Middle censoring in the multinomial distribution with applications.

In a multinomial set-up with k possible outcomes, we develop estimation under a "middle censoring" paradigm, which is as defined in Jammalamadaka and Mangalam (2003). This problem has many special features because of the inter-dependent probabilities, which we explore here.

Focused analysis of RNFL decay in glaucomatous eyes using circular statistics on high-resolution OCT data.

We generated Optical Coherence Tomography (OCT) data of much higher resolution than usual on retinal nerve fiber layer (RNFL) thickness of a given eye. These consist of measurements made at hundreds of angular-points defined on a circular coordinate system. Traditional analysis of OCT RNFL data does not utilize insightful characteristics such as its circularity and granularity for common downstream applications. To address this, we present a new circular statistical framework that defines an Angular Decay function and thereby provides a directionally precise representation of an eye with attention to patterns of focused RNFL loss. By applying to a clinical cohort of Asian Indian eyes, the generated circular data were modeled with a finite mixture of von Mises distributions, which led to an unsupervised identification in different age-groups of recurrent clusters of glaucomatous eyes with distinct directional signatures of RNFL decay. New indices of global and local RNFL loss were computed for comparing the structural differences between these glaucoma clusters across the age-groups and improving classification. Further, we built a catalog of directionally precise statistical distributions of RNFL thickness for the said population of normal eyes as stratified by their age and optic disc size.

A Two-sample Nonparametric Test for Circular Data- its Exact Distribution and Performance.

A nonparametric test labelled 'Rao Spacing-frequencies test' is explored and developed for testing whether two circular samples come from the same population. Its exact distribution and performance relative to comparable tests such as the Wheeler-Watson test and the Dixon test in small samples, are discussed. Although this test statistic is shown to be asymptotically normal, as one would expect, this large sample distribution does not provide satisfactory approximations for small to moderate samples. Exact critical values for small samples are obtained and tables provided here, using combinatorial techniques, and asymptotic critical regions are assessed against these. For moderate sample sizes in-between i.e. when the samples are too large making combinatorial techniques computationally prohibitive but yet asymptotic regions do not provide a good approximation, we provide a simple Monte Carlo procedure that gives very accurate critical values. As is well-known, the large number of usual rank-based tests are not applicable in the context of circular data since the values of such ranks depend on the arbitrary choice of origin and the sense of rotation used (clockwise or anti-clockwise). Tests that are invariant under the group of rotations, depend on the data through the so-called 'spacing frequencies', the frequencies of one sample that fall in between the spacings (or gaps) made by the other. The Wheeler-Watson, Dixon, and the proposed Rao tests are of this form and are explicitly useful for circular data, but they also have the added advantage of being valid and useful for comparing any two samples on the real line. Our study and simulations establish the 'Rao spacing-frequencies test' as a desirable, and indeed preferable test in a wide variety of contexts for comparing two circular samples, and as a viable competitor even for data on the real line. Computational help for implementing any of these tests, is made available online "TwoCircles" R package and is part of this paper.

Competing risks survival data under middle censoring-An application to COVID-19 pandemic.

Survival data is being analysed here under the middle censoring scheme, using specifically quantile function modelling under competing risks. The use of middle censoring scheme has been shown to be very appropriate under the COVID-19 pandemic scenario. Cause-specific quantile inference under middle censoring is employed. Such quantile inferences are obtained through cumulative incidence function based on cause-specific proportional hazards model. The baseline lifetime is assumed to follow a very general parametric model namely the Weibull distribution, and is independent of the censoring mechanism. We obtain estimates of the unknown parameters and cause specific quantile functions under classical as well as a Bayesian set-up. A Monte Carlo simulation study assesses the relative performance of the different estimators. Finally, a real life data analysis is given for illustration of the proposed methods.

Circular functional analysis of OCT data for precise identification of structural phenotypes in the eye.

Progressive optic neuropathies such as glaucoma are major causes of blindness globally. Multiple sources of subjectivity and analytical challenges are often encountered by clinicians in the process of early diagnosis and clinical management of these diseases. In glaucoma, the structural damage is often characterized by neuroretinal rim (NRR) thinning of the optic nerve head, and other clinical parameters. Baseline structural heterogeneity in the eyes can play a key role in the progression of optic neuropathies, and present challenges to clinical decision-making. We generated a dataset of Optical Coherence Tomography (OCT) based high-resolution circular measurements on NRR phenotypes, along with other clinical covariates, of 3973 healthy eyes as part of an established clinical cohort of Asian Indian participants. We introduced CIFU, a new computational pipeline for CIrcular FUnctional data modeling and analysis. We demonstrated CIFU by unsupervised circular functional clustering of the OCT NRR data, followed by meta-clustering to characterize the clusters using clinical covariates, and presented a circular visualization of the results. Upon stratification by age, we identified a healthy NRR phenotype cluster in the age group 40-49 years with predictive potential for glaucoma. Our dataset also addresses the disparity of representation of this particular population in normative OCT databases.