Survey Methodology
A note on the concept of invariance in two-phase sampling designs

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by Jean-François Beaumont and David HazizaNote 1

  • Release date: December 20, 2016


Two-phase sampling designs are often used in surveys when the sampling frame contains little or no auxiliary information. In this note, we shed some light on the concept of invariance, which is often mentioned in the context of two-phase sampling designs. We define two types of invariant two-phase designs: strongly invariant and weakly invariant two-phase designs. Some examples are given. Finally, we describe the implications of strong and weak invariance from an inference point of view.

Key Words: Double expansion estimator; Horvitz-Thompson estimator; Strong invariance; Two-phase sampling; Weak invariance.

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How to cite

Beaumont, J.-F., and Haziza, D. (2016). A note on the concept of invariance in two-phase sampling designs. Survey Methodology, Statistics Canada, Catalogue No. 12-001-X, Vol. 42, No. 2. Paper available at


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