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With respect to FWER control, the Bonferroni correction can be conservative if there are a large number of tests and/or the test statistics are positively correlated. [9] Multiple-testing corrections, including the Bonferroni procedure, increase the probability of Type II errors when null hypotheses are false, i.e., they reduce statistical power.
Multiple testing correction refers to making statistical tests more stringent in order to counteract the problem of multiple testing. The best known such adjustment is the Bonferroni correction, but other methods have been developed.
The procedures of Bonferroni and Holm control the FWER under any dependence structure of the p-values (or equivalently the individual test statistics).Essentially, this is achieved by accommodating a `worst-case' dependence structure (which is close to independence for most practical purposes).
The Holm–Bonferroni method is "uniformly" more powerful than the classic Bonferroni correction, meaning that it is always at least as powerful. There are other methods for controlling the FWER that are more powerful than Holm–Bonferroni. For instance, in the Hochberg procedure, rejection of () …
It is less stringent than the Bonferroni correction, but only slightly. For example, for α {\displaystyle \alpha } = 0.05 and m = 10, the Bonferroni-adjusted level is 0.005 and the Šidák-adjusted level is approximately 0.005116.
A procedure that goes from a small test-statistic to a large one will be called a step-up procedure. In a similar way, in a "step-down" procedure we move from a large corresponding test statistic to a smaller one.
The t-test p-value for the difference in means, and the regression p-value for the slope, are both 0.00805. The methods give identical results. This example shows that, for the special case of a simple linear regression where there is a single x-variable that has values 0 and 1, the t-test gives the same results as the linear regression. The ...
(This is a simple type of cross-validation and is often termed training-test or split-half validation.) Another remedy for data dredging is to record the number of all significance tests conducted during the study and simply divide one's criterion for significance (alpha) by this number; this is the Bonferroni correction. However, this is a ...