Bootstrap Procedures for Estimating Standard Errors of Estimated Variance Components for Two-Facet Designs∗

The estimation of standard errors of estimated variance components has long been a challenging task in generalizability theory. For two decades researchers have speculated about the potential applicability of the bootstrap for obtaining such estimates, but the same researchers have identified problems in using the bootstrap. Some of these problems are associated with the fact that bootstrap procedures generally produce biased estimates of variance components and their standard errors. Other problems are associated with the fact that, with ANOVA-type designs, there are many possibilities for obtaining bootstrap samples, and it has not been demonstrated that there is a single optimal bootstrap procedure for obtaining the “best” estimates for all variance components and standard errors. Using normal-data simulations, this paper suggests solutions to both of these problems for the p× i×h and p×(i :h) designs. Specifically, this paper proposes explicit bias-correction formulas and suggests rules for choosing a particular bootstrap procedure for particular variance components. The recommendations proposed are tentative, but they are largely confirmed by the simulations. Further investigations need to be conducted to extend these recommendations to dichotomous and polytomous data, and to more complicated designs.