Survey Methodology
Variance estimation in multi-phase calibration

by Noam Cohen, Dan Ben-Hur and Luisa BurckNote 1

  • Release date: June 22, 2017


The derivation of estimators in a multi-phase calibration process requires a sequential computation of estimators and calibrated weights of previous phases in order to obtain those of later ones. Already after two phases of calibration the estimators and their variances involve calibration factors from both phases and the formulae become cumbersome and uninformative. As a consequence the literature so far deals mainly with two phases while three phases or more are rarely being considered. The analysis in some cases is ad-hoc for a specific design and no comprehensive methodology for constructing calibrated estimators, and more challengingly, estimating their variances in three or more phases was formed. We provide a closed form formula for the variance of multi-phase calibrated estimators that holds for any number of phases. By specifying a new presentation of multi-phase calibrated weights it is possible to construct calibrated estimators that have the form of multi-variate regression estimators which enables a computation of a consistent estimator for their variance. This new variance estimator is not only general for any number of phases but also has some favorable characteristics. A comparison to other estimators in the special case of two-phase calibration and another independent study for three phases are presented.

Key Words: Calibration; Multi-phase sampling; Generalized regression.

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How to cite

Cohen, N., Ben-Hur, D. and Burck, L. (2017). Variance estimation in multi-phase calibration. Survey Methodology, Statistics Canada, Catalogue No. 12-001-X, Vol. 43, No. 1. Paper available at


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