A new face on two-phase sampling with calibration estimators - ARCHIVED

This paper provides a framework for estimation by calibration in two phase sampling designs. This work grew out of the continuing development of generalized estimation software at Statistics Canada. An important objective in this development is to provide a wide range of options for effective use of auxiliary information in different sampling designs. This objective is reflected in the general methodology for two phase designs presented in this paper.

We consider the traditional two phase sampling design. A phase one sample is drawn from the finite population and then a phase two sample is drawn as a sub sample of the first. The study variable, whose unknown population total is to be estimated, is observed only for the units in the phase two sample. Arbitrary sampling designs are allowed in each phase of sampling. Different types of auxiliary information are identified for the computation of the calibration weights at each phase. The auxiliary variables and the study variables can be continuous or categorical.

The paper contributes to four important areas in the general context of calibration for two phase designs:
(1) Three broad types of auxiliary information for two phase designs are identified and used in the estimation. The information is incorporated into the weights in two steps: a phase one calibration and a phase two calibration. We discuss the composition of the appropriate auxiliary vectors for each step, and use a linearization method to arrive at the residuals that determine the asymptotic variance of the calibration estimator.
(2) We examine the effect of alternative choices of starting weights for the calibration. The two "natural" choices for the starting weights generally produce slightly different estimators. However, under certain conditions, these two estimators have the same asymptotic variance.
(3) We re examine variance estimation for the two phase calibration estimator. A new procedure is proposed that can improve significantly on the usual technique of conditioning on the phase one sample. A simulation in section 10 serves to validate the advantage of this new method.
(4) We compare the calibration approach with the traditional model assisted regression technique which uses a linear regression fit at two levels. We show that the model assisted estimator has properties similar to a two phase calibration estimator.