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Inference on overlap index: with an application to cancer data

  • Raju Dey , Arne C. Bathke und Somesh Kumar EMAIL logo
Veröffentlicht/Copyright: 5. September 2025
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The International Journal of Biostatistics
Aus der Zeitschrift The International Journal of Biostatistics

Abstract

The quantification of overlap between two distributions has applications in various fields of biology, medical, genetic, and ecological research. In this article, new overlap and containment indices are considered for quantifying the niche overlap between two species/populations. Some new properties of these indices are established and the problem of estimation is studied, when the two distributions are exponential with different scale parameters. We propose several estimators and compare their relative performance with respect to different loss functions. The asymptotic normality of the maximum likelihood estimators of these indices is proved under certain conditions. We also obtain confidence intervals of the indices based on three different approaches and compare their average lengths and coverage probabilities. The point and confidence interval procedures developed here are applied on a breast cancer data set to analyze the similarity between the survival times of patients undergoing two different types of surgery. Additionally, the similarity between the relapse free times of these two sets of patients is also studied.

Keywords: confidence intervals; containment indices; maximum likelihood estimators; overlap index; parametric bootstrap
Corresponding author: Somesh Kumar, Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India, E-mail: 

Funding source: Salzburger Landesregierung

Award Identifier / Grant number: 20204-WISS/225/197-2019, 20102-F1901166-KZP

Funding source: Council of Scientific and Industrial Research, India

Award Identifier / Grant number: 09/0081(11244)/2021-EMR-I

Acknowledgments

All three authors thank an associate editor and two anonymous reviewers for their valuable comments and constructive suggestions that have significantly enhanced the quality and the content of the manuscript. The first author gratefully acknowledges the financial assistance provided by the ``Council of Scientific and Industrial Research (CSIR), Government of India’’ under grant No. 09/0081(11244)/2021-EMR-I. The second author gratefully acknowledges the generous support through the State of Salzburg WISS 2025 project “IDA-Lab Salzburg” (20204-WISS/225/197–2019 and 20102-F1901166-KZP).

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: Council of Scientific and Industrial Research (CSIR), Government of India. State of Salzburg WISS 2025 project “IDA-Lab Salzburg” (20204-WISS/225/197–2019 and 20102-F1901166-KZP).

  7. Data availability: In this article, a real data set on breast cancer which consists of 2,509 breast cancer patients is considered to illustrate the applicability and importance of estimating the overlap and containment indices. Two sub-parts of this data set is taken for studying the survival times and relapse free times of breast cancer patients who have taken either of the two surgeries: Breast conserving surgery (with radio therapy) or Mastectomy (with radio therapy). One can access these data sets through the hyperlink: https://github.com/RajuProbStat/Overlap-Index-Data.

Appendix A

Proof. In order to prove that K 2(η) = 0 has only one root in (0, ∞), it is enough to show

(A.1) 2 1 − η η ⁡ log e ( 2 ) η + 1 − 2 η 1 − 2 − η η + 1 2 = 0 ,

has only one root in (0, ∞). Since the above equation reduces to ϕ 2 ′ ( η ) = 0 and this has only one root in (0, ∞) as already proven in Lemma 2.4, η* is the only solution of K 2(η) = 0 in (0, ∞). In a similar way, K 1(η) = 0 reduces to the equation ϕ 1 ′ ( η ) = 0 , and hence 1 η * is the only root of K 1(η) = 0 in (0, ∞).

We write K (θ 1, θ 2) as a function of η in the following way

(A.2) K ( η ) = 256 η 2 λ 1 ( 1 − λ 1 ) η 1 − 2 − 1 η 2 − η − 1 η ( 1 − 2 − η ) 2 − 1 η log e ( 2 ) ( η + 1 ) 2 + ( 1 − η ) 1 − 2 − 1 η ( 1 − 2 − η ) ( η + 1 ) 3 2 .

For 0 < η < 1, ( 1 − η ) ( 1 − 2 − 1 η ) ( 1 − 2 − η ) > 0 and ( η ( 1 − 2 − 1 η ) 2 − η − 1 η ( 1 − 2 − η ) 2 − 1 η ) > 0 , because 2 x − 1 x is increasing in (0, ∞) and x < 1 x for 0 < x < 1. Again, for η > 1, ( 1 − η ) ( 1 − 2 − 1 η ) ( 1 − 2 − η ) < 0 and ( η ( 1 − 2 − 1 η ) 2 − η − 1 η ( 1 − 2 − η ) 2 − 1 η ) < 0 , because 2 x − 1 x is increasing in (0, ∞) and x > 1 x for x > 1. Therefore, K(η) > 0 for η ∈ (0, 1) ∪ (1, ∞). Again K (1) = 0 and hence, K(η) = 0 has only one root η = 1 in (0, ∞). □

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/ijb-2024-0106).

Received: 2024-11-23
Accepted: 2025-07-23
Published Online: 2025-09-05

© 2025 Walter de Gruyter GmbH, Berlin/Boston

https://doi.org/10.1515/ijb-2024-0106
Schlagwörter für diesen Artikel
confidence intervals; containment indices; maximum likelihood estimators; overlap index; parametric bootstrap
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