Three defensible approaches are:
  • Use simultaneous calibration, for example Bonferroni correction: give each of eight hospitals outside probability at most . Replace 1.96 by in the normal bands, giving familywise error at most 5% when the marginal tests are calibrated.
  • Apply the Holm step-down procedure to valid two-sided binomial p-values. Order them and compare the jth smallest with until the first nonrejection. This controls familywise error while often being less conservative than giving every hospital the same Bonferroni threshold.
  • Treat an initial signal as a screen and require prespecified confirmation in independent new data before declaring a persistent deviation, alongside checking case mix and coding. Calibration must include both stages; unrestricted repeated testing until an alarm appears would increase, rather than solve, the problem.
A control-limit crossing prompts investigation; it is not itself a verdict on a hospital. Wider or adjusted thresholds also reduce sensitivity, so the false-alarm objective and the cost of missed genuine problems should be made explicit.

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