Standard Errors of Variance Components, Measurement Errors and Generalizability Coefficients for Crossed Designs

Abstract

AbstractEstimates of various variance components, universe score variance, measurement error variances, and generalizability coefficients, like all statistics, are subject to sampling variability, particularly in small samples. Such variability is quantified traditionally through estimated standard errors and/or confidence intervals. The paper derived new standard errors for all estimated statistics for two crossed designs (single‐facet design and two‐facet design) in generalizability theory. The derivation was based on the assumption of multivariate normal distribution for observation scores using delta method. The derivation was differentiating between fixed and random facets. The adequacy of the derived standard errors was examined using Mont Carlo simulation for the two designs under several test conditions and compared to the traditional existing methods as well as implementation on real data. Results showed that the derived standard errors for all estimators are converging to the empirical standard errors for all simulation conditions with the two designs for both normal and non‐normal continuous data.