  Simultaneous testing of one hypothesis at multiple alpha levels can be
performed within a conventional Neyman-Pearson framework. This is achieved by
treating the hypothesis as a family of hypotheses, each member of which
explicitly concerns test level as well as effect size. Such testing encourages
researchers to think about error rates and strength of evidence in both the
statistical design and reporting stages of a study. Here, we show that these
multi-alpha level tests can deliver acceptable expected total error costs. We
first present formulas for expected error costs from single alpha and multiple
alpha level tests, given prior probabilities of effect sizes that have either
dichotomous or continuous distributions. Error costs are tied to decisions,
with different decisions assumed for each of the potential outcomes in the
multi-alpha level case. Expected total costs for tests at single and multiple
alpha levels are then compared with optimal costs. This comparison highlights
how sensitive optimization is to estimated error costs and to assumptions about
prevalence. Testing at multiple default thresholds removes the need to formally
identify decisions, or to model costs and prevalence as required in
optimization approaches. Although total expected error costs with this approach
will not be optimal, our results suggest they may be lower, on average, than
when so-called optimal test levels are based on mis-specified models.
