Estimation Under Cross-Classified Sampling With Application to a Childhood Survey

ABSTRACT

The cross-classified sampling design consists in drawing samples from a two-dimensional population, independently in each dimension. Such design is commonly used in consumer price index surveys and has been recently applied to draw a sample of babies in the French Longitudinal Survey on Childhood, by crossing a sample of maternity units and a sample of days. We propose to derive a general theory of estimation for this sampling design. We consider the Horvitz–Thompson estimator for a total, and show that the cross-classified design will usually result in a loss of efficiency as compared to the widespread two-stage design. We obtain the asymptotic distribution of the Horvitz–Thompson estimator and several unbiased variance estimators. Facing the problem of possibly negative values, we propose simplified nonnegative variance estimators and study their bias under a super-population model. The proposed estimators are compared for totals and ratios on simulated data. An application on real data from the French Longitudinal Survey on Childhood is also presented, and we make some recommendations. Supplementary materials for this article are available online.

KEYWORDS:

• Analysis of variance
• Horvitz–Thompson estimator
• Independence
• Invariance
• Sen–Yates–Grundy estimator
• Two-stage sampling

Supplementary Materials

readme: description of the supplemental files. (txt file)

CodeR_functions: basic functions required to calculate estimators. (R file)

CodeR_Tables: commands that calculate and display the results in Table 1 and Table 2 (call the CodeR_functions). (R file)

Acknowledgment

The authors would like to thank the Editor, the Associate Editor, and the referee for useful comments that led to an improvement of the article.