Optimal linear estimation in two-phase sampling
Section 1. Introduction
The two-phase sampling design, also called double sampling, has traditionally been used in sample surveys as a cost-effective survey method. In the first phase, a relatively large sample is drawn from the target population to provide auxiliary information that is inexpensive to collect. This sample forms a highly informative frame from which a subsample is drawn in the second phase to collect information on the items of interest. Also, two-phase sampling has been increasingly used as a mechanism for handling nonresponse. Särndal, Swensson and Wretman (1992) provide an extensive account of such uses of two-phase sampling. Groves and Heeringa (2006), and Brick and Tourangeau (2017) discuss the important role of two-phase sampling in responsive designs when costly actions are taken for reduction of non-response bias. Other applications of two-phase sampling, which have emerged in recent survey practice, involve various forms of integration of separate surveys. In one such case, a first-phase sample serves as a frame for the second-phase sample for a multitude of similar surveys (Turmelle and Beaucage, 2013). In another case, a primary large survey is used as a frame for another smaller survey with a larger set of survey items (Australian Bureau of Statistics, 2004).
Auxiliary information in two-phase sampling may be available at different levels. Some information is at the level of the whole population, and other information is at the level of the first-phase sample or the second-phase sample. Much research has been devoted to the use of such information for improved estimation of population totals or means; see Särndal et al. (1992), Hidiroglou and Särndal (1998), Hidiroglou (2001), Estevao and Särndal (2002, 2009), Wu and Luan (2003), Chen and Kim (2014), and references therein. In general, two approaches are identified in this literature for incorporating auxiliary information into the estimation process. The generalized regression approach and the calibration approach; the two phases of sampling imply two regression fits or two successive calibrations. Under certain conditions the two approaches lead to identical estimators, but this is not so in general. Variance estimation of these two-phase estimators has been studied extensively; see, for example, Sitter (1997), Fuller (1998), Kim and Sitter (2003), Kim, Navarro and Fuller (2006), Hidiroglou, Rao and Haziza (2008), Kim and Yu (2011), Beaumont, Beliveau and Haziza (2015).
Irrespective of the regression or calibration formulation of the existing estimation procedures, the resulting estimators for a target variable are in effect linear combinations of Horvitz-Thompson estimators of various totals (or means), including the estimator for the target variable derived from the second-phase sample and estimators for auxiliary variables derived from both first-phase and second-phase sample. Taking a formal approach to optimal estimation, in this paper we consider the most efficient linear combination of available estimators from both phases, based on the principle of best linear unbiased estimation. We show that the derived, in analytic form, best linear unbiased estimator (BLUE) possesses a useful orthogonality property and that it can be alternatively constructed as calibration estimator, linear in the values of the associated variable and incorporating the auxiliary information into the calibrated design weights. Estimation of the unknown coefficients of this BLUE, using all available auxiliary information from both phases of sampling, gives an “optimal” estimator, analogous to the single-phase optimal regression estimator of Montanari (1987) and Rao (1994). This estimator is a large-sample approximation of the BLUE, with the estimated coefficients minimizing its estimated approximate (large sample) variance, and preserving the orthogonality property of the BLUE. With a proper reformulation of the BLUE, the optimal estimator can also be obtained through a suitable calibration procedure. The distinctive feature of such calibration is the convenient one-step procedure of aligning estimates from the two phases using the combined first-and-second phase samples. Optimal estimation is feasible for certain two-phase designs that are used often in large scale surveys. For general designs, an alternative one-step calibration procedure gives a novel generalized regression estimator as a convenient approximation to the optimal estimator.
The proposed general method of estimation guides the construction of calibration estimators in any particular case of two-phase survey, making the most effective use of the available auxiliary information. It also provides an insight into existing less efficient estimation methods when these are placed into the framework of optimal estimation. The advantages of the proposed method over existing methods are shown both theoretically and through a simulation study.
The paper is organized as follows. The structure of the two-phase sampling design, and notation, are introduced in Section 2. The derivation of the BLUE for the standard type of auxiliary information in two-phase sampling, and its alternative construction as a calibration estimator, are described in Section 3. The two-phase optimal estimator and its calibration equivalent are presented in Section 4. The approximation of the optimal estimator by a generalized regression estimator is discussed in Section 5. Comparisons with existing methods are presented in Section 6. A simulation study is presented in Section 7. The paper concludes with a discussion in Section 8.
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