Survey Methodology
Improving the coverage of confidence intervals with respect to degrees of freedom: Application to the Canadian Census

Abstract

Confidence intervals are very often constructed based on a probability distribution that uses a certain number of degrees of freedom as a parameter. This is the case with the Student and the modified Wilson confidence intervals, discussed in this article, which use quantiles from the Student distribution where the number of degrees of freedom is generally unknown. For the length of a confidence interval to be representative of the reliability of an estimate, the actual coverage rate must match the nominal rate. To that end, the number of degrees of freedom in the probability distribution used in practice to calculate the confidence interval must be estimated as precisely as possible. An approximate rule is often used, although it tends to overestimate the actual number of degrees of freedom. In this article, a more precise version of degrees of freedom, derived from the Satterthwaite approximation, is obtained in the context of the Canadian Census of Population. The sampling design is equivalent to a simple random design without replacement, cluster-stratified, and the variance estimation method is an adaptation of the balanced repeated replication method. An explicit expression of the degrees of freedom is obtained under these conditions, enabling the factors influencing them to be identified. For comparison, the degree of freedom formula is also established for the conventional variance estimator. A simulation study shows that using this version of degrees of freedom corrects the undercoverage problem observed with the approximate rule, showing the importance of accurately assessing this number.

Key Words: Balanced repeated replication method; Canadian Census of Population; Confidence intervals; Coverage; Degrees of freedom; Small area estimation.

Table of contents

• Section 1. Introduction
• Section 2. Satterthwaite approximation
• Section 3. Framework and notation
• Section 4. Construction of a confidence interval
• Section 5. Partially balanced repeated replication epsilon (PBRR- ℇ) method
• Section 6. Degrees of freedom
• Section 7. Simulations
• Section 8. Conclusion
• Acknowledgements
• Appendix
• References

How to cite

Toupin, M.-H. and Martin, V. (2026). Improving the coverage of confidence intervals with respect to degrees of freedom: Application to the Canadian Census. Survey Methodology, 52(1), 187-211. Available at: http://www.statcan.gc.ca/pub/12-001-x/2026001/article/00005-eng.pdf.

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