An investigation of two-stage tests. Two-stage tests may be defined in terms of a combination function for the $p$-values of the separate stages, or alternatively by specifying a conditional error function, i.e., the conditional probability for an erroneous rejection given the first stage. Examples have been published suggesting that these two approaches are essentially equivalent. We provide a formal link between them that yields a general framework for two-stage tests. Our viewpoint leads to an overall $p$-value notion that covers different previously proposed concepts, and it allows an easy construction of new two-stage tests. One particular test is further characterized.
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