Family-wise error rate

In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests.

Tukey coined the terms experimentwise error rate and "error rate per-experiment" to indicate error rates that the researcher could use as a control level in a multiple hypothesis experiment.

Within the statistical framework, there are several definitions for the term "family":

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).

The following table defines the possible outcomes when testing multiple null hypotheses. Suppose we have a number m of null hypotheses, denoted by: H₁, H₂, ..., H_m. Using a statistical test, we reject the null hypothesis if the test is declared significant. We do not reject the null hypothesis if the test is non-significant. Summing each type of outcome over all H_i  yields the following random variables:

In ${\displaystyle m}$ hypothesis tests of which ${\displaystyle m_{0}}$ are true null hypotheses, ${\displaystyle R}$ is an observable random variable, and ${\displaystyle S}$, ${\displaystyle T}$, ${\displaystyle U}$, and ${\displaystyle V}$ are unobservable random variables.

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