Non‐negative Gaussian estimation of variance components in random effects models

When used to estimate variance components (VCs), confidence intervals (CIs) can be truncated at zero, have a point estimate not in the quoted CI, be empty with positive probability, or be all‐inclusive. This is because they have conflicting dual roles, since they are considered to cover the parameter with a specified probability while also indicating the precision of an estimate. Such results provide counterexamples to the belief that CIs indicate precise estimation and are a validating principle. Repetitions with similar results do not provide a scientific proof—which is disproven by a single counterexample. Our Gaussian theory of error, initiated by J.W. Tukey, only indicates the precision of an estimate, and provides a useful estimate for a unique sample. Simulated distributions of propagated error and margins of error can estimate VCs and non‐negative functions of VCs in balanced random effects models, and in one‐way unbalanced random effects models, providing useful estimates. Decisions made after Gaussian estimation are independent acts.