We present a family of tests for detecting initialization bias in the mean of a simulation output series using a hypothesis testing framework. The null hypothesis is that the output mean does not change throughout the simulation run. The alternative hypothesis specifies a general transient mean function. The tests are asymptotically optimal based on cumulative sums of deviations about the sample mean. A particular test in this family is applied to a variety of simulation models. The test requires very modest computation and appears to be both robust and powerful.