be
tested on a simple realistic head model is about 3,000 ( N = 3,000) and the number of

EEG trials analyzed for the subject is 10 ( M = 10), the value for a single univariate

Student t -threshold will be t 0.05 = 2.26. This means that during the statistical compar-

isons on each of the 3,000 dipoles, all of the t- values higher than 2.26 will be
declared statistically significant at 5% (i.e., with a p 0.05 ). However, because we know
that many false positives could occurs due to the execution of multiple t -tests, by
adopting the Bonferroni procedure we will instead declare statistically significant at
5% Bonferroni corrected all of the statistical tests that returns a p value higher than

P 0.5Bonf = 005 3000 . = 0000016 .

Here is a value of t that generates such a probability for a population of 10 EEG
trials:

t 005 * = 11

.

This means that by using the Bonferroni correction, we can declare as statisti-
cally significant at 5% Bonferroni corrected all of those cortical areas that generate a