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Article A Proportional Hazards Mixture Cure Model for Subgroup Analysis: Inferential Method and an Application to Colon Cancer Data(MDPI, 2025) Liu, Kai; Balakrishnan, Narayanaswamy; Peng, YingweiWhen determining subgroups with heterogeneous treatment effects in cancer clinical trials, the threshold of a variable that defines subgroups is often pre-determined by physicians based on their experience, and the optimality of the threshold is not well studied, particularly when the mixture cure rate model is considered. We propose a mixture cure model that allows optimal subgroups to be estimated for both the time to event for uncured subjects and the cure status. We develop a smoothed maximum likelihood method for the estimation of model parameters. An extensive simulation study shows that the proposed smoothed maximum likelihood method provides accurate estimates. Finally, the proposed mixture cure model is applied to a colon cancer study to evaluate the potential differences in the treatment effect of levamisole plus fluorouracil therapy versus levamisole alone therapy between younger and older patients. The model suggests that the difference in the treatment effect on the time to cancer recurrence for uncured patients is significant between patients younger than 67 and patients older than 67, and the younger patient group benefits more from the combined therapy than the older patient group.Article Robust Divergence-Based Tests of Hypotheses for Simple Step-Stress Accelerated Life-Testing Under Gamma Lifetime Distributions(Elsevier, 2026) Balakrishnan, Narayanaswamy; Jaenada, Maria; Pardo, LeandroMany modern devices are highly reliable, with long lifetimes before their failure. Conducting reliability tests under actual use conditions may require therefore impractically long experimental times to gather sufficient data for developing accurate inference. To address this, Accelerated Life Tests (ALTs) are often used in industrial experiments to induce product degradation and eventual failure more quickly by increasing certain environmental stress factors. Data collected under such increased stress conditions are analyzed, and results are then extrapolated to normal operating conditions. These tests typically involve a small number of devices and so pose significant challenges, such as interval-censoring. As a result, the outcomes are particularly sensitive to outliers in the data. Additionally, a comprehensive analysis requires more than just point estimation; inferential methods such as confidence intervals and hypothesis testing are essential to fully assess the reliability behaviour of the product. This paper presents robust statistical methods based on minimum divergence estimators for analyzing ALT data of highly reliable devices under step-stress conditions and Gamma lifetime distributions. Robust test statistics generalizing the Rao test and divergence-based tests for testing linear null hypothesis are then developed. These hypotheses include in particular tests for the significance of the identified stress factors and for the validity of the assumption of exponential lifetimes.Article On the Notion of Discrete-Time Signature and Some Associated Properties and Results(Cambridge University Press, 2026) Balakrishnan, Narayanaswamy; Yi, He; Goroncy, AgnieszkaIn this work, by considering coherent systems comprising independent components with discrete lifetimes, we introduce the notion of discrete-time signature and then discuss some of its properties. With the use of the introduced signature, a stochastic ordering result is also established. We then introduce transformation formulas for the discrete-time signature to facilitate the comparison of systems of different sizes. Some examples are also presented to illustrate all the results developed here.Article Multi-State Linear Three-Dimensional Consecutive k-Type Systems(Cambridge Univ Press, 2026) Yi, He; Balakrishnan, Narayanaswamy; Li, XiangConsecutive $k$-type systems have become important in both reliability theory and applications; in spite of a large literature existing on them, three-dimensional consecutive $k$-type systems have not yet been studied for multi-state case. In this paper, we introduce several different types of multi-state linear three-dimensional consecutive $k$-type systems for the first time, with due consideration to possible overlapping of failure blocks. The finite Markov chain imbedding approach is then used for the derivation of their reliability functions with state spaces and transition matrices provided in a novel way, and the involved computational process is illustrated through several numerical examples. Finally, some possible applications of the work and potential extensions are pointed out.Article Social Determinants of Health and Risk of Lower Extremity Amputation in Patients with Peripheral Artery Disease in Canada: Protocol for a Systematic Review and Meta-Analysis(BMJ Publishing Group, 2026) Chowdhury, Abhiroop; Sheikh, Fatima; Azab, Sandi M.; De Souza, Russell J.; Banfield, Laura; Balakrishnan, Narayanaswamy; Anand, Sonia S.Introduction Peripheral artery disease (PAD) affected approximately 800 000 Canadians aged 25 years or older in 2015 and it poses a substantial risk of lower extremity amputation (LEA). While clinical risk factors for amputation are well-established, the impact of social determinants of health (SDoH) on amputation risk remains unclear, particularly in a Canadian context.Objectives This systematic review aims to: (1) synthesise evidence on the associations between multilevel SDoH domains and LEA (both major and/or minor) risk in Canadian PAD patients including intersectional effects of race and ethnicity with another SDoH domain, and (2) evaluate the statistical methodologies used in the researched literature to inform future study design and analysis approaches.Methods and analysis We will systematically search MEDLINE, Embase, EmCare, Global Health, Cumulative Index to Nursing and Allied Health Literature and Web of Science for studies examining SDoH and LEA in Canadian patients with PAD (including chronic limb-threatening ischaemia which is a severe form of PAD). Date limits for each database will be from inception through December 2025. SDoH will be categorised using a modified Healthy People 2030 SDoH framework under six domains: economic stability, education, food, neighbourhood and physical environment, healthcare system and community and social context. Two reviewers will independently screen titles, abstracts and full texts, with discrepancies resolved by a third reviewer. Data will be extracted on study characteristics, SDoH measures, outcomes and statistical methods. Risk of bias will be assessed using RoB 2 for randomised trials, ROBINS-I for non-randomised studies of interventions and ROBINS-E for studies investigating exposures. A narrative synthesis, and where data permit, a Bayesian hierarchical meta-analysis using both effect size and contingency table approaches will be conducted. Statistical heterogeneity will be explored through subgroup analyses and meta-regression, examining study design, SDoH measurement approaches and population characteristics.Ethics and dissemination As a systematic review and meta-analysis, ethics approval is not required. For institutional oversight, we provide the contact of Dr Sonia Anand (Associate Vice-President, Global Health, McMaster University; anands@mcmaster.ca). Results will be reported following PRISMA guidelines and disseminated through a peer-reviewed publication.PROSPERO registration number CRD420251115759.
