Trigonometry development in ancient and medieval India

This paper explores the history and possible mathematical methods behind the development of trigonometry in ancient and medieval India. Specifically, it describes possible methods of interpolating sine values from these tables

Novel Statistical Methodology Development and Applications in ALS Research

Being able to predict, with accuracy, the disease progression of patients with a given disease is extremely useful from the perspectives of clinicians, patients, and clinical trial investigators. We introduce a novel method of reducing the expected prediction error when using linear models, given approximate monotonicity of the response; we refer to this method as utilizing an “anchor.” We justify this method mathematically, and then show how to improve predictions arising from standard ordinary least squares (OLS) models when modelling disease progression in a population of patients with amyotrophic lateral sclerosis (ALS). We go on to show that using an anchor can be used in conjunction with more complex modelling schemes to further improve the predictions of ALS patients; an anchor improves both Bayesian hierarchical linear models and Bayesian mixture models. Furthermore, we explore potential covariates that may be included in the models to improve predictions, but find that only time of disease onset results in improved model performance. We also explore how well these models work in a clinical setting, rather than in a clinical trial. We first demonstrate the feasibility of automatically extracting patients’ data, pertaining to survival and disease progression, from the electronic medical record, as well as showing that our disease progression model is feasible for clinical patients. We then compare survival rates between the two populations and determine that, even after adjusting for several important covariates, there is a large difference between survival in the clinic setting and survival in ALS clinical trials. We assert that the two patient groups’ differences in disease progression and survival highlight the needs to understand better disease variability in the clinical setting and to refine the inclusion criteria in ALS trials. We determine an anchor can be used to improve predictive models in ALS disease progression, for both simple independent OLS regressions and for far more complicated Bayesian hierarchical linear models. We conclude that using a Bayesian hierarchical linear model with an anchor is useful in both a clinical trial population of ALS patients as well as a dissimilar population seen in the Midwestern academic medical center ALS clinic

Using an Anchor to Improve Linear Predictions with Application to Predicting Disease Progression

Linear models are some of the most straightforward and commonly used modelling approaches. Consider modelling approximately monotonic response data arising from a time-related process. If one has knowledge as to when the process began or ended, then one may be able to leverage additionalassumed data to reduce prediction error. This assumed data, referred to as the anchor, is treated as an additional data-point generated at either the beginning or end of the process. The response value of the anchor is equal to an intelligently selected value of the response (such as the upper bound, lower bound, or 99th percentile of the response, as appropriate). The anchor reduces the variance of prediction at the cost of a possible increase in prediction bias, resulting in a potentially reduced overall mean-square prediction error. This can be extremely eective when few individual data-points are available, allowing one to make linear predictions using as little as a single observed data-point. We develop the mathematics showing the conditions under which an anchor can improve predictions, and also demonstrate using this approach to reduce prediction error when modelling the disease progression of patients with amyotrophic lateral sclerosis.Modelos lineales son los modelos más fáciles de usar y comunes en modelamiento. Si se considera el modelamiento de una respuesta aprosimadamente monótona que surge de un proceso relacionado al tiempo y se sabe cuándo el proceso inició o terminó, es posible asumir datos adicionales como palanca para reducir el error de predicción. Estos datos adicionales son llamados de ``anclaje'' y son datos generados antes del inicion o después del final del proceso. El valor de respuesta del anclaje es igual a un valor de respuesta escogido de manera inteligente (como por ejemplo la cota superior, iferior o el percentil 99, según conveniencia). Este anclaje reduce la varianza de la predicción a costo de un posible sesgo en la misma, lo cual resulta en una reducción potencial del error medio de predicción. Lo anterior puede ser extremadamente efectivo cuando haypocos datos individuales, permitiendo hacer predicciones con muy pocos datos. En este trabajo presentamos en desarrollo matemático demostrando las condiciones bajo las cuales el anclaje puede mejorar predicciones y también demostramos una reducción del error de predicción aplicando el método a la modelación de progresión de enfermedad en pacientes con esclerosis lateral amiotrófica

Using an onset-anchored Bayesian hierarchical model to improve predictions for amyotrophic lateral sclerosis disease progression

Abstract Background Amyotrophic Lateral Sclerosis (ALS), also known as Lou Gehrig’s disease, is a rare disease with extreme between-subject variability, especially with respect to rate of disease progression. This makes modelling a subject’s disease progression, which is measured by the ALS Functional Rating Scale (ALSFRS), very difficult. Consider the problem of predicting a subject’s ALSFRS score at 9 or 12 months after a given time-point. Methods We obtained ALS subject data from the Pooled Resource Open-Access ALS Clinical Trials Database, a collection of data from various ALS clinical trials. Due to the typical linearity of the ALSFRS, we consider several Bayesian hierarchical linear models. These include a mixture model (to account for the two potential classes of “fast” and “slow” ALS progressors) as well as an onset-anchored model, in which an additional artificial data-point, using time of disease onset, is utilized to improve predictive performance. Results The onset-anchored model had a drastically reduced posterior predictive mean-square-error distributions, when compared to the Bayesian hierarchical linear model or the mixture model under a cross-validation approach. No covariates, other than time of disease onset, consistently improved predictive performance in either the Bayesian hierarchical linear model or the onset-anchored model. Conclusions Augmenting patient data with an additional artificial data-point, or onset anchor, can drastically improve predictive modelling in ALS by reducing the variability of estimated parameters at the cost of a slight increase in bias. This onset-anchored model is extremely useful if predictions are desired directly after a single baseline measure (such as at the first day of a clinical trial), a feat that would be very difficult without the onset-anchor. This approach could be useful in modelling other diseases that have bounded progression scales (e.g. Parkinson’s disease, Huntington’s disease, or inclusion-body myositis). It is our hope that this model can be used by clinicians and statisticians to improve the efficacy of clinical trials and aid in finding treatments for ALS

Using automated electronic medical record data extraction to model ALS survival and progression

Abstract Background To assess the feasibility of using automated capture of Electronic Medical Record (EMR) data to build predictive models for amyotrophic lateral sclerosis (ALS) outcomes. Methods We used an Informatics for Integrating Biology and the Bedside search discovery tool to identify and extract data from 354 ALS patients from the University of Kansas Medical Center EMR. The completeness and integrity of the data extraction were verified by manual chart review. A linear mixed model was used to model disease progression. Cox proportional hazards models were used to investigate the effects of BMI, gender, and age on survival. Results Data extracted from the EMR was sufficient to create simple models of disease progression and survival. Several key variables of interest were unavailable without including a manual chart review. The average ALS Functional Rating Scale – Revised (ALSFRS-R) baseline score at first clinical visit was 34.08, and average decline was − 0.64 per month. Median survival was 27 months after first visit. Higher baseline ALSFRS-R score and BMI were associated with improved survival, higher baseline age was associated with decreased survival. Conclusions This study serves to show that EMR-captured data can be extracted and used to track outcomes in an ALS clinic setting, potentially important for post-marketing research of new drugs, or as historical controls for future studies. However, as automated EMR-based data extraction becomes more widely used there will be a need to standardize ALS data elements and clinical forms for data capture so data can be pooled across academic centers

Stratification of amyotrophic lateral sclerosis patients: a crowdsourcing approach

Amyotrophic lateral sclerosis (ALS) is a fatal neurodegenerative disease where substantial heterogeneity in clinical presentation urgently requires a better stratification of patients for the development of drug trials and clinical care. In this study we explored stratification through a crowdsourcing approach, the DREAM Prize4Life ALS Stratification Challenge. Using data from >10,000 patients from ALS clinical trials and 1479 patients from community-based patient registers, more than 30 teams developed new approaches for machine learning and clustering, outperforming the best current predictions of disease outcome. We propose a new method to integrate and analyze patient clusters across methods, showing a clear pattern of consistent and clinically relevant sub-groups of patients that also enabled the reliable classification of new patients. Our analyses reveal novel insights in ALS and describe for the first time the potential of a crowdsourcing to uncover hidden patient sub-populations, and to accelerate disease understanding and therapeutic development