Many analyses of microwave background experiments neglect the correlation of noise in different frequency or polarization channels. We show that these correlations, should they be present, can lead to severe misinterpretation of an experiment. In particular, correlated noise arising from either electronics or atmosphere may mimic a cosmic signal. We quantify how the likelihood function for a given experiment varies with noise correlation, using both simple analytic models and actual data. For a typical microwave background anisotropy experiment, noise correlations at the level of 1% of the overall noise can seriously reduce the significance of a given detection.
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