Survey Methodology
Statistical methods for sampling cross-classified populations under constraints

by Louis-Paul Rivest^{Note 1}

Abstract

The article considers sampling designs for populations that can be represented as a $N\times M.$ matrix. For instance when investigating tourist activities, the rows could be locations visited by tourists and the columns days in the tourist season. The goal is to sample cells $\mathrm{<mo\; stretchy="false">(</mo>}i\mathrm{,}j\mathrm{<mo\; stretchy="false">)</mo>}$ of the matrix when the number of selections within each row and each column is fixed a priori. The $i^{\text{th}}$ row sample size represents the number of selected cells within row $i;$ the $j^{\text{th}}$ column sample size is the number of selected cells within column $j.$ A matrix sampling design gives an $N\times M$ matrix of sample indicators, with entry 1 at position $\mathrm{<mo\; stretchy="false">(</mo>}i\mathrm{,}j\mathrm{<mo\; stretchy="false">)</mo>}$ if cell $\mathrm{<mo\; stretchy="false">(</mo>}i\mathrm{,}j\mathrm{<mo\; stretchy="false">)</mo>}$ is sampled and 0 otherwise. The first matrix sampling design investigated has one level of sampling, row and column sample sizes are set in advance: the row sample sizes can vary while the column sample sizes are all equal. The fixed margins can be seen as balancing constraints and algorithms available for selecting such samples are reviewed. A new estimator for the variance of the Horvitz-Thompson estimator for the mean of survey variable $y$ is then presented. Several levels of sampling might be necessary to account for all the constraints; this involves multi-level matrix sampling designs that are also investigated.

Key Words: Balanced sampling; Creel surveys; Cube method; Multi-level sampling; Monte Carlo simulation; Variance estimation.

Table of contents

• Section 1. Introduction
• Section 2. The properties of the matrix sampling design with fixed row and column totals
• Section 3. A conditional matrix sampling design
• Section 4. A new estimator for the between row covariance
• Section 5. A multi-level matrix design
• Section 6. Discussion
• Acknowledgements
• Appendix
• References

How to cite

Rivest, L.-P. (2023). Statistical methods for sampling cross-classified populations under constraints. Survey Methodology, Statistics Canada, Catalogue No. 12-001-X, Vol. 49, No. 2. Paper available at http://www.statcan.gc.ca/pub/12-001-x/2023002/article/00011-eng.htm.

Note

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