The Ten Cases of Auxiliary Information for Calibration in TwoPhase Sampling Victor M. Estevao and CarlErik Särndal Abstract: Calibration is commonly used to produce estimation weights in sample surveys. Calibration weights satisfy a set of calibration equations that make use of the specified auxiliary information. In a twophase design, the information used for calibration can take different forms. The case that we call cromplete auxiliary infomation arises when information is available at the level of the population for one set of auxiliary variables and at the lower level of the firstphase sample for another set of auxiliary variables. In practice, we may be restricted to a calibration on a subset of the complete auxiliary information, or we may decide to discard some of the complete information if no significant loss of efficiency occurs. We show that there are exactly nine different subsets of the complete information, for a total of ten different cases of auxiliary information. We propose one calibration estimator in each of these ten cases. In general, the more extensive the auxiliary information, the better the precision of the resulting estimates. However, there are sometimes surprising exceptions to this, as illustrated both by our theoretical results and by our simulation. We study the precision of the calibration estimators in the ten cases, both theoretically (by deriving the sum of the two variance components) and empirically (by repeated sampling from different types of populations). We suggest a simple approach to determine the best use of auxiliary information. Keywords: Designbased inference; linear regression representation; calibrated weights; regression residuals; variance estimation.
