Design and relative efficiency in two-phase studies

Many two-phase sampling designs have been applied in practice to obtain efficient estimates of regression parameters while minimizing the cost of data collection. This research investigates two-phase sampling designs for so-called expensive variable problems, and compares them with one-phase designs. Closed form expressions for the asymptotic relative efficiency of maximum likelihood estimators from the two designs are derived for parametric normal models, providing insight into the available information for regression coefficients under the two designs. We further discuss when we should apply the two-phase design and how to choose the sample sizes for two-phase samples. Our numerical study indicates that the results can be applied to more general settings.