Accurate saddlepoint approximations for clustered rank tests under randomized block urn design

Haidy A. Newer; Bader S. Alanazi; Haidy A. Newer; Bader S. Alanazi

doi:10.3934/math.2026220

AIMS Mathematics

2026, Volume 11, Issue 3: 5340-5364. doi: 10.3934/math.2026220

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Accurate saddlepoint approximations for clustered rank tests under randomized block urn design

Haidy A. Newer ^{1
,
,},
Bader S. Alanazi ²

1.
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo 11511, Egypt
2.
Department of Mathematics, College of Science, Northern Border University, Arar 73222, Saudi Arabia

Received: 03 December 2025 Revised: 20 January 2026 Accepted: 02 February 2026 Published: 02 March 2026
MSC : 62F12, 62G10, 62F03, 62J12, 62N01, 62C12, 62G09, 60K35

Randomized block urn design is a clinical trial design that is increasingly being used as a means of treatment allocation to balance the treatment allocation. With repeated measures or multi-centre trials though, the corresponding complex structures of dependency invalidate the standard asymptotic approximations, even in small to moderate samples. In this paper, a high-accuracy saddlepoint approximation model of a linear rank test is established with the dual conditions of cluster sampling and adaptive urn randomization. We obtained the joint cumulant generating function of the test statistic conditional in the realized block allocation counts, and the accurate calculation of mid-$ p $-values that takes into consideration the discreteness of rank scores. We have a wide variety of classes of score functions, such as the log-rank, GehanWilcoxon and MannWhitney statistics. Wide-scale simulation experiments showed that the suggested saddlepoint algorithm is much more effective at controlling Type-Ⅰ error rates and preserving nominal coverage probabilities of confidence intervals, and is comparable to computationally-intensive permutation benchmarks even with strong intra-cluster correlation. Its approach was demonstrated by applying it to oncology and ophthalmology trial data demonstrating its soundness in finite-sample cases.
- saddlepoint approximation,
- clustered data,
- linear rank tests,
- randomized block urn design,
- log-rank test,
- Gehan–Wilcoxon test,
- Mann–Whitney test,
- Prentice–Wilcoxon test,
- confidence intervals
Citation: Haidy A. Newer, Bader S. Alanazi. Accurate saddlepoint approximations for clustered rank tests under randomized block urn design[J]. AIMS Mathematics, 2026, 11(3): 5340-5364. doi: 10.3934/math.2026220

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Abstract

Randomized block urn design is a clinical trial design that is increasingly being used as a means of treatment allocation to balance the treatment allocation. With repeated measures or multi-centre trials though, the corresponding complex structures of dependency invalidate the standard asymptotic approximations, even in small to moderate samples. In this paper, a high-accuracy saddlepoint approximation model of a linear rank test is established with the dual conditions of cluster sampling and adaptive urn randomization. We obtained the joint cumulant generating function of the test statistic conditional in the realized block allocation counts, and the accurate calculation of mid-$ p $-values that takes into consideration the discreteness of rank scores. We have a wide variety of classes of score functions, such as the log-rank, GehanWilcoxon and MannWhitney statistics. Wide-scale simulation experiments showed that the suggested saddlepoint algorithm is much more effective at controlling Type-Ⅰ error rates and preserving nominal coverage probabilities of confidence intervals, and is comparable to computationally-intensive permutation benchmarks even with strong intra-cluster correlation. Its approach was demonstrated by applying it to oncology and ophthalmology trial data demonstrating its soundness in finite-sample cases.

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