Familywise and Experimentwise Error Rates
Tukey (1953) developed the concept of a familywise error rate as the probability of making a Type I error among a specified group, or "family," of tests. Based on Tukey (1953), Ryan (1959) proposed the related concept of an ''experimentwise error rate'', which is the probability of making a Type I error in a given experiment. Hence, an experimentwise error rate is a familywise error rate for all of the tests that are conducted within an experiment. As Ryan (1959, Footnote 3) explained, an experiment may contain two or more families of multiple comparisons, each of which relates to a particular statistical inference and each of which has its own separate familywise error rate. Hence, familywise error rates are usually based on theoretically informative collections of multiple comparisons. In contrast, an experimentwise error rate may be based on a co-incidental collection of comparisons that refer to a diverse range of separate inferences. Consequently, some have argued that it may not be useful to control the experimentwise error rate. Indeed, Tukey was against the idea of experimentwise error rates (Tukey, 1956, personal communication, in Ryan, 1962, p. 302).Background
Within the statistical framework, there are several definitions for the term "family": * Hochberg & Tamhane (1987) defined "family" as "any collection of inferences for which it is meaningful to take into account some combined measure of error". * According to Cox (1982), a set of inferences should be regarded a family: # To take into account the selection effect due to data dredging # To ensure simultaneous correctness of a set of inferences as to guarantee a correct overall decision To summarize, a family could best be defined by the potential selective inference that is being faced: A family is the smallest set of items of inference in an analysis, interchangeable about their meaning for the goal of research, from which selection of results for action, presentation or highlighting could be made ( Yoav Benjamini).Classification of multiple hypothesis tests
Definition
The FWER is the probability of making at least oneControlling procedures
Some classical solutions that ensure strong level FWER control, and some newer solutions exist.The Bonferroni procedure
* Denote by the ''p''-value for testing * reject ifThe Šidák procedure
* Testing each hypothesis at level is Sidak's multiple testing procedure. * This procedure is more powerful than Bonferroni but the gain is small. * This procedure can fail to control the FWER when the tests are negatively dependent.Tukey's procedure
* Tukey's procedure is only applicable forHolm's step-down procedure (1979)
* Start by ordering the ''p''-values (from lowest to highest) and let the associated hypotheses be * Let be the minimal index such that * Reject the null hypotheses . If then none of the hypotheses are rejected. This procedure is uniformly more powerful than the Bonferroni procedure. The reason why this procedure controls the family-wise error rate for all the m hypotheses at level α in the strong sense is, because it is a closed testing procedure. As such, each intersection is tested using the simple Bonferroni test.Hochberg's step-up procedure
Hochberg's step-up procedure (1988) is performed using the following steps: * Start by ordering the ''p''-values (from lowest to highest) and let the associated hypotheses be * For a given , let be the largest such that * Reject the null hypotheses Hochberg's procedure is more powerful than Holms'. Nevertheless, while Holm’s is a closed testing procedure (and thus, like Bonferroni, has no restriction on the joint distribution of the test statistics), Hochberg’s is based on the Simes test, so it holds only under non-negative dependence.Dunnett's correction
Charles Dunnett (1955, 1966) described an alternative alpha error adjustment when ''k'' groups are compared to the same control group. Now known as Dunnett's test, this method is less conservative than the Bonferroni adjustment.Scheffé's method
Resampling procedures
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). But such an approach is conservative if dependence is actually positive. To give an extreme example, under perfect positive dependence, there is effectively only one test and thus, the FWER is uninflated. Accounting for the dependence structure of the ''p''-values (or of the individual test statistics) produces more powerful procedures. This can be achieved by applying resampling methods, such as bootstrapping and permutations methods. The procedure of Westfall and Young (1993) requires a certain condition that does not always hold in practice (namely, subset pivotality). The procedures of Romano and Wolf (2005a,b) dispense with this condition and are thus more generally valid.Harmonic mean ''p''-value procedure
The harmonic mean ''p''-value (HMP) procedure provides a multilevel test that improves on the power of Bonferroni correction by assessing the significance of ''groups'' of hypotheses while controlling the strong-sense family-wise error rate. The significance of any subset of the tests is assessed by calculating the HMP for the subset, where are weights that sum to one (i.e. ). An approximate procedure that controls the strong-sense family-wise error rate at level approximately rejects the null hypothesis that none of the ''p''-values in subset are significant when (where ). This approximation is reasonable for small (e.g. ) and becomes arbitrarily good as approaches zero. An asymptotically exact test is also available (see main article).Alternative approaches
FWER control exerts a more stringent control over false discovery compared to false discovery rate (FDR) procedures. FWER control limits the probability of ''at least one'' false discovery, whereas FDR control limits (in a loose sense) the expected proportion of false discoveries. Thus, FDR procedures have greaterReferences
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